Relativistic cosmological perturbation scheme on a general background: scalar perturbations for irrotational dust

In standard perturbation approaches and N-body simulations, inhomogeneities are described to evolve on a predefined background cosmology, commonly taken as the homogeneous-isotropic solutions of Einstein's field equations (Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmologies). In order to make physical sense, this background cosmology must provide a reasonable description of the effective, i.e. spatially averaged, evolution of structure inhomogeneities also in the nonlinear regime. Guided by the insights that (i) the average over an inhomogeneous distribution of matter and geometry is in general not given by a homogeneous solution of general relativity, and that (ii) the class of FLRW cosmologies is not only locally but also globally gravitationally unstable in relevant cases, we here develop a perturbation approach that describes the evolution of inhomogeneities on a general background being defined by the spatially averaged evolution equations. This physical background interacts with the formation of structures. We derive and discuss the resulting perturbation scheme for the matter model `irrotational dust' in the Lagrangian picture, restricting our attention to scalar perturbations.Comment: 18 pages. Matches published version in CQ

Improving the Lagrangian perturbative solution for cosmic fluid: Applying Shanks transformation

We study the behavior of Lagrangian perturbative solutions. For a spherical void model, the higher order the Lagrangian perturbation we consider, the worse the approximation becomes in late-time evolution. In particular, if we stop to improve until an even order is reached, the perturbative solution describes the contraction of the void. To solve this problem, we consider improving the perturbative solution using Shanks transformation, which accelerates the convergence of the sequence. After the transformation, we find that the accuracy of higher-order perturbation is recovered and the perturbative solution is refined well. Then we show that this improvement method can apply for a $\Lambda$CDM model and improved the power spectrum of the density field.Comment: 17 pages, 7 figures; accepted for publication in Phys.Rev.D; v2: Evolution of power spectrum in LCDM model is added; v3: References are correcte

Lagrangian theory of structure formation in relativistic cosmology I: Lagrangian framework and definition of a nonperturbative approximation

In this first paper we present a Lagrangian framework for the description of structure formation in general relativity, restricting attention to irrotational dust matter. As an application we present a self-contained derivation of a general-relativistic analogue of Zel'dovich's approximation for the description of structure formation in cosmology, and compare it with previous suggestions in the literature. This approximation is then investigated: paraphrasing the derivation in the Newtonian framework we provide general-relativistic analogues of the basic system of equations for a single dynamical field variable and recall the first-order perturbation solution of these equations. We then define a general-relativistic analogue of Zel'dovich's approximation and investigate its implications by functionally evaluating relevant variables, and we address the singularity problem. We so obtain a possibly powerful model that, although constructed through extrapolation of a perturbative solution, can be used to put into practice nonperturbatively, e.g. problems of structure formation, backreaction problems, nonlinear properties of gravitational radiation, and light-propagation in realistic inhomogeneous universe models. With this model we also provide the key-building blocks for initializing a fully relativistic numerical simulation.Comment: 21 pages, content matches published version in PRD, discussion on singularities added, some formulas added, some rewritten and some correcte

How is the local-scale gravitational instability influenced by the surrounding large-scale structure formation?

We develop the formalism to investigate the relation between the evolution of the large-scale (quasi) linear structure and that of the small-scale nonlinear structure in Newtonian cosmology within the Lagrangian framework. In doing so, we first derive the standard Friedmann expansion law using the averaging procedure over the present horizon scale. Then the large-scale (quasi) linear flow is defined by averaging the full trajectory field over a large-scale domain, but much smaller than the horizon scale. The rest of the full trajectory field is supposed to describe small-scale nonlinear dynamics. We obtain the evolution equations for the large-scale and small-scale parts of the trajectory field. These are coupled to each other in most general situations. It is shown that if the shear deformation of fluid elements is ignored in the averaged large-scale dynamics, the small-scale dynamics is described by Newtonian dynamics in an effective Friedmann-Robertson-Walker (FRW) background with a local scale factor. The local scale factor is defined by the sum of the global scale factor and the expansion deformation of the averaged large-scale displacement field. This means that the evolution of small-scale fluctuations is influenced by the surrounding large-scale structure through the modification of FRW scale factor. The effect might play an important role in the structure formation scenario. Furthermore, it is argued that the so-called {\it optimized} or {\it truncated} Lagrangian perturbation theory is a good approximation in investigating the large-scale structure formation up to the quasi nonlinear regime, even when the small-scale fluctuations are in the non-linear regime.Comment: 15pages, Accepted for publication in Gravitation and General Relativit

On average properties of inhomogeneous fluids in general relativity II: perfect fluid cosmologies

