# Seminars and Round Tables

## 08.09.2017 /
Jan. J. Ostrowski (Postdoc LIO, CRAL) :
Cosmological backreaction conjecture:
recent developments and future prospects

##
Inhomogeneous, relativistic cosmology has recently observed a rise in popularity among
the scientific community. In particular, the scalar averaging approach has been intensively
examined both analytically and numerically, giving some new insights into the problem of
cosmological backreaction, i.e. the conjectured influence of small scale density
inhomogeneities on the large scale evolution of the Universe. In my talk, I will summarize
these recent efforts including the Green and Wald theorem and several results from N-body
simulations, and I present future prospects of cosmological backreaction investigations.

## 04-08.12.2017 /
Thomas Buchert (ERC PI, CRAL) :
ARTHUS ROUND TABLE I

##
This first round table focusses on (i) nonperturbative modeling aspects of backreaction, (ii) the fundamental question of closure of the averaged equations, (iii) the generalization of the averaged equations themselves, (iv) topological and geometrical issues, both in mathematical cosmology approaches and in statistical measurements of observational data,
and finally on other fundamentals concerning, e.g., strategies for light cone averaging and light cone smoothing.
The general idea of this round table is to give the core team members and our guests a complete overview over the project structure and its workpackages, emphasizing open problems in some selected cases, and to provide near-future perspectives. The headline is to pose the good questions that allow to find the good answers.
Participants (alphabetic): Léo Brunswic, Thomas Buchert, Mauro Carfora, Martin J. France, Étienne Jaupart,
Pierre Mourier, Jan J. Ostrowski, Pratyush Pranav, Nezihe Uzun, Quentin Vigneron, Rolf Walder, David L. Wiltshire.

## 04.12.2017 /
Jan. J. Ostrowski (Postdoc LIO, CRAL) :
The mass function in relativistic cosmology

##
The realistic description of cosmological structure formation is an important challenge from both theoretical and numerical point of views. This can be partially answered via the cosmological mass function, which is a statistical tool describing a number density of collapsed objects as a function of initial conditions, mass and redshift.
In my presentation I will give a prescription for a semi-analytic treatment of structure formation and a resulting mass function on galaxy cluster scales for a highly generic scenario. This approach relies on scalar averaging of Einstein's equations together with the relativistic generalization of Zel'dovich's approximation serving as a closure condition, and thus allows one to calculate in addition mass-dependent distributions of kinematical backreaction and averaged scalar curvature. Comparison with the N-body-inferred results will also be presented.

## 04.12.2017 /
Quentin Vigneron (ERC Internship ENS, CRAL) :
Cosmological Dark Matter as a morphon field

##
Backreaction terms can be viewed as arising from an effective scalar field, called morphon field. A simple example of such a field is the scaling solution (kinematical backreaction and average scalar curvature being powers of the volume scale factor). This solution can describe models of quintessence for Dark Energy. It, however, always requires Cold Dark Matter in addition to the baryonic matter. Thus, trying to describe Dark Energy and Dark Matter with a single morphon field using a scaling solution is not possible. I describe here, how a partitioning of both the space and the morphon field could solve the problem and allow us to describe the expansion rate without Dark Energy and Dark Matter in a unified way.

## 04.12.2017 /
Jan. J. Ostrowski (Postdoc LIO, CRAL) :
The Green-Wald conjecture and its aftermath

##
In a series of papers by Green and Wald, the authors claimed to have developed a
formalism to properly describe the influence of density inhomogeneities on
background properties of the Universe, i.e., the cosmological backreaction effect.
Their main conclusion is that backreaction is trace-free and obeys the weak energy condition, i.e. it cannot mimic
Dark Energy. The applicability of Green and Wald's formalism to cosmology was criticized by a
number of cosmologists and relativists, which launched a debate among the wider community. Recent numerical investigations, as well as some explicit analytic examples show that Green and Wald's main theorem is at least not general, and in any case not applicable to cosmological backreaction in its present form.
In my talk I give a brief introduction into the Green and Wald formalism, point out the major drawbacks and attempts to put this formalism into work, and I summarize the current status of the `backreaction debate'.

