The Universe is homogeneous and isotropic on large scales, so on those scales it is usually modelled as a Friedmann-Lemaître-Robertson-Walker (FLRW) space-time. The non-linearity of the Einstein field equations raises concern over averaging over small-scale deviations form homogeneity and isotropy, with possible implications on the applicability of the FLRW metric to the Universe, even on large scales. Here I present a technique, based on the multiple-scales method of singular perturbation theory, to handle the small-scale inhomogeneities consistently. I obtain a leading order effective Einstein equation for the large-scale space-time metric, which contains a back-reaction term. The derivation relies on a series of consistency conditions, that ensure that the growth of deviations from the large-scale space-time metric do not grow unboundedly; criteria for their satisfiability are discussed, and it is shown that they are indeed satisfied if matter is non-relativistic on small scales. The analysis is performed in harmonic gauge, and conversion to other gauges is discussed. I estimate the magnitude of the back-reaction term relative to the critical density of the Universe in the example of an NFW halo, and find it to be of the order of a few percent. In this example, the back-reaction term is interpreted as a contribution of the energy-density of gravitational potential energy, averaged over the small-scale, to the total energy-momentum tensor.
We introduce a generalization of the 4-dimensional averaging window function of Gasperini, Marozzi and Veneziano (2010) that may prove useful for a number of applications. The covariant nature of spatial scalar averaging schemes to address the averaging problem in relativistic cosmology is an important property that is implied by construction, but usually remains implicit. We employ here the approach of Gasperini et al. for two reasons. First, the formalism and its generalization presented here are manifestly covariant. Second, the formalism is convenient for disentangling the dependencies on …
General K=-1 Friedman-Lemaître models and the averaging problem in cosmology
我们介绍了一般的K=-1弗里德曼-勒梅特（紧凑）宇宙学的概念和平均演化的概念，通过一个平均映射。然后，我们分析了弗里德曼-勒梅特方程和引力能在宇宙演化中的作用。我们区分两种渐近行为：辐射和质量间隙。我们通过精确的定义讨论了它们在宇宙学中的平均问题。然后，我们定量详细地描述了辐射的情况下，强调精确估计的引力能量的演变及其在宇宙的减速的影响。同样在辐射情况下，我们提出了一个平滑性质，它告诉我们平坦K=-1 FL模型的长时间H^{3} x H^{2}稳定性意味着H^{i+1} x H^{i}稳定性与H^{i+1} x H^{i}中的初始状态有多大无关，即存在时空的长时间平滑。最后，我们讨论了大引力能的初始“大爆炸”状态的存在，表明没有数学限制来假设它在时间开始时是低的。
The Hubble tension cast a blight on the standard cosmology. As a possible attitude to the problem, the local variation of the expansion rate in an inhomogeneous cosmology has been proposed where the spatial averaging over a finite domain was introduced in order to construct local Friedmann spacetime. However, it is not clear that the concept of the spatial averaging itself is gauge invariant or not. Namely the conclusion obtained by the averaging in a particular gauge is physically meaningful or not. In this paper we address this question, namely the gauge invariance of the spatial averaging i…
The Hubble constant problem is the discrepancy between different measurements of the Hubble constant in different scales. We show that this problem can be resolved within the general relativistic framework of the perturbation theory in the inhomogeneous universe, with the help of spatial averaging procedure over a finite local domain in the $t=\mbox{const.}$ hypersurface. The idea presented in this paper is unique in the sense that it has all of the following properties. a) It is based on the general relativistic perturbation theory, with ordinary dust matter only. No strange matter nor energy…
Quantum cosmology is traditionally formulated in a minisuperspace setting, implicitly averaging fields over space to obtain homogeneous models. For universal reasons related to the uncertainty principle, quantum corrections then depend on the size of the averaging volume. In minisuperspace truncations, the value of this volume remains an arbitrary parameter devoid of physical meaning, but in an effective field theory it is identified with the infrared scale of inhomogeneous modes. Moreover, the infrared scale is running during gravitational collapse, when regions in which homogeneity remains a…
This thesis concerns research undertaken in two related topics concerning high-energy gravitational physics. The first is the construction of a manifestly diffeomorphism-invariant Exact Renormalization Group (ERG). This is a procedure that constructs effective theories of gravity by integrating out high-energy modes down to an ultraviolet cutoff scale without gauge-fixing. The manifest diffeomorphism invariance enables us to construct a fully background-independent formulation. This thesis will explore both the fixed-background and background-independent forms of the manifestly diffeomorphism-…
