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
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
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
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.
18.09.2018 /
Jan. J. Ostrowski (Warsaw, Poland) :
The most massive virialized objects in the Universe (Invited Talk at 'Inhomogeneous Cosmologies III', Jagiellonian University, Krakow, Poland)
The existence of the most massive virialized objects can be used as a tool to study properties of the Universe, such as the nature of gravity and the dark sector, as well as the Gaussianity of primordial density fluctuations. In my talk I will show the predictions for the most massive objects in the Universe, using the silent universe model (Einstein field equations with no rotation and energy transfer), and the scalar averaging formalism with the Relativistic Zel'dovich approximation serving as a closure condition. I will present the cosmological mass function of galaxy clusters for both cases and put them into the context of current and future astronomical sky surveys.
21.09.2018 /
Thomas Buchert (ERC PI, CRAL) :
Prismatic Summary (Invited Talk at 'Inhomogeneous Cosmologies III', Jagiellonian University, Krakow, Poland)
Closing talk summarizing the sessions at this International Workshop.
29.11.2018 /
Ismaël Delgado Gaspar (National Autonomous University of Mexico, Mexico City) :
Modeling of cosmic structures from exact solutions of Einstein's equations
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
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
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-LIP Informatics 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 (Invited Seminar at Monash University, Melbourne, Australia)
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.
21.03.2019 (Université Ouverte de Lyon) & 06.04.2019 (Société Astronomique de Lyon) /
Thomas Buchert (ERC PI, CRAL) :
L'Univers de Newton et Einstein - Énergie et matière noires démystifiées (Public Outreach, in French)
Le modèle de la cosmologie standard ne donne qu'une vision réduite de la structure de l'Univers, au regard de la généralité de la théorie de la gravitation d'Einstein. Nous exposerons d'abord les éléments de la cosmologie newtonienne qui sont suffisants pour comprendre les modèles cosmologiques actuels, puis nous considérerons la théorie de la gravitation d'Einstein dans toute sa généralité. Un survol de l'évolution historique du modèle standard de cosmologie nous permettra de comprendre comment sont apparus les concepts d'énergie noire et de matière noire. Nous discuterons ensuite les principes généraux de la construction d'une cosmologie physique qui ouvre une nouvelle perspective pour expliquer les énergies obscures à partir des propriétés géométriques de l'Univers. En particulier, nous essayerons de comprendre le rôle de l'espace et du temps dans le contexte d'une cosmologie inhomogène.
11.04.2019 /
Quentin Vigneron (PhD, CRAL) :
Relativistic cosmological simulations and Cosmic Topology
Cosmological simulations have been around for years. They 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 with Friedmann's equations as a background and matter is evolved around it. This approach does not allow for the investigation of the potential effects of inhomogeneities, called backreaction, on the global expansion and its link to dark energy. 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 & Ehlers, 1997), this justifies the need for fully general relativistic cosmological simulations. This field of cosmology has been developing in the last few years from setups with simple symmetries (e.g. Bentivegna & 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. Thus, probing the influence of Cosmic Topology on backreaction is an important step to quantify 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.
08.05.2019 /
Nezihe Uzun (ERC Postdoc, CRAL) :
Symplectic evolution of light beams in general relativity (Invited Seminar, ZARM, Bremen, Germany)
We present a symplectic evolution of thin null bundles of generic, curved spacetimes on a reduced phase space. Following an action principle for the deviation of null curves and identifying the observable, screen projected components of the Jacobi fields allows one to have a Hamiltonian flow which is similar to the one of a time-dependent harmonic oscillator. We consider only the first order part of the evolution and obtain the ray bundle transfer matrices analogous to the ABCD transfer matrices of the Newtonian paraxial optics. Later, it is shown that the the distance reciprocity in relativity follows from the symplecticity of the transfer matrices whose algebraic properties sheds light on the optical properties of the given spacetime. Other potential applications of this formalism for astrophysical and cosmological scenarios will also be discussed.
14.05.2019 /
Pratyush Pranav (ERC postdoc, CRAL) :
Algebraic topology of data sets: unexpected topology of the Cosmic Microwave Background
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
(p = 0 ... d). Intuitively, in three 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 datasets arising in Cosmology: 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 occurring in the nascent stages of the Universe. The accepted model for the CMB is a homogeneous, isotropic Gaussian random field. We find that the observations depart significantly from the theoretical model.
23.05.2019 /
Léo Brunswic (ERC Postdoc, CRAL) :
Topological spacetimes: from causality to Choquet-Bruhat-Geroch
Usually, by spacetime we mean a manifold endowed with an infinitely differentiable Lorentzian metric. However, several research directions lead to considering spacetimes which are slightly more complex: on the partial differential equation side, we tend to introduce weak solutions and therefore a metric which might not be differentiable at all before proving a regularity statement; on the geometrical side, we approximate a smooth manifold by constant curvature manifolds with singularities. Both sides use weaker versions of the usual theory of Lorentzian spacetimes. We present a unified framework that heads toward encapsulating causal intuition, and we infer from this new causal concepts.
11.06.2019 /
Mikhail Tsitsvero (ENS-LIP Informatics Department) :
Laplacian operators over simplicial complexes and localization properties of their spectra
In this talk first I will review Hodge and Bochner Laplacian operators over simplicial complexes. Secondly, I will describe the localization properties of Laplacian eigenvectors. Finally, I will present the tool for computing Laplacians and
higher-order diffusion in Python (see github project).
03.07.2019 /
Pravabati Chingangbam (Indian Institute of Astrophysics, Bangalore, India) :
Minkowski tensors as probes of cosmological fields
We present Minkowski tensors as a statistical tool that exploit the morphology of random fields and can be applied generically to a variety of physical fields. Our main interest is application to cosmological fields. We show how they can be used to trace the properties of quantum fluctuations from the epoch of inflation to the late stages of the Universe. We demonstrate cosmological insights about the statistical nature and isotropy of the fields by applying them to the temperature and polarization fields of the Cosmic Microwave Background. We also discuss how they can be used to probe the epochs of reionization and structure formation.
18.07.2019 /
Rémi Faure (ENS-M1) :
Averages in Newtonian cosmology
Cosmology aims at describing the whole Universe, facing a very complex problem. The dynamic equations of a fluid on cosmological scales can be averaged over domains for the sake of simplification. This is possible for scalar equations, but also for vectorial and tensorial equations in a Newtonian space. Our goal is to write these averaged equations to describe domain dynamics, not necessarily isolated, kinetically and energetically. Then, these results can be specified to the case of a 3-torus, commonly employed in simulations, noting differences compared with isolated systems.
29.08.2019 /
Pierre Mourier (PhD, CRAL) :
Inhomogeneous relativistic cosmology --