## 05.12.2017 /
David L. Wiltshire (Univ. of Canterbury, New Zealand) :
Cosmological averages, observables and spacetime structure

##
Solving the fitting problem requires a geometrical understanding
of the averages in inhomogeneous cosmology as they affect light
propagation, and their relation to measurements. In my view
foundational physical questions must be addressed. The appropriate
mathematical ingredients are an open question.

## 05.12.2017 /
Martin J. France (ERC Technician, CRAL) :
Non-Gaussianity and signatures of cosmic topology: toward a generalized model-independent analysis of the CMB?

##
Our recent Minkowski functional analyses of the CMB show that, seen in the framework of the LCDM model, the observed CMB is still highly Gaussian. But, model-independent Hermite expansions fit better the CMB Non-Gaussianity than the prescriptions of perturbation theory. We find also that more than 1 percent of a huge LCDM model sample of CMB maps is 2 to 3 times less Gaussian than the Planck map.

In order to widen the view on interpretation of both the CMB Non-Gaussianity and the CMB signatures of cosmic topology, I discuss as a proposal whether a part of the CMB dipole or/and some multipoles could be considered as being a consequence of the deviation of the CMB support manifold from the ideal sphere.

## 05.12.2017 /
Pratyush Pranav (ERC Postdoc, CRAL) :
Topological holes and their persistence

##
Topological and morphological studies of cosmic density fields have a long history, with the Euler characteristic and Minkowski functionals playing the key role in analysis. However, the information contained in the Euler characteristic is compressed, due to the Euler-Poincare formula, which states that the Euler characteristic is the alternating sum of another topological invariant called the Betti numbers. The rest of the Minkowski functionals are more geometric in nature and provide morphological assessment of the cosmic fields. Furthermore, these descriptors are not equipped to handle the hierarchical nature of structure formation and evolution in the Universe.

In view of these observations, I will present a brief introduction to homology and persistence, and Morse theory. Stemming from algebraic topology, homology quantifies the topological characteristics of a manifold in terms of the presence of topological holes of different dimensions present in it. The topology of a d-dimensional manifold can be expressed in terms of k-dimensional holes, where k runs from 0 to d. Mathematically, these holes represent the homology groups of the manifold, and the Betti numbers are the ranks of the homology groups, counting the number of independent holes. Persistence homology is an extension of the regular homology theory in hierarchical settings. The hierarchical aspects of persistence are achieved by creating a filtering of the manifold, such that the sub-manifold at higher density threshold is contained in sub-manifolds at lower thresholds through inclusion, thereby creating a continuous map. The central tenet of persistence lies in the idea of formation and destruction of topological holes, as the density threshold changes continuously. Due to its hierarchical nature, persistence homology may be a powerful descriptor of the topology of cosmic fields, when model discrimination is the primary focus.
At the end I present an analysis of the Planck CMB temperature field.

## 06.12.2017 /
Pierre Mourier (PhD, CRAL) :
Fluid-comoving generalizations of the GR scalar spatial averaging formalism to arbitrary foliations

##
In a first part I will present a generalization of the spatial averaging formalism for inhomogeneous scalars as exposed by T. Buchert in two papers in 2000-2001. This averaging process was carried out in a fluid-orthogonal foliation for a universe model filled with irrotational dust or an irrotational perfect fluid with pressure. I will show how this can be extended to averaging in arbitrary spatial hypersurfaces, which also allows the treatment of a fully general fluid, that is, in general, non-perfect and with vorticity. Although several proposals for such a generalization have already been introduced in the literature, contrary to these our formalism sticks to the crucial concept of an averaging domain following the fluid flow and, thus, preserving its fluid rest-mass content along its evolution.

In a second part I will present the averaging operation and the system of averaged scalar Einstein equations for two different averaging definitions. The first one is directly inspired from the existing literature, modifying mostly the domain's propagation, while the second approach roots instead the averaging operator itself to the fluid flow. This latter choice allows for simple and transparent averaged and effective equations without loss of generality. I will then make use of the foliation and shift freedom to highlight an especially physically relevant particular choice, dubbed the Lagrange picture, which makes the averaged equations even simpler and removes an otherwise generally present interpretation subtlety. I will, however, point out a mathematical difficulty lying in the construction of the corresponding foliation.

