# Seminars and Round Tables

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

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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

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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.

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

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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

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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

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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

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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?

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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

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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

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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?

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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, Tchechoslovakia) :
On the reciprocity relation for light propagation through multiple geometries

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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.

### Last Update: November 27, 2017