Kinematical Backreaction Research Articles

Cosmic backreaction and the mean redshift drift from symbolic regression

The possibility of obtaining symbolic expressions for cosmic backreaction is explored through a case study of so-called 2-region models. By using the publicly available symbolic regression algorithm AI Feynman, it is shown that the kinematical backreaction from a single 2-region model can be well described as a function of the mean redshift (or, equivalently, the volume averaged scale factor). A single expression depending on the redshift/scale factor as well as a model parameter, $f$, that can accurately describe the backreaction for a significant range of models is naturally more complicated but is also achieved with percent-level accuracy. \newline\indent Data sets of redshift drift in the 2-region models are also considered. Again utilizing AI Feynman, expressions for the redshift drift are found. In particular, an expression for the difference between the mean redshift drift and the drift of the mean redshift in terms of the kinematical backreaction is easily obtained for a single 2-region model. An accurate symbolic expression that describes this difference for an array of 2-region models is achieved by using the redshift as a feature instead of the kinematical backreaction.

Gauss–Bonnet–Chern approach to the averaged Universe

The standard model of cosmology with postulated dark energy and dark matter sources may be considered as a fairly successful fitting model to observational data. However, this model leaves the question of the physical origin of these dark components open. Fully relativistic contributions that act like dark energy on large scales and like dark matter on smaller scales can be found through generalization of the standard model by spatially averaging the inhomogeneous Universe within general relativity. The spatially averaged 3 + 1 Einstein equations are effective balance equations that need a closure condition. Heading for closure we here explore topological constraints. Results are straightforwardly obtained for averaged 2 + 1 model universes. For the relevant 3 + 1 case, we employ a method based on the Gauss–Bonnet–Chern theorem generalized to Lorentzian spacetimes and implement a sandwich approach to obtain spatial average properties. The 3 + 1 topological approach supplies us with a new equation linking evolution of scalar invariants of the expansion tensor to the norm of the Weyl tensor. From this we derive general evolution equations for averaged scalar curvature and kinematical backreaction, and we discuss related evolution equations on this level of the hierarchy of averaged equations. We also discuss the relation between topological properties of cosmological manifolds and dynamical topology change, e.g. as resulting from the formation of black holes.

Observations in statistically homogeneous, locally inhomogeneous cosmological toy models without FLRW backgrounds

ABSTRACT We study observations in toy models that constitute exact cosmological solutions to the Einstein equation. These models are statistically homogeneous but locally inhomogeneous, without an a priori introduced Friedmann–Lemaître–Roberston–Walker (FLRW) background and with ‘structures’ evolving fairly slowly. The mean redshift–distance relation and redshift drift along 500 light rays in each of two models are compared with relations based on spatial averages. The relations based on spatial averages give a good reproduction of the mean redshift–distance relation, although most convincingly in the model where the kinematical backreaction and average spatial curvature cancel each other to a subpercentage precision. In both models, the mean redshift drift clearly differs from the drift of the mean redshift. This indicates that redshift drift could be an important tool for testing the backreaction conjecture as redshift drift appears to distinguish between local and global effects. The method presented for computing the redshift drift is straightforward to generalize and can thus be utilized to fairly easily compute this quantity in a general space–time.

A Few Numbers from the Turnaround Epoch of Collapse

The turnaround epoch of gravitational collapse is examined by means of relativistic Lagrangian perturbation theory. Averaged, scalar equations applied to the fluid's evolution reveal some scale-independent universality of parameters for a wide variety of initial conditions. In particular, the density contrast of the collapsing domain at the turnaround is shown to be significantly smaller than the value provided by Eulerian perturbative (homogeneous and spherical) model. Combined curvature and kinematical backreaction are shown to be of the order of the energy density. Possible improvements of our treatment are put into perspective.