## 07.12.2017 /
Léo Brunswic (ERC Postdoc 08.01.2018, CRAL) :
Closure with the Gauss-Bonnet-Chern-Avez theorem?

##
Following Magni, we present how the Gauss-Bonnet formula closes the 2+1 averaging framework of self-gravitating dust, and how the topology relates to the universal expansion: the Euler characteristic behaves as a mass which can be negative. Then, we generalize the Gauss-Bonnet-Chern theorem to 3+1 self-gravitating dust using a sandwich approach and a method of Avez. The formula we obtain does not close the averaging framework in dimension 3+1 but gives an interesting relation between a Weyl tensor invariant, the second moment of the density and extrinsic curvature scalar invariants.

## 07.12.2017 /
Nezihe Uzun (Univ. of Prague, Czech Republic) :
On the reciprocity relation for light propagation through multiple geometries

##
The reciprocity theorem of Etherington is used in cosmology in order to relate angular diameter distance to luminosity distance. It is applicable for light propagation within a single spacetime geometry and it holds irrespective of the distribution of the matter fields. Here I will consider the light propagation within multiple geometries which are not isometric to each other. I will mainly focus on the applicability of the reciprocity relation for Swiss-cheese-like cosmologies and its observational outcomes.

## 26.01.2018 /
Jan. J. Ostrowski (Postdoc LIO, CRAL) :
Cosmological mass function

##
The realistic description of cosmological structure formation is an important challenge from both theoretical and
numerical point of views. In my presentation I will give a brief prescription for a semi-analytic treatment of structure formation and a resulting mass function on galaxy cluster scales in a highly generic scenario. This will be obtained by an exact scalar averaging scheme together with the relativistic generalization of Zel'dovich's approximation (RZA) that serves as a closure condition for the averaged equations. Some results related to the
'silent universe' model will be also presented.

## 30.01.2018 /
Michel Mizony (Univ. Lyon1, Camille Jordan Institute of Mathematics, France) :
What is our Universe now? - For a century of a formula written by Willem de Sitter

##
Starting with the Friedmann-Lemaître metric of an isotropic model universe,
we give the radially inertial form of this metric which is a generalized Gullstrand-Painlevé form of metric. The equivariant stress-energy tensor is convenient because it has straightforward interpretation in terms of velocity and potential. For each model for the Universe, the osculating manifold is a de Sitter model. Moreover, if the model universe is the open cone of a big bang event, then this de Sitter model is an open, accelerated, one. We confront this model with local observations, taking into account the observed SNIa and the Hubble parameter H(z) for redshifts z up to 2. The recent data on H(z) provide a tool to estimate cosmological parameters for the de Sitter models and their Milne limits; we find:
Ho = 65 + 2 km/s/Mpc, Omega_o = 0.05 + 0.02 and an age = 15.2 + 0.3 Gyr. In other words, our model universe would only contain baryonic matter. This work can be viewed as an update of a marvelous de Sitter paper (1917-1918) about his metric.

## 09.05.2018 /
Thomas Buchert (ERC PI, CRAL) and Jan. J. Ostrowski (Postdoc LIO, CRAL) :
Tutorial: mass function

##
This tutorial focusses on themes at the interface of the ERC ARThUs group and the AstroENS Team. The aim is to
initialize a PhD topic. Foundations of Lagrangian effective models and the application to Newtonian and
relativistic backreaction models, in particular concentrating on the mass function, will be discussed.
Participants (alphabetic): Thomas Buchert, Gilles Chabrier, Étienne Jaupart, Jan. J. Ostrowski.

## 14.06.2018 /
Quentin Vigneron (CRAL) :
Topological acceleration in non-flat spaces

##
The topological acceleration is the effect of a nontrivial topology of the Universe on the gravitational field created by a point mass. The calculation of this acceleration can be done either with a Newtonian approach or a relativistic approach (e.g. Korotkin et al. (1994), Roukema et al. (2006), Ostrowski et al. (2012), Steiner (2016)). The former has the advantage of being easily implemented for various types of topological spaces. However, the Newtonian theory of gravitation is only valid for flat spaces, thus we should not be able to describe topological acceleration in spherical or hyperbolic spaces with this theory. A generalized theory of Newtonian gravity would be, however, interesting to develop and could have a number of applications, e.g. for N-body simulations, fast analytical estimates of gravitational dynamics in non-flat spaces. From this perspective, Roukema et al. (2009) used an heuristic approach in order to increase the domain of validity of Newton's theory for any spaces and calculated the topological acceleration in spherical spaces.
In a first part, I show that the approach of Roukema et al. leads to unphysical results. Then, I present in the second part what I think is a proof that a Newtonian theory on non-flat spaces is not possible. The last part presents a fully relativistic calculation of topological acceleration in a specific spherical space. The calculation is not finished but attempts to show that static topological acceleration in spherical spaces is possible, contrary to the results of Bentivegna et al. (2012).

