Thomas Buchert

On average properties of inhomogeneous fluids in general relativity. 1. Dust cosmologies

For general relativistic spacetimes filled with irrotational ‘dust’ a generalized form of Friedmanns equations for an ‘effective’ expansion factor aD of inhomogeneous cosmologies is derived. Contrary to the standard Friedmann equations, which hold for homogeneous-isotropic cosmologies, the new equations include the ‘backreaction effect’ of inhomogeneities on the average expansion of the model. A universal relation between ‘backreaction’ and average scalar curvature is also given. For cosmologies whose averaged spatial scalar curvature is proportional to aD-2, the expansion law governing a generic domain can be found. However, as the general equations show, ‘backreaction’ acts as to produce average curvature in the course of structure formation, even when starting with space sections that are spatially flat on average.

Beyond genus statistics: A Unifying approach to the morphology of cosmic structure

The genus statistics of isodensity contours has become a well-established tool in cosmology. In this Letter we place the genus in the wider framework of a complete family of morphological descriptors. These are known as the Minkowski functionals, and we here apply them for the first time to isodensity contours of a continuous random field. By taking two equivalent approaches, one through differential geometry, the other through integral geometry, we derive two complementary formulae suitable for numerically calculating the Minkowski functionals. As an example, we apply them to simulated Gaussian random fields and compare the outcome to the analytically known results, demonstrating that both are indeed well suited for numerical evaluation. The code used for calculating all Minkowski functionals is available from the authors.

The universe seen at different scales

Abstract A large-scale smoothed-out model of the universe ignores small-scale inhomogeneities, but the averaged effects of those inhomogeneities may alter both observational and dynamical relations at the larger scale. This Letter discusses these effects, and comments briefly on the relation to gravitational entropy.

Lagrangian theory of gravitational instability of Friedman–Lemaître cosmologies – a generic third-order model for non-linear clustering

The Lagrangian perturbation theory of Friedman-Lemaitre cosmologies, which was investigated and solved up to the second order in earlier papers, is evaluated up to the third order. Based on this, a model for non-linear clustering applicable to the modelling of large-scale structure in the Universe for generic initial conditions is formulated. A truncated model is proposed which represents the main body of the perturbation sequence in the early non-linear regime by neglecting all gravitational sources that describe interaction of the perturbations. I also, however, give the irrotational solutions generated by the interaction terms to the third order, which induce vorticity in Lagrangian space. The consequences and applicability of the solutions are put into perspective

Disentangling the Cosmic Web. I. Morphology of Isodensity Contours

We apply Minkowski functionals and various derived measures to decipher the morphological properties of large-scale structure seen in simulations of gravitational evolution. Minkowski functionals of isodensity contours serve as tools to test global properties of the density field. Furthermore, we identify coherent objects at various threshold levels and calculate their partial Minkowski functionals. We propose a set of two derived dimensionless quantities, planarity and filamentarity, which reduce the morphological information in a simple and intuitive way. Several simulations of the gravitational evolution of initial power-law spectra provide a framework for systematic tests of our method.

On the curvature of the present-day Universe

We discuss the effect of curvature and matter inhomogeneities on the averaged scalar curvature of the present-day universe. Motivated by studies of averaged inhomogeneous cosmologies, we contemplate on the question of whether it is sensible to assume that curvature averages out on some scale of homogeneity, as implied by the standard concordance model of cosmology, or whether the averaged scalar curvature can be largely negative today, as required for an explanation of dark energy from inhomogeneities. We confront both conjectures with a detailed analysis of the kinematical backreaction term and estimate its strength for a multi-scale inhomogeneous matter and curvature distribution. Our main result is a formula for the spatially averaged scalar curvature involving quantities that are all measurable on regional (i.e. up to 100 Mpc) scales. We propose strategies to quantitatively evaluate the formula, and pinpoint the assumptions implied by the conjecture of a small or zero averaged curvature. We reach the conclusion that the standard concordance model needs fine tuning in the sense of an assumed equipartition law for curvature in order to reconcile it with the estimated properties of the averaged physical space, whereas a negative averaged curvature is favoured, independent of the prior on the value of the cosmological constant.

Back reaction of inhomogeneities on the expansion: The evolution of cosmological parameters

Averaging and evolving inhomogeneities are non-commuting operations. This implies the existence of deviations of an averaged model from the standard Friedmann-Lemaitre cosmologies. We quantify these deviations, encoded in a backreaction parameter, in the framework of Newtonian cosmology. We employ the linear theory of gravitational instability in the Eulerian and Lagrangian approaches, as well as the spherically- and plane-symmetric solutions as standards of reference. We propose a model for the evolution of the average characteristics of a spatial domain for generic initial conditions that contains the spherical top-hat model and the planar collapse model as exact sub cases. A central result is that the backreaction term itself, calculated on sufficiently large domains, is small but, still, its presence can drive the cosmological parameters on the averaging domain far away from their global values of the standard model. We quantify the variations of these parameters in terms of the fluctuations in the initial data as derived from the power spectrum of initial cold dark matter density fluctuations. E.g. in a domain with a radius of 100Mpc today and initially one-sigma fluctuations, the density parameters deviate from their homogeneous values by 15%; three-sigma fluctuations lead to deviations larger than 100%.

Newtonian Cosmology in Lagrangian Formulation: Foundations and Perturbation Theory

The “Newtonian” theory of spatially unbounded, self-gravitating, pressureless continua in Lagrangian form is reconsidered. Following a review of the pertinent kinematics, we present alternative formulations of the Lagrangian evolution equations and establish conditions for the equivalence of the Lagrangian and Eulerian representations. We then distinguish open models based on Euclidean space R3 from closed models based (without loss of generality) on a flat torus T3. Using a simple averaging method we show that the spatially averaged variables of an inhomogeneous toroidal model form a spatially homogeneous “background” model and that the averages of open models, if they exist at all, in general do not obey the dynamical laws of homogeneous models. We then specialize to those inhomogeneous toroidal models whose (unique) backgrounds have a Hubble flow, and derive Lagrangian evolution equations which govern the (conformally rescaled) displacement of the inhomogeneous flow with respect to its homogeneous background. Finally, we set up an iteration scheme and prove that the resulting equations have unique solutions at any order for given initial data, while for open models there exist infinitely many different solutions for given data.

Minkowski functionals of Abell/ACO clusters

We determine the Minkowski functionals for a sample of Abell/ACO clusters, 401 with measured and 16 with estimated redshifts. The four Minkowski functionals (including the void probability function and the mean genus) deliver a global description of the spatial distribution of clusters on scales from

Multiscale cosmology and structure-emerging dark energy: A plausibility analysis

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