The Cosmological Constant From Planckian Fluctuations and the Averaging Procedure

In this paper I continue the investigation in \cite{1,1b} concerning my proposal on the nature of the cosmological constant. In particular, I study both mathematically and physically the quantum Planckian context and I provide, in order to depict quantum fluctuations and in absence of a complete quantum gravity theory, a semiclassical solution where an effective inhomogeneous metric at Planckian scales or above is averaged. In such a framework, a generalization of the well known Buchert formalism \cite{2} is obtained with the foliation in terms of the mean value $s(\hat{t})$ of the time operator $\hat{t}$ in a maximally localizing state $\{s\}$ of a quantum spacetime \cite{3,4,5,6} and in a cosmological context \cite{7}. As a result, after introducing a decoherence length scale $L_D$ where quantum fluctuations are averaged on, a classical de Sitter universe emerges with a small cosmological constant depending on $L_D$ and frozen in a true vacuum state (lowest energy), provided that the kinematical backreaction is negligible at that scale $L_D$. Finally, I analyse the case with a non-vanishing initial spatial curvature $\mathcal{R}$ showing that, for a reasonable large class of models, spatial curvature and kinematical backreation $\mathcal{Q}$ are suppressed by the dynamical evolution of the spacetime.

Dark matter from backreaction? Collapse models on galaxy cluster scales

In inhomogeneous cosmology, restricting attention to an irrotational dust matter model, backreaction arises in terms of the deviation of the averaged spatial scalar curvature from a constant-curvature model on some averaging domain , , and the kinematical backreaction . These backreaction variables can be modeled as an effective scalar field, called the ‘morphon field’. The general cosmological equations still need a closure condition to be solved. A simple example is the class of scaling solutions where and are assumed to follow a power law of the volume scale factor . But while they can describe models of quintessence, these and other models still assume the existence of dark matter in addition to the known sources. Going beyond scaling solutions by using a model for structure formation that we argue is reasonably generic, we investigate the correspondence between the morphon field and fundamental scalar field dark matter models, in order to describe dark matter as an effective phenomenon arising from kinematical backreaction and the averaged spatial curvature of the inhomogeneous Universe. While we find significant differences with those fundamental models, our main result is that the energy budget on typical collapsing domains is provided by curvature and matter in equal parts already around the turn-around time, leading to curvature dominance thereafter and increasing to a curvature contribution of 3/4 of the energy budget at the onset of virialization. Kinematical backreaction is subdominant at early stages, but its importance rises quickly after turn-around and dominates the curvature contribution in the final phase of the collapse. We conclude that backreaction can indeed mimic dark matter (in the energy budget) during the collapse phase of megaparsec-scale structures.

On backreaction effects in geometrical destabilisation of inflation

We study the geometrical instability arising in multi-field models of inflation with negatively-curved field space. We analyse how the homogeneous background evolves in presence of geometrical destabilisation, and show that, in simple models, a kinematical backreaction effect takes place that shuts off the instability. We also follow the evolution of the unstable scalar fluctuations and show that, in most situations, they must remain in the perturbative regime in order to satisfy observational constraints. We conclude that, in the simplest models of geometrical destabilisation, inflation does not end prematurely, but rather proceeds along a modified, sidetracked, field-space trajectory.

Replacing dark energy by silent virialisation

Context.Standard cosmologicalN-body simulations have background scale factor evolution that is decoupled from non-linear structure formation. Prior to gravitational collapse, kinematical backreaction (QD) justifies this approach in a Newtonian context.Aims.However, the final stages of a gravitational collapse event are sudden; a globally imposed smooth expansion rate forces at least one expanding region to suddenly and instantaneously decelerate in compensation for the virialisation event. This is relativistically unrealistic. A more conservative hypothesis is to allow non-collapsed domains to continue their volume evolution according to theQDZel’dovich approximation (QZA). We aim to study the inferred average expansion under this “silent” virialisation hypothesis.Methods.We set standard (MPGRAFIC) EdS 3-torus (T3) cosmologicalN-body initial conditions. UsingRAMSES, we partitioned the volume into domains and called theDTFElibrary to estimate the per-domain initial values of the three invariants of the extrinsic curvature tensor that determine the QZA. We integrated the Raychaudhuri equation in each domain using theINHOMOGlibrary, and adopted the stable clustering hypothesis to represent virialisation (VQZA). We spatially averaged to obtain the effective global scale factor. We adopted an early-epoch–normalised EdS reference-model Hubble constantH1EDS= 37.7km s-1∕Mpc and an effective Hubble constantHeff,0= 67.7km s-1∕Mpc.Results.From 2000 simulations at resolution 2563, we find that reaching a unity effective scale factor at 13.8 Gyr (16% above EdS), occurs for an averaging scale ofL13.8= 2.5−0.4+0.1Mpc∕heff. Relativistically interpreted, this corresponds to strong average negative curvature evolution, with the mean (median) curvature functionalΩRDgrowing from zero to about 1.5–2 by the present. Over 100 realisations, the virialisation fraction and super-EdS expansion correlate strongly at fixed cosmological time.Conclusions.Thus, starting from EdS initial conditions and averaging on a typical non-linear structure formation scale, the VQZA dark-energy–free average expansion matchesΛCDM expansion to first order. The software packages used here are free-licensed.

