We prove the existence of a large class of one-parameter families of cosmological solutions to the Einstein-Euler equations that have a Newtonian limit. This class includes solutions that represent a finite, but otherwise arbitrary, number of compact fluid bodies. These solutions provide exact cosmological models that admit Newtonian limits but, are not, either implicitly or explicitly, averaged.
The basic concepts and equations of Newtonian Cosmology are presented in the form necessary for the derivation and analysis of the averaged Navier-Stokes-Poisson equations. A particular attention is paid to the physical and cosmological hypotheses about the structure of Newtonian universes. The system of the Navier-Stokes-Poisson equations governing the cosmological dynamics of Newtonian universes is presented and discussed. A reformulation of the Navier-Stokes-Poisson equations in terms of the fluid kinematic quantities is given and the structure of this system of equations is analyzed.
The basic concepts and equations of classical fluid mechanics are presented in the form necessary for the formulation of Newtonian cosmology and for derivation and analysis of a system of the averaged Navier-Stokes-Poisson equations. A special attention is paid to the analytic formulation of the definitions and equations of moving fluids and to their physical content.
The basic concepts and hypotheses of Newtonian Cosmology necessary for a consistent treatment of the averaged cosmological dynamics are formulated and discussed in details. The space-time, space, time and ensemble averages for the cosmological fluid fields are defined and analyzed with a special attention paid to their analytic properties. It is shown that all averaging procedures require an arrangement for a standard measurement device with the same measurement time interval and the same space region determined by the measurement device resolution to be prescribed to each position and each mo…
Idealizing matter as a pressureless fluid and representing its motion by a peculiar--velocity field superimposed on a homogeneous and isotropic Hubble expansion, we apply (Lagrangian) spatial averaging on an arbitrary domain $\cal D$ to the (nonlinear) equations of Newtonian cosmology and derive an exact, general equation for the evolution of the (domain dependent) scale factor $a_{\cal D}(t)$. We consider the effect of inhomogeneities on the average expansion and discuss under which circumstances the standard description of the average motion in terms of Friedmann's equation holds. We find th…
Jürgen Ehlers developed \emph{frame theory} to better understand the relationship between general relativity and Newtonian gravity. Frame theory contains a parameter $λ$, which can be thought of as $1/c^2$, where $c$ is the speed of light. By construction, frame theory is equivalent to general relativity for $λ>0$, and reduces to Newtonian gravity for $λ=0$. Moreover, by setting $\ep=\sqrtλ$, frame theory provides a framework to study the Newtonian limit $\ep \searrow 0$ (i.e. $c\to \infty$). A number of ideas relating to frame theory that were introduced by Jürgen have subsequently found impo…
The fourth post-Newtonian (4PN) two-body dynamics has been recently tackled by several different approaches: effective field theory, Arnowitt-Deser-Misner Hamiltonian, action-angle-Delaunay averaging, effective-one-body, gravitational self-force, first law of dynamics, and Fokker action. We review the achievements of these approaches and discuss the complementarity of their results. Our main conclusions are: (i) the results of the first complete derivation of the 4PN dynamics [T.Damour, P.Jaranowski, and G.Schäfer, Phys. Rev. D 89, 064058 (2014)] have been, piecewise, fully confirmed by severa…
The averaging problem in cosmology is of considerable importance for the correct interpretation of cosmological data. We review cosmological observations and discuss some of the issues regarding averaging. We present a precise definition of a cosmological model and a rigorous mathematical definition of averaging, based entirely in terms of scalar invariants.
Applying the recently developed dynamical perturbation formalism on cosmological background to scalar-tensor theory, we provide a solid theoretical basis and a rigorous justification for phenomenological models of orbital dynamics that are currently used to interpret experimental measurements of the time-dependent gravitational constant. We derive the field equations for the scalar-tensor perturbations and study their gauge freedom associated with the cosmological expansion. We find a new gauge eliminating a prohibitive number of gauge modes in the field equations and significantly simplifying…
The Shapiro effect, also known as the gravitational time delay, is close kin to the gravitational deflection of light that was the central topic of our Summer School. It is also an interesting test bed for exploring a topic that provides the foundations for most of the calculations we have done in this school, yet is highly complex when treated more rigorously: the question of the Newtonian limit, and of the post-Newtonian corrections that must be applied to include the leading-order effects of general relativity. This contribution discusses simplified derivations for the gravitational redshif…
Due to the nonlinearity of the Euler{Poisson equations, it is possible that the nonlinear Jeans instability may lead to a faster density growing rate than the rate in the standard theory of linearized Jeans instability, which motivates us to study the nonlinear Jeans instability. The aim of this article is to develop a method proving the Jeans instability for slightly nonlinear Euler-Poisson equations in the expanding Newtonian universe. The standard proofs of the Jeans instability rely on the Fourier analysis. However, it is difficult to generalize Fourier method to a nonlinear setting, and t…
We hereby derive the Newtonian metric potentials for the fourth-derivative gravity including the one-loop logarithm quantum corrections. It is explicitly shown that the behavior of the modified Newtonian potential near the origin is improved respect to the classical one, but this is not enough to remove the curvature singularity in $r=0$. Our result is grounded on a rigorous proof based on numerical and analytic computations.
