02-03.09.2019 / Thomas Buchert (ERC PI, CRAL) : ARTHUS ROUND TABLE III

This third round table focusses on (i) topological constraints for averaged model universes, (ii) light cone averages and fundamental aspects of formalisms for cosmological distance measurement, and (iii) analytical frameworks for general-relativistic simulations including topological aspects. The general idea of this round table is to communicate current new ideas of the team members in the domains of the three subjects. This round table is wider in scope to allow for an overview but it also aims at understanding technical problems. Participants (alphabetic): Léo Brunswic, Thomas Buchert, Mauro Carfora, Martin J. France, Étienne Jaupart, Colin MacLaurin, Pierre Mourier, Jan J. Ostrowski, Dennis Stock, Nezihe Uzun, Quentin Vigneron, Rolf Walder.




02.09.2019 / Léo Brunswic (ERC Postdoc, CRAL) : Update on the GBC approach to the averaged Universe

The Gauss-Bonnet formula gives a topological constraint on the average of the scalar curvature on a 2D Riemannian manifold. Such a constraint closes the 2+1 Buchert scalar averaging equations for irrotational dust fluids on a space-time admitting a compact without boundary Cauchy-surface. We explore what can be learned from this special (unphysical?) situation for higher dimensions: first, the Euler characteristic behaves as a mass that generically favours expansion. Mess-like theorems show that multi-connectedness acts qualitatively the same way in any dimension for empty universe models, multi-connectedness to which singularities like black holes contribute; second, one notices that almost any constraint on the averaged scalar curvature closes the system of averaged scalar equations. It is therefore straightforward to look for a natural ansatz on the averaged scalar curvature, leading to so-called scaling solutions; third, one can apply the 3+1 Gauss-Bonnet-Chern formula to find a new equation supplementing the known system. We give this new equation in the form of a second-order evolution equation for kinematical backreaction.




02.09.2019 / Thomas Buchert (ERC PI, CRAL) : Some remarks on the closure problem

I will discuss aspects of evolution equations for kinematical backreaction, thereby discussing a result of Léo's talk. In particular, we will try to understand the status of the closure problem of the scalar-averaged equations, and I will put other work into perspective: (i) the investigation of the Penrose Weyl-entropy conjecture within the Relativistic Zel'dovich Approximation, and (ii) the conservation of the Euler characteristic of averaging domains related to a conservation law for the averaged third principal scalar invariant of the expansion tensor in Newtonian cosmology. This may pave the way to obtain a physically sound closure condition.




02.09.2019 / Mauro Carfora (University of Pavia, Italy) : Light cone averaging and causal diamonds

I discuss the notion of causal diamonds and its potential relevance for addressing light cone averaging from a cosmographic point of view. Causal diamonds are the Lorentzian analog of geodesic balls in Riemannian geometry. They can be associated with a sufficiently small proper time span of an observer (i.e. with proper time segments of a chronological line). Their geometric properties allow proving the Lorentzian version of the familiar Bertrand-Puiseaux formulae of Riemannian geometry, relating Riemann, Ricci and scalar curvatures to the length, area, and volume associated with the Riemannian measure. In the Lorentzian case, these curvature-measure relations are subtler than in the Riemannian case. I will argue that they are relevant for addressing light cone averaging from a cosmographic point of view. The talk will be mainly a list of open problems and suggestions for possible answers.




02.09.2019 / Dennis Stock (ZARM, Bremen, Germany) : The Hawking energy on the past light cone

In the most general set-up in general relativity, it is difficult to define a reasonable energy notion. Amongst other candidates, Hawking's quasi-local proposal satisfies many natural limits, however, fails in general to be positive and monotonous. In this talk, I will discuss how one could use the Hawking energy on the past light cone in cosmology and under which circumstances it can be shown to be positive and monotonously increasing.




03.09.2019 / Jan. J. Ostrowski (Warsaw, Poland) : Spatial curvature in the Relativistic Zel'dovich Approximation

As in Newtonian Lagrangian perturbation theory, relativistic Lagrangian perturbations allow us to get insight into mildly nonlinear stages of structure formation, substantially exceeding the Eulerian regime. I will mainly focus on the spatial curvature estimates utilizing the Relativistic Zel'dovich approximation. Several theoretical and numerical results will be presented regarding the value of scalar curvature and averaged scalar curvature at the turnaround epoch for a wide set of initial conditions.
I will also address an inherent ambiguity when it comes to evaluating certain functionals (e.g. density, spatial curvature, expansion) within the Lagrangian framework, coming from the fact that there is more than one definition of these fields in terms of the governing equations. One possible work-around is to compare the perturbative expectations with some exact relativistic solutions. I will put into perspective such comparison for spherically symmetric dust solutions, the Lemaître-Tolman-Bondi metrics, with an emphasis on the evaluation of spatial curvature, extending previous works focusing mostly on the density field.