For general relativistic spacetimes filled with an irrotational perfect fluid a generalized form of Friedmann's equations governing the expansion factor of spatially averaged portions of inhomogeneous cosmologies is derived. The averaging problem for scalar quantities is condensed into the problem of finding an `effective equation of state' including kinematical as well as dynamical `backreaction' terms that measure the departure from a standard FLRW cosmology. Applications of the averaged models are outlined including radiation-dominated and scalar field cosmologies (inflationary and dilaton/string cosmologies). In particular, the averaged equations show that the averaged scalar curvature must generically change in the course of structure formation, that an averaged inhomogeneous radiation cosmos does not follow the evolution of the standard homogeneous-isotropic model, and that an averaged inhomogeneous perfect fluid features kinematical `backreaction' terms that, in some cases, act like a free scalar field source. The free scalar field (dilaton) itself, modelled by a `stiff' fluid, is singled out as a special inhomogeneous case where the averaged equations assume a simple form.Comment: TeX 21 pages, matches published version: G.R.G., in pres

Hydrodynamic approach to the evolution of cosmological structures

A hydrodynamic formulation of the evolution of large-scale structure in the Universe is presented. It relies on the spatially coarse-grained description of the dynamical evolution of a many-body gravitating system. Because of the assumed irrelevance of short-range (``collisional'') interactions, the way to tackle the hydrodynamic equations is essentially different from the usual case. The main assumption is that the influence of the small scales over the large-scale evolution is weak: this idea is implemented in the form of a large-scale expansion for the coarse-grained equations. This expansion builds a framework in which to derive in a controlled manner the popular ``dust'' model (as the lowest-order term) and the ``adhesion'' model (as the first-order correction). It provides a clear physical interpretation of the assumptions involved in these models and also the possibility to improve over them.Comment: 14 pages, 3 figures. Version to appear in Phys. Rev.

A weak acceleration effect due to residual gravity in a multiply connected universe

Could cosmic topology imply dark energy? We use a weak field (Newtonian) approximation of gravity and consider the gravitational effect from distant, multiple copies of a large, collapsed (virialised) object today (i.e. a massive galaxy cluster), taking into account the finite propagation speed of gravity, in a flat, multiply connected universe, and assume that due to a prior epoch of fast expansion (e.g. inflation), the gravitational effect of the distant copies is felt locally, from beyond the naively calculated horizon. We find that for a universe with a $T^1xR^2$ spatial section, the residual Newtonian gravitational force (to first order) provides an anisotropic effect that repels test particles from the cluster in the compact direction, in a way algebraically similar to that of dark energy. For a typical test object at comoving distance $\chi$ from the nearest dense nodes of the cosmic web of density perturbations, the pressure-to-density ratio $w$ of the equation of state in an FLRW universe, is w \sim - (\chi/L)^3, where $L$ is the size of the fundamental domain, i.e. of the universe. Clearly, |w|<<1. For a T^3 spatial section of exactly equal fundamental lengths, the effect cancels to zero. For a T^3 spatial section of unequal fundamental lengths, the acceleration effect is anisotropic in the sense that it will *tend to equalise the three fundamental lengths*. Provided that at least a modest amount of inflation occurred in the early Universe, and given some other conditions, multiple connectedness does generate an effect similar to that of dark energy, but the amplitude of the effect at the present epoch is too small to explain the observed dark energy density and its anisotropy makes it an unrealistic candidate for the observed dark energy.Comment: 12 pages, 8 figures, accepted by Astronomy & Astrophysics; v2 includes 3D calculation and result; v3 includes analysis of numerical simulation, matches accepted versio

Lagrangian description of fluid flow with pressure in relativistic cosmology

The Lagrangian description of fluid flow in relativistic cosmology is extended to the case of flow accelerated by pressure. In the description, the entropy and the vorticity are obtained exactly for the barotropic equation of state. In order to determine the metric, the Einstein equation is solved perturbatively, when metric fluctuations are small but entropy inhomogeneities are large. Thus, the present formalism is applicable to the case when the inhomogeneities are small in the large scale but locally nonlinear.Comment: 11 pages (RevTeX); accepted for publication in Phys. Rev.

Lagrangian description of the fluid flow with vorticity in the relativistic cosmology

We develop the Lagrangian perturbation theory in the general relativistic cosmology, which enables us to take into account the vortical effect of the dust matter. Under the Lagrangian representation of the fluid flow, the propagation equation for the vorticity as well as the density is exactly solved. Based on this, the coupling between the density and vorticity is clarified in a non-perturbative way. The relativistic correspondence to the Lagrangian perturbation theory in the Newtonian cosmology is also emphasized.Comment: 14 pages (RevTeX); accepted for publication in Phys. Rev.