One of the outstanding problems in general relativistic cosmology is that of the averaging. That is, how the lumpy universe that we observe at small scales averages out to a smooth Friedmann-Lemaitre-Robertson-Walker (FLRW) model. The root of the problem is that averaging does not commute with the Einstein equations that govern the dynamics of the model. This leads to the well-know question of backreaction in cosmology. In this work, we approach the problem using the covariant framework of Macroscopic Gravity (MG). We use its cosmological solution with a flat FLRW macroscopic background where …
Cosmology relies on a coarse-grained description of the universe, assumed to be valid on large length scales. However, the nonlinearity of general relativity makes coarse-graining extremely difficult. We here address this problem by extending the Mori-Zwanzig projection operator formalism, a highly successful coarse-graining method from statistical mechanics, towards general relativity. Using the Buchert equations, we derive a new dynamic equation for the Hubble parameter which captures the effects of averaging through a memory function. This gives an empirical prediction for the cosmic jerk.
The current cosmological model ($Λ$CDM) with the underlying FLRW metric relies on the assumption of local isotropy, hence homogeneity of the Universe. Difficulties arise when one attempts to justify this model as an average description of the Universe from first principles of general relativity, since in general, the Einstein tensor built from the averaged metric is not equal to the averaged stress--energy tensor. In this context, the discrepancy between these quantities is called "cosmological backreaction" and has been the subject of scientific debate among cosmologists and relativists for m…
Cosmological models typically neglect the complicated nature of the spacetime manifold at small scales in order to hypothesize idealized general relativistic solutions for describing the average dynamics of the Universe. Although these solutions are remarkably successful in accounting for data, they introduce a number of puzzles in cosmology, and their foundational assumptions are therefore important to test. In this paper, we go beyond the usual assumptions in cosmology and propose a formalism for averaging the local general relativistic spacetime on an observer's past null cone: we formulate…
In relativistic inhomogeneous cosmology, structure formation couples to average cosmological expansion. A conservative approach to modelling this assumes an Einstein--de Sitter model (EdS) at early times and extrapolates this forward in cosmological time as a "background model" against which average properties of today's Universe can be measured. This requires adopting an early-epoch--normalised background Hubble constant $H_1^{bg}$. Here, we show that the $Λ$CDM model can be used as an observational proxy to estimate $H_1^{bg}$ rather than choose it arbitrarily. We assume (i) an EdS model at …
We investigate anisotropic fluid cosmology in a situation where the spacetime metric back-reacts in a local, time-dependent way to the presence of inhomogeneities. We derive exact solutions to the Einstein field equations describing Friedmann-Lemaitre-Robertson-Walker (FLRW) large scale cosmological evolution in the presence of local inhomogeneities and time-dependent back reaction. We use our derivation to tackle the cosmological constant problem. A cosmological constant emerges by averaging the back reaction term on spatial scales of the order of $100 \ \text{Mpc}$, at which our universe beg…
No. In a number of papers Green and Wald argue that the standard FLRW model approximates our Universe extremely well on all scales, except close to strong field astrophysical objects. In particular, they argue that the effect of inhomogeneities on average properties of the Universe (backreaction) is irrelevant. We show that this latter claim is not valid. Specifically, we demonstrate, referring to their recent review paper, that (i) their two-dimensional example used to illustrate the fitting problem differs from the actual problem in important respects, and it assumes what is to be proven; (i…
We discuss dynamical systems approaches and methods applied to flat Robertson-Walker models in $f(R)$-gravity. We argue that a complete description of the solution space of a model requires a global state space analysis that motivates globally covering state space adapted variables. This is shown explicitly by an illustrative example, $f(R) = R + αR^2$, $α> 0$, for which we introduce new regular dynamical systems on global compactly extended state spaces for the Jordan and Einstein frames. This example also allows us to illustrate several local and global dynamical systems techniques involving…