Testing backreaction effects with observational Hubble parameter data

The spatially averaged inhomogeneous Universe includes a kinematical backreaction term {mathcal {Q}}_{mathcal {D}} that is relate to the averaged spatial Ricci scalar {langle mathcal {R}} rangle _{mathcal {D}} in the framework of general relativity. Under the assumption that {mathcal {Q}}_{mathcal {D}} and {langle mathcal {R}} rangle _{mathcal {D}} obey the scaling laws of the volume scale factor a_{mathcal {D}}, a direct coupling between them with a scaling index n is remarkable. In order to explore the generic properties of a backreaction model for explaining the accelerated expansion of the Universe, we exploit two metrics to describe the late time Universe. Since the standard FLRW metric cannot precisely describe the late time Universe on small scales, the template metric with an evolving curvature parameter kappa _{mathcal {D}}(t) is employed. However, we doubt the validity of the prescription for kappa _{mathcal {D}}, which motivates us apply observational Hubble parameter data (OHD) to constrain parameters in dust cosmology. First, for FLRW metric, by getting best-fit constraints of varOmega ^{{mathcal {D}}_0}_m = 0.25^{+0.03}_{-0.03}, n = 0.02^{+0.69}_{-0.66}, and H_{mathcal {D}_0} = 70.54^{+4.24}_{-3.97} mathrm{km s^{-1} Mpc^{-1}}, the evolutions of parameters are explored. Second, in template metric context, by marginalizing over H_{mathcal {D}_0} as a prior of uniform distribution, we obtain the best-fit values of n=-1.22^{+0.68}_{-0.41} and {{varOmega }_{m}^{mathcal {D}_{0}}}=0.12^{+0.04}_{-0.02}. Moreover, we utilize three different Gaussian priors of H_{mathcal {D}_0}, which result in different best-fits of n, but almost the same best-fit value of {{varOmega }_{m}^{mathcal {D}_{0}}}sim 0.12. Also, the absolute constraints without marginalization of parameter are obtained: n=-1.1^{+0.58}_{-0.50} and {{varOmega }_{m}^{mathcal {D}_{0}}}=0.13pm 0.03. With these constraints, the evolutions of the effective deceleration parameter q^{mathcal {D}} indicate that the backreaction can account for the accelerated expansion of the Universe without involving extra dark energy component in the scaling solution context. Nevertheless, the results also verify that the prescription of kappa _{mathcal {D}} is insufficient and should be improved.

On Backreaction in Newtonian cosmology

Abstract We clarify that a result recently stated by Kaiser is contained in a theorem of Buchert and Ehlers that is widely known for its main result: that there is no global kinematical backreaction in Newtonian cosmology. Kaiser cites this paper, re-derives parts of the theorem, but incompletely restates its content. He makes further claims, which cannot be proven beyond the limited context of Newtonian cosmology. We also discuss recent papers of Rácz et al. and Roukema, who claim the existence of global backreaction within the Newtonian framework.

Lagrangian theory of structure formation in relativistic cosmology. II. Average properties of a generic evolution model