## 05.07.2018 /
Francesco Sartini (ENS-M1) :
Globally static and stationary inhomogeneous cosmologies

##
Einstein's static model, which first has motivated the introduction of the cosmological constant, was abandoned soon after its formulation, as a result of the observation of the expansion of the Universe. This decision has been based on the simplistic view that any part of the Universe should reproduce the global behaviour. This is no longer valid in inhomogeneous cosmologies, where we can still have a globally static, and also stationary solution (i.e., expanding at constant speed), even without a cosmological constant. Using Buchert's averaging formalism, and a volume-partitioning method, we try to investigate the possibility of a regional expansion that could comply with observation, within a globally static or stationary Universe, ending by pointing out some remarkable differences between the two cases.

## 10.07.2018 /
Célia Desgrange (ENS-L3) :
Averaged inhomogeneous cosmology: fit to Supernovae without Dark Energy

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Unlike the standard model of cosmology which considers the Universe to be locally homogeneous, the averaged inhomogeneous model takes into account these inhomogeneities called backreaction in order to be closer to the reality. We tested the averaged model with one of the largest currently available Type Ia supernovae data catalogue, the JLA (Joint Light-curve Analysis) catalogue. This kind of supernovae is still the only direct evidence for Dark Energy in the Universe according to the standard model.
By using exact scaling solutions parametrized with the scaling index n for the averaged curvature
and backreaction, we find n = -1.05 and a dimensionless contribution to the energy budget of
the matter parameter Ω_D0 = 0.237, results independent of the Hubble function on the domain
of averaging. In this work, we conclude that the averaged model without Dark Energy is indistinguishable from the ΛCDM model on these data, as the AIC (Akaike Information Criterion) is equal to 0.35.

##
## 04-05.09.2018 /
Thomas Buchert (ERC PI, CRAL) :
ARTHUS ROUND TABLE II

##
This second round table focusses on (i) aspects of map generation and statistical characterization of the Cosmic Microwave Background (CMB) including non-trivial topologies, and (ii) other roles of topology including topological acceleration and dynamical topology change.
The general idea of this round table is to intensify collaboration between the Ulm group and the core team members,
to concretize current collaborations, and to setup future projects. This round table is narrowed in scope to allow
for specific interaction.
Participants (alphabetic): Ralf Aurich, Léo Brunswic, Thomas Buchert, Martin J. France, Étienne Jaupart,
Pierre Mourier, Jan J. Ostrowski, Pratyush Pranav, Frank Steiner, Quentin Vigneron.

## 04.09.2018 /
Ralf Aurich (Ulm University, Germany) :
On the computation of CMB anisotropies

##
The simulation of the anisotropies of the CMB is discussed. The numerical computation of the brightness function is outlined, which allows the calculation of CMB sky maps and the computation of the angular power spectrum. Applications to unified dark matter cosmologies and to non-trivial topologies of the Universe are presented.

## 04.09.2018 /
Martin J. France (ERC Technician, CRAL) :
Non-spherical CMB support manifold: some cosmological implications

##
During the hydrogen recombination process, exact spherical symmetry is assumed for the support manifold of the CMB. Thus, contemporary CMB anisotropies are supposed equidistant (at the same redshift) to an observer today and, angularly, differential effects are treated in a global and perturbative way in the CMB simulation maps. These anisotropies are also interpreted with respect to a 2-sphere. Except for the masked regions, the CMB observed today displays in each point the behavior of an ideal black body and, thanks to Wien's law, we infer that the CMB is a primordial black body as well. In a realistic description, the Universe is inhomogeneous and in this talk I am interested to interpret the CMB anisotropies over an inhomogeneous support manifold. The components of our boost relative to the surface of last scattering are not all known. Here, I consider first that the intrinsic CMB dipole originates from an ovoid deviation with respect to the ideal sphere (a first order inhomogeneity), and I calculate and discuss some implications for the CMB interpretation and the cosmology.