03.09.2019 / Quentin Vigneron (PhD, CRAL) : Relativistic cosmological simulations and Cosmic topology

Cosmological simulations have been around for years and seek to probe the relation between the formation of large-scale structures and the global expansion of the Universe. This relation is mostly studied in one way, i.e. the global expansion law is fixed by Friedmann's equations as a background and matter is evolved around it. This approach does not allow for the investigation of backreaction. Such a study must not have any background. As backreaction is known to be strictly zero in Newtonian simulations due to periodic boundary conditions (Buchert and Ehlers 1997), this justifies the need for fully general relativistic cosmological simulations. This latter field has been developing in the last few years from setups with simple symmetries (e.g. Bentivegna and Bruni 2016) to a setup with the full CMB power spectrum (Macpherson et al. 2018-19). However, as for Newtonian simulations, the relativistic ones all assume the topology of a flat 3-Torus for the Universe, i.e. a cubic simulation box with periodic boundary conditions. Hence, probing the influence of cosmic topology on backreaction is an important step to quantifying the effects of inhomogeneities on the global expansion.
The goal of this presentation is to introduce the first steps towards relativistic simulations in any topology. After reviewing the current status of relativistic cosmological simulations, I will go into the numerical methods in general relativity and how they could be adapted depending on the topology.




03.09.2019 / Léo Brunswic (ERC Postdoc, CRAL) : Iterative scheme for solving the Einstein equation

Most schemes used in the physics literature to solve Einstein's equation more or less involve improvement on the Euler scheme (or finite difference method). Such methods are crippled by three main issues: (i) one cannot use the result of a previous computation and improve on it directly; (ii) ensuring conservation of some first integral (like the total mass for a dust fluid) is not easy; (iii) numerical stability is difficult to ensure (one of the main interest of the known BSSN scheme is its seemingly numerical stability). We shall present a sketch of an iterative scheme based on a fixed point method for solving Einstein's equation presented by Friedrich and Rendall, as well as progress on quasilinear wave equations on a Lorentzian manifold. We shall present the following reduction steps: first, a transformation to reduce Einstein's equation to a nonlinear wave equation following a trick of Rendall and Friedrich; second, a construction of a solution to a nonlinear wave equation as a fixed point of a sequence of linear wave equations---this reduces a nonlinear equation to infinitely many linear wave equations; third, progress on linear wave equations on a Lorentzian manifold presented by Bär, Ginoux and Pfäffle allow us to compute the fundamental solution of such a linear wave equation by reduction to infinitely many transport equations which in turn can be reduced to a system of ODEs.




16-17.09.2019 / Nezihe Uzun (ERC Postdoc, CRAL) : Cosmology: Simplistic versus Realistic (Invited Lectures, Nicolaus Copernicus University, Torun, Poland)

In this lecture series we give a brief introduction to cosmology via identifying two approaches: simplistic and realistic. The first lecture, which summarizes the simplistic route, outlines the basic constructions of standard cosmology. We start with identifying the important concepts and mathematical objects within General Relativity and discuss on the simplified assumptions on the Einstein equations in order to derive the standard cosmological metric. We outline different evolution scenarios of the Universe and introduce important observational phenomena which constrain the parameters of the standard model. Those include (i) distance calculations and Supernova Ia experiments, (ii) Hot Big Bang and the primordial nucleosynthesis, and (iii) cosmic microwave background radiation (CMB). The second lecture is about a more realistic approach that outlines the problems of the standard model. Those include (i) dark energy, (ii) Hubble parameter discrepancy, (iii) flatness and horizon, (iv) dark matter, (v) CMB anomalies and (vi) lithium abundance problems. In this lecture, we introduce some approaches to attack those problems such as modified gravity and inhomogeneous cosmological models. The latter is given in further detail by identifying its sub-branches which gained much popularity over the last few decades within the relativistic cosmology community.