Kinematical and dynamical properties of a generic inhomogeneous cosmological model, spatially averaged with respect to free-falling (generalized fundamental) observers, are investigated for the matter model irrotational dust. Paraphrasing a previous Newtonian investigation, we present a relativistic generalization of a backreaction model based on volume-averaging the Relativistic Zeldovich Approximation. In this model we investigate the effect of kinematical backreaction on the evolution of cosmological parameters as they are defined in an averaged inhomogeneous cosmology, and we show that the backreaction model interpolates between orthogonal symmetry properties by covering subcases of the plane-symmetric solution, the Lemaitre-Tolman-Bondi solution and the Szekeres solution. We so obtain a powerful model that lays the foundations for quantitatively addressing curvature inhomogeneities as they would be interpreted as Dark Energy or Dark Matter in a quasi-Newtonian cosmology. The present model, having a limited architecture due to an assumed FLRW background, is nevertheless capable of replacing 1/4 of the needed amount for Dark Energy on domains of 200 Mpc in diameter for typical (one-sigma) fluctuations in a CDM initial power spectrum. However, the model is far from explaining Dark Energy on larger scales (spatially), where a 6% effect on 400 Mpc domains is identified that can be traced back to an on average negative intrinsic curvature today. One drawback of the quantitative results presented is the fact that the epoch when backreaction is effective on large scales and leads to volume acceleration lies in the future. We discuss this issue in relation to the initial spectrum, the Dark Matter problem, the coincidence problem, and the fact that large-scale Dark Energy is an effect on the past light cone (not spatial), and we pinpoint key elements of future research.

Relative information entropy and Weyl curvature of the inhomogeneous Universe

Penrose conjectured a connection between entropy and Weyl curvature of the Universe. This is plausible, as the almost homogeneous and isotropic Universe at the onset of structure formation has negligible Weyl curvature, which then grows (relative to the Ricci curvature) due to the formation of large-scale structure, and thus reminds us of the second law of thermodynamics. We study two scalar measures to quantify the deviations from a homogeneous and isotropic space-time: the relative information entropy and a Weyl tensor invariant, and show their relation to the averaging problem. We calculate these two quantities up to second order in standard cosmological perturbation theory and find that they are correlated and can be linked via the kinematical backreaction of a spatially averaged universe model.

Post-Newtonian cosmological dynamics of plane-parallel perturbations and back-reaction

We study the general relativistic non-linear dynamics of self-gravitating irrotational dust in a cosmological setting, adopting the comoving and synchronous gauge, where all the equations can be written in terms of the metric tensor of spatial hyper-surfaces orthogonal to the fluid flow. Performing an expansion in inverse powers of the speed of light, we obtain the post-Newtonian equations, which yield the lowest-order relativistic effects arising during the non-linear evolution. We then specialize our analysis to globally plane-parallel configurations, i.e. to the case where the initial perturbation field depends on a single coordinate. The leading order of our expansion, corresponding to the ``Newtonian background'', is the Zel'dovich approximation, which, for plane-parallel perturbations in the Newtonian limit, represents an exact solution. This allows us to find the exact analytical form for the post-Newtonian metric, thereby providing the post-Newtonian extension of the Zel'dovich solution: this accounts for some relativistic effects, such as the non-Gaussianity of primordial perturbations. An application of our solution in the context of the back-reaction proposal is eventually given, providing a post-Newtonian estimation of kinematical back-reaction, mean spatial curvature, average scale-factor and expansion rate.

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.

Correspondence between kinematical backreaction and scalar field cosmologies—the ‘morphon field’

Spatially averaged inhomogeneous cosmologies in classical general relativity can be written in the form of effective Friedmann equations with sources that include backreaction terms. In this paper, we propose to describe these backreaction terms with the help of a homogeneous scalar field evolving in a potential; we call it the ‘morphon field’. This new field links classical inhomogeneous cosmologies to scalar field cosmologies, allowing to reinterpret, e.g., quintessence scenarios by routing the physical origin of the scalar field source to inhomogeneities in the universe. We investigate a one-parameter family of scaling solutions to the backreaction problem. Subcases of these solutions (all without an assumed cosmological constant) include scale-dependent models with Friedmannian kinematics that can mimic the presence of a cosmological constant or a time-dependent cosmological term. We explicitly reconstruct the scalar field potential for the scaling solutions and discuss those cases that provide a solution to the dark energy and coincidence problems. In this approach, dark energy emerges from morphon fields, a mechanism that can be understood through the proposed correspondence: the averaged cosmology is characterized by a weak decay (quintessence) or growth (phantom quintessence) of kinematical fluctuations, fed by ‘curvature energy’ that is stored in the averaged 3-Ricci curvature. We find that the late-time trajectories of those models approach attractors that lie in the future of a state that is predicted by observational constraints.