The averaging problem in cosmology is of considerable importance for the correct interpretation of cosmological data. A rigorous mathematical definition of averaging in a cosmological model is necessary. In general, a spacetime is completely characterized by its scalar curvature invariants, and this suggests a particular spacetime averaging scheme based entirely on scalars. We clearly identify the problems of averaging in a cosmological model. We then present a precise definition of a cosmological model, and based upon this definition, we propose an averaging scheme in terms of scalar curvatur…
We construct cosmological models consisting of large numbers of identical, regularly spaced masses. These models do not rely on any averaging procedures, or on the existence of a global Friedmann-Robertson-Walker (FRW) background. They are solutions of Einstein's equations up to higher order corrections in a perturbative expansion, and have large-scale dynamics that are well modelled by the Friedmann equation. We find that the existence of arbitrarily large density contrasts does not change either the magnitude or scale of the background expansion, at least when masses are regularly arranged, …
We construct a framework to probe the effect of non-linear structure formation on the large-scale expansion of the universe. We take a bottom-up approach to cosmological modelling by splitting our universe into cells. The matter content within each cell is described by the post-Newtonian formalism. We assume that most of the cell is in the vicinity of weak gravitational fields, so that it can be described using a perturbed Minkowski metric. Our cells are patched together using the Israel junction conditions. We impose reflection symmetry across the boundary of these cells. This allows us to ca…
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 $\R^3$ from closed models based (without loss of generality) on a flat torus $\T^3$. Using a simple averaging method we show that the spatially averaged variables of an inhomogeneous toroidal model form…
We develop a new approach to building cosmological models, in which small pieces of perturbed Minkowski space are joined together at reflection-symmetric boundaries in order to form a global, dynamical space-time. Each piece of this patchwork universe is described using post-Newtonian gravitational physics, with the large-scale expansion of the universe being an emergent phenomenon. This approach to cosmology does not require any assumptions about non-local averaging processes. Our framework clarifies the relation between the weak-field limit of general relativity, and the cosmological solutio…
Due to the non-commutation of spatial averaging and temporal evolution, inhomogeneities and anisotropies (cosmic structures) influence the evolution of the averaged Universe via the cosmological backreaction mechanism. We study the backreaction effect as a function of averaging scale in a perturbative approach up to higher orders. We calculate the hierarchy of the critical scales, at which 10% effects show up from averaging at different orders. The dominant contribution comes from the averaged spatial curvature, observable up to scales of 200 Mpc. The cosmic variance of the local Hubble rate i…
Averaged inhomogeneous cosmologies lie at the forefront of interest, since cosmological parameters like the rate of expansion or the mass density are to be considered as volume-averaged quantities and only these can be compared with observations. For this reason the relevant parameters are intrinsically scale-dependent and one wishes to control this dependence without restricting the cosmological model by unphysical assumptions. In the latter respect we contrast our way to approach the averaging problem in relativistic cosmology with shortcomings of averaged Newtonian models. Explicitly, we in…
Post-Newtonian secular dynamics of hierarchical triples. I. Eccentricity-, inclination- and node-dependence of the multiple-scale formulation, and its canonical consistency
We present a formalism for spatial averaging in cosmology applicable to general spacetimes and coordinates, and allowing the easy incorporation of a wide variety of matter sources. We apply this formalism to a Friedmann-LeMaitre-Robertson-Walker universe perturbed to second-order and present the corrections to the background in an unfixed gauge. We then present the corrections that arise in uniform curvature and conformal Newtonian gauges.
Action-angle variables of a binary black hole with arbitrary eccentricity, spins, and masses at 1.5 post-Newtonian order
Observatoire de Paris, Université PSL, Sorbonne Université, CNRS（巴黎天文台，巴黎文理研究大学，索邦大学，法国国家科学研究中心）；Department of Physics and Astronomy, The University of Mississippi（密西西比大学物理与天文学系）；Universidad de Ingenieria y Tecnologia – UTEC（秘鲁科技工程大学）；Department of Physics and Astronomy, Seoul National University（首尔大学物理与天文学系）
A persistent theme in the study of dark energy is the question of whether it really exists or not. It is often claimed hat we are mis-calculating the cosmological model by neglecting the effects associated with averaging over large-scale structures. In the Newtonian approximation this is clear: there is no effect. Within the full relativistic picture this remains an important open question however, owing to the complex mathematics involved. We study this issue using particle numerical simulations which account for all relevant relativistic effects without any problems from shell crossing. In t…
We give a rigorous derivation of the general-relativistic formula for the two-way Doppler tracking of a spacecraft in Friedmann-Lemaitre-Robertson-Walker and in McVittie spacetimes. The leading order corrections of the so-determined acceleration to the Newtonian acceleration are due to special-relativistic effects and cosmological expansion. The latter, although linear in the Hubble constant, is negligible in typical applications within the Solar System.
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 co…