17.10.2019 / Nezihe Uzun (ERC Postdoc, CRAL) : Symplectic ray bundle transfer in general relativity (Invited Seminar, INAF Turin Astrophysics Observatory, Italy)

Reciprocity relations in physics signal the existence of potentiality of a system. Maxwell-Betti reciprocity for virtual work in elasticity, Onsager's reciprocity in thermodynamics or quantum mechanical reciprocity of the received signal all state that the observables are unchanged when the input and output agents are traversed. Those distinct systems share a similar property: they can be linked to a well-defined symplectic potential. The work we will present here grew out of questioning what kind of potentiality Etherington's distance reciprocity in relativity corresponds to. We will present the outcome of such an investigation which turns out to be a symplectic phase space reformulation of first order geometric optics in relativity. Potential applications of this formalism for astrophysical and cosmological scenarios will also be discussed.




07.11.2019 / Nezihe Uzun (ERC Postdoc, CRAL) : Reciprocity and Symplecticity: why are they relevant for cosmological distances?

Reciprocity relations in physics signal the existence of potentiality of a system. Maxwell-Betti reciprocity for virtual work in elasticity, Onsager's reciprocity in thermodynamics or quantum mechanical reciprocity of the received signal all state that the observables are unchanged when the input and output agents are traversed. Those distinct systems share a similar property: they can be linked to a well-defined symplectic potential. The work we will present here grew out of questioning what kind of potentiality Etherington's distance reciprocity in general relativity corresponds to. We will present the outcome of such an investigation which turns out to be a symplectic reduced phase space reformulation of first-order geometric optics in general relativity. Potential applications of this formalism and its relevance for realistic astrophysical and cosmological scenarii will also be briefly discussed.




19.11.2019 / Pratyush Pranav (ERC Postdoc, CRAL) : 2D projections of 3D galaxy distributions for redshift bins of finite thickness

Various observational studies of the 3D large-scale structure and the galaxy distribution in the Universe often do not concentrate on the evolution of structure, i.e. that structure further out in space also corresponds to earlier times, and hence are at different evolutionary timelines. Any full 3D study of galaxy distributions will always have contributions from a mixture of objects at various evolutionary stages. To clearly segregate the signatures of various observables, such as tracers of the BAO, it may therefore be desirable to analyze the structure of the galaxy distributions by collapsing/projecting objects within 2D shells of finite thickness. While, a priori, this projection is in redshift space and on 2-dimensional spherical surfaces, it is a challenge for the interpretation within inhomogeneous cosmology, since the redshift does no longer correspond to the evolution "time", as in the standard model of cosmology. In this talk, I will discuss a strategy to implement this procedure from SDSS data. The final product is a set of 2D maps in HealPix format, which can then be subjected to our tools from Geometry and Topology, as well as more. Characteristics of these maps at different redshift bins will contain information about different evolutionary stages of the Universe. The aim of this exercise is to further discuss how we can extract meaningful information from the maps, as well as to formulate improvements in the methods.




11.12.2019 / Asta Heinesen (ERC Postdoc, CRAL) : Planck evidence for a closed Universe and a possible crisis for cosmology

We discuss the paper "Planck evidence for a closed Universe and a possible crisis for cosmology" (arXiv:1911.02087), which suggests the removal of LCDM tensions for the Planck CMB power spectrum by introducing positive curvature to the base LCDM model. The introduction of curvature as a free parameter has implications for the consistency with other observations, such as BAO measurements. We discuss the results of the paper in relation to inhomogeneous universe models with evolving curvature.




17.12.2019 / Stephen Appleby (KIAS, Seoul, Korea) : Cosmological parameter estimation from topological statistics

The genus of the matter density field, as traced by galaxies, contains cosmological information. In particular, this statistic provides a measurement of the shape of the linear matter power spectrum. As the genus is a topological quantity, it is insensitive to galaxy bias and only weakly sensitive to non-linear gravitational collapse. Furthermore, as it approximately traces the linear matter power spectrum, it is a conserved quantity with redshift. Hence the genus amplitude is a standard ruler that can be used to test the distance-redshift relation. In this talk I discuss methods by which we can extract information from the genus, and more generally the Minkowski functionals, from current and future large-scale structure catalogs. I also introduce the Minkowski tensors, a rank-p generalization of the Minkowski functionals, which are a class of statistics sensitive to anisotropic signals. I explain how these quantities can be used to measure the anisotropy generated by redshift space distortion.




20.01.2020 / Thomas Buchert (ERC PI, CRAL) : Averaged inhomogeneous geometry in relativistic cosmology (Invited Seminar, Colloqium ICJ, Mathematical Institute Camille Jordan, Lyon)

The standard model of cosmology rests on a homogeneous-isotropic solution of Einstein's laws of gravitation. The so-called "concordance model" with homogeneous geometry fixes the parameters of this model in conformity with available observational data. Accepting this model leads to a number of unresolved issues related to the conjecture of existence of Dark Matter and Dark Energy. To resolve these, a large community either seeks to generalize the laws of gravitation, or assumes the existence of new fundamental fields providing challenges for particle physics.
In this talk we focus on a third possibility that is conservative by not generalizing the laws of Einstein and by not including any new fundamental field. We present and motivate from first principles a set of effective (i.e. spatially averaged) Einstein equations that govern the regional and global dynamics of inhomogeneous cosmological models. In this framework there are new terms that arise from curvature invariants of the inhomogeneous geometry of spatial hypersurfaces. These terms qualitatively play the role of Dark Matter and Dark Energy.
I will first recall basic principles that lead to the standard model of cosmology and discuss its governing cornerstones (without assuming prerequisites in general relativity from the audience). I will also recall how a cosmological model is built from the splitting of spacetime into spatial hypersurfaces that evolve in time. By introducing a spatial averaging operation we will arrive at a set of equations that govern general cosmologies. Due to their generality this set of equations is not closed, and I will outline recent work that investigates a topological approach, based on the Gauss-Bonnet-Chern theorem, to achieve closure.




30.01.2020 / Thomas Buchert (ERC PI, CRAL) : ARTHUS ROUND TABLE IV

This small round table mainly focusses on discussions with our visitors Jan Ostrowski (Poland), Roberto Sussman (Mexico) and David Wiltshire (New Zealand) during two weeks from the end of January to the beginning of February. We only scheduled the following two talks. Participants (alphabetic): Léo Brunswic, Thomas Buchert, Martin J. France, Asta Heinesen, Jan J. Ostrowski, Pratyush Pranav, Roberto A. Sussman, Nezihe Uzun, Quentin Vigneron, and David L. Wiltshire.




30.01.2020 / Roberto A. Sussman (UNAM, Mexico) : Relativistic interpretation and cosmological signature of Milgrom's acceleration

We propose in this work a relativistic coordinate-independent interpretation of Milgrom's acceleration, a_0=1.2 times 10^{-8} {cm/s}^2, through a geometric constraint obtained from the product of the Kretschmann invariant scalar times the surface area of 2-spheres defined through suitable characteristic length scales for local and cosmic regimes, described by Schwarzschild and FLRW geometries, respectively. By demanding consistency between these regimes we obtain an appealing expression for the empirical (so far unexplained) relation between the accelerations a_0 and c H_0. Imposing this covariant geometric criterion upon a FLRW model, yields a dynamical equation for the Hubble scalar whose solution matches, to a very high accuracy, the cosmic expansion rate of the LCDM concordance model fit for cosmic times close to the present epoch. While these results are very preliminary and strictly valid only at a toy model level, we believe that they could provide relevant information in the search of alternative gravity theories or even within General Relativity itself.




30.01.2020 / Quentin Vigneron (PhD, CRAL) : Post-Newtonian theories and topological acceleration

I will show that Newton's equations can be written in a ADM 3+1-like form where the spacetime manifold is taken to be the Minkowski manifold. In this picture, the fluid 4-velocity is not normalized and not necessarily time-like, allowing for spatial velocity higher than c, as expected in Newton. Moreover, a fundamental foliation has to be defined, called the Galilean foliation. This (new?) formulation shows that expanding Newtonian simulations are not Newtonian, but already post-Newtonian as the Galileiian respectively Minkowski manifold has to be changed into a FLRW manifold. More generally, the formulation provides a (new?) way to define post-Newtonian theories coherently with GR. We will be interested in non-flat post-Newtonian theories as they could allow a simpler study of the effect of global topology on backreaction and structure formation.




05.02.2020 and 10.02.2020 / Asta Heinesen (ERC Postdoc, CRAL) : Inhomogeneous Cosmology (Invited Seminars at Univ. Stavanger, Norway, and Univ. Copenhagen, Denmark)

I will give an introduction to the field of inhomogeneous cosmology focusing on averaging schemes for formulating a large-scale cosmological theory from a general relativistic spacetime. An important insight from the averaging formalism discussed is that structure in a relativistic spacetime induces violation of the spatial curvature conservation law of the FLRW class of models usually used for interpreting cosmological data. I will mention some theoretical and observational problems of interest in inhomogeneous cosmology and talk about some of my own work in relation to these questions.




31.03.2020 / Thomas Buchert (ERC PI, CRAL) : ARTHUS VIRTUAL ROUND TABLE V

This online round table is supposed to bridge the gap during the corona crisis. We'll make the tour over all subjects. Participants (alphabetic): Léo Brunswic, Thomas Buchert, Asta Heinesen, Pierre Mourier, Jan J. Ostrowski, Pratyush Pranav, Nezihe Uzun, Quentin Vigneron.




15.05.2020 / Pratyush Pranav (ERC Postdoc, CRAL) : Topology and Geometry: Application to cosmological datasets (Online Seminar: CRAL)

In the last couple of decades, topology and geometry have matured from purely theoretical fields towards strong focus on applicability in various research domains. The principal tool from Geometry has been the development of integral geometric quantifiers, viz. the Minkowski functionals. On the side of topology, a combination of Morse theory, homology and persistent homology, has enabled a new branch in data analysis called topological data analysis (TDA). The central tenet is based on the identification and assessment of geometrical and 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. The geometric quantifiers are associated with the notions of d-dimensional volume, area etc. These methods provide additional and complementary information with regards to traditional measures like n-point correlation functions. In the first part of my talk, I will present a non-technical summary of the methods.
In the second part, I will present two examples highlighting the application component. The first example concerns theoretical Gaussian random field models with power-law power spectra. We find that a topological characterization through Betti numbers and persistence diagrams provide information that is missed by traditional topological measures like the Euler characteristic and the more familiar geometric Minkowski functionals. The second example concerns the analysis of the topological characteristics of the temperature fluctuations in the Cosmic Microwave Background (CMB) from the temperature anisotropy maps measured by the Planck satellite. We find that the observed maps differ significantly from the simulations modeled as isotropic, homogeneous Gaussian random fields.




25.05.2020 / Asta Heinesen (ERC Postdoc, CRAL) : Inhomogeneous Cosmology (Online Seminar: Torun, Poland)

I will give an introduction to the field of inhomogeneous cosmology focusing on averaging schemes for formulating a large-scale cosmological theory from a general relativistic spacetime. An important insight from the averaging formalism discussed is that structure in a relativistic spacetime induces violation of the spatial curvature conservation law of the FLRW class of models usually used for interpreting cosmological data. Such a violation, if significant, would have important physical and observational consequences. I will mention some questions of interest in inhomogeneous cosmology and talk about some of my own work in relation to these questions.




03.06.2020 / Jan. J. Ostrowski (Warsaw, Poland) : On the most massive objects in the Universe (Online Seminar: National Centre for Nuclear Research, Warsaw, Poland)

Galaxy clusters and superclusters can be used to test cosmological models. In particular, big enough objects at low redshifts would be a strong indication of the failure of the Concordance Model. It is for this reason among others, why precise predictions on how many gravitationally bound objects are we expecting to observe, are of great importance. In my presentation I will sketch a theoretical pipeline leading from the early Universe initial conditions, via the general relativistic model of evolution, to the expected number of compact objects as a function of their masses and redshifts. Several of the so-obtained results will be put into the context of our current observational knowledge.




25.06.2020 / Pratyush Pranav (ERC Postdoc, CRAL) : Topo-geometric methods for geophysical applications (Interdisciplinary Online Seminar: XLVIII International Summer School, St. Petersburg, Russia)

Topological and geometric statistics carry a wide range of possibilities for interdisciplinary applications. In geophysics, the study of geological structures is important from the perspective of investigating tectonic structures, as well as understanding the characteristics of oil, gas and mineral deposits. In the context of oil and gas exploration, the study of fractal models engenders novel insights about efficient distribution and extraction mechanisms. In this talk, I present an application of topo-geometric tools to fractal models. We employ a fractal model that was developed in cosmology by Soneira and Peebles. Through adaptation to a distribution of objects projected on a sphere, we can with these tools also characterize and understand natural gas and oil reserves.