## 04.09.2018 /
Pratyush Pranav (ERC Postdoc, CRAL) :
Homology of the Cosmic Microwave Background

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We study the topology generated by the temperature fluctuations of the Cosmic
Microwave Background radiation (CMB), as quantified by the number of loops in the growing excursion sets. We compare CMB maps observed by the Planck satellite with a thousand simulated maps generated according to the LCDM paradigm with Gaussian distributed fluctuations. The comparison is multi-scale, being performed on a sequence of degraded maps with mean pixel separation ranging from 0.05 to 7.33 degrees.
The parametric chi^2 test shows differences between observations and simulations, yielding p-values at per mil level at the scale of roughly 3 -7 degrees, with the difference in the number of loops peaking at more than 3 sigma. There are reports of mildly unusual behaviour of the Euler characteristic at this scale, which is phenomenologically related to the strongly anomalous behaviour of loops. It is also the scale at which the observed maps have low variance compared to the simulations, and roughly the scale at which the power spectrum exhibits a dip with respect to the theoretical model. Non-parametric tests show even stronger differences at almost all scales. Barring a trivial secondary or systematic effect, it motivates a closer look at the standard paradigm. The significant difference between the observations and simulations leaves a few possibilities, including primordial non-Gaussianity and observation of non-standard manifolds.

## 05.09.2018 /
Frank Steiner (CRAL) :
On the gravitational field of a star in a flat 3-torus universe

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The exact first-order solution to the Einstein equations is derived which determines the exterior static gravitational field of an isolated non-rotating star in a spatially finite universe having the topology of a general flat 3-torus. The solution is given in terms of Appell's and Epstein's zeta function, respectively, and implies the existence of a topological dark energy. The anisotropy of the field is made explicit by giving its exact multipole expansion.

## 05.09.2018 /
Quentin Vigneron (PhD, CRAL) :
Newtonian theory in non-flat spaces

##
Newtonian and relativistic simulations in cosmology are currently realized assuming periodic boundary conditions in order to have a finite simulation volume. This induces a compact topology for the Universe. In relativistic simulations, any topology could be implemented, but to the best of my knowledge, the only one used throughout the literature is the flat 3-torus T^3 for reasons of simplicity. The same applies to Newtonian cosmology, however for a different reason: the flatness of the space is imposed by the theory, thus the flat and orientable 3-torus is the simplest among the compact homogeneous topologies available. Due to this choice of topology, the underlying results of the simulations might be dependent on this topology. Probing other compact topologies could provide clues on this dependence, especially on the resulting backreaction effects.
Although, it is possible in relativistic cosmology to probe topologies for non-flat spaces, it is in practice much more complicated to implement them. Thus, developing a Newtonian theory on a non-flat space would be an easy tool, using N-body simulations, to probe different topologies.
For this purpose, Roukema et al. (2009) used a heuristic approach in order to increase the domain of validity of Newton's theory for any spaces. I will show that this approach leads to unphysical results. Then, I will present two attempts to develop such a theory. The first one uses the Schwarzschild geometry in a specific spatial coordinate system to derive the new equations of motion; the second one starts from an exact relativistic solution of N black holes in spherical space S^3, using geometrostatics.
So far, no physical Newtonian theory on a non-flat space has been found, but it seems that in order to achieve this goal, one should consider the non-flat space to be non-static.

## 05.09.2018 /
Léo Brunswic (ERC Postdoc, CRAL) :
Topology dynamics in globally hyperbolic singular Ricci flat 2+1 spacetimes

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In 2+1 Buchert's dust framework, the Gauss-Bonnet Theorem allows us to identify the Euler characteristic with a mass of topological origin, with positive or negative sign. We show how the Euler characteristic may evolve in a `sticky particle model': in dimension 2+1, a dust spacetime may be discretized to a Ricci flat spacetime with singular lines (massive point-particles); these particles may collide and we assume that at most one massive particle arises from the collision (the particles are `sticky'). Such a spacetime is a singular E^{1,2}-manifold and a collision can be described as a `blow-up' of a singular HS^2-manifold. Its holonomy provides conservation laws, thus, constraining outgoing particles.
If the ingoing `energy' is above a threshold, collision does not allow for a unique outgoing massive particle, and extra particles have to come out of the collision. 2+1 analogues of black/white holes are natural candidates for such collisions changing the topological type of the spacetime and, thus, the Euler characteristic.
Ways to generalize this approach to 3+1 dimensions will be discussed. In such a scenario, Dark Energy arises through the evolution of the Euler characteristic to strongly negative values, corresponding to on average negative spatial curvature.

## 29.11.2018 /
Ismaël Delgado Gaspar (National Autonomous University of Mexico) :
Modeling of cosmic structures from exact solutions of Einstein's equations

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We show that the full dynamical freedom of the Szekeres models allows for the description of elaborated 3-dimensional networks of CDM structures (overdensities and density voids), with the spatial comoving location of each structure uniquely specified by the initial conditions. This type of structure modeling provides a coarse-grained but fully relativistic description of evolving large-scale cosmic structures before their virialization. We also examine the gravitational collapse and black hole formation of these nonspherical configurations, smoothly matched to a Schwarzschild exterior. Finally, we discuss a model for the evolution of cosmic voids made up of a mixture of two non-comoving dust components, namely CDM and baryonic matter.

## 10.01.2019 /
Eleonora Villa (CFT PAN, Warsaw, Poland) :
The distance-redshift relation in the inhomogeneous Universe: improving on perturbation theory

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We consider an approximation scheme aiming at accounting for the effect of non-linear inhomogeneities in relativistic cosmology at all scales of interest: the perturbations in the space-time metric are considered to be small, but a different weight is given to their first and second spatial derivatives, taking into account that peculiar velocities - corresponding to first spatial derivatives - are small, but allowing the spatial curvature and the density perturbations - corresponding to second spatial derivatives - to be large. We apply this framework to light propagation and discuss the possible improvement of our approach with respect to the standard perturbative description.

## 15.02.2019 /
Pratyush Pranav (ERC Postdoc, CRAL) :
Algebraic topology and its application to cosmological data sets

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In the last couple of decades, algebraic topology has matured from a purely theoretical field to one with strong applicability in various research domains. Combining Morse theory, Homology and Persistent homology, this has enabled a new branch in data analysis called topological data analysis (TDA). The central tenet is based on the identification of topological changes that occur in a manifold as a function of the excursion sets of the field. The topological changes are accounted for by tracking the creation and destruction of p-dimensional topological holes in a d-dimensional manifold. Intuitively, in 3 spatial dimensions, these changes correspond to creation and destruction of connected components, loops/tunnels and voids. Towards the end, I will present an example of application of the formalism to the Cosmic Microwave Background (CMB) radiation. The CMB is the earliest visible light in the universe, and studying its properties has the potential to reveal information about the physical processes occuring in the nascent stages of the Universe.

## 14.03.2019 /
Mikhail Tsitsvero (ENS Physics Department) :
Analytic signal and filtering in several dimensions
(with potential applications to study the CMB data on a sphere)

##
In this talk I briefly present two main objects of mathematical signal processing - analytic signal and space-varying filtering. First of all a method for calculation of instantaneous amplitude and phases of a multidimensional oscillating process will be presented. Second, starting with 1-dimensional filtering theory, I provide possible extension for the multidimensional filtering theory. To conclude the discussion some remarks on space-frequency/phase-space methods will be given (Wigner-Ville transform and Segal-Bargmann transform).

## 19.03.2019 /
Pierre Mourier (PhD, CRAL) :
Cosmology at the Antipodes --- foliation-dependence in averaging and a little bit of numerical relativity

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I will first recall the context of averaging in general foliations of spacetime in cosmology, and two possible averaging schemes with their resulting averaged Einstein equations. I will then discuss the covariant formulation for these averages that we have been investigating with Asta Heinesen in Christchurch and some of the early results regarding the dependence of averages (or integrated quantities) in the foliation choice. Finally, after a very brief review of the recent developments in numerical simulations for relativistic cosmology, I will discuss the application of the general averaging schemes to one of these simulations in collaboration with Hayley Macpherson, Daniel Price and Paul Lasky at Monash University.

### Last Update: March 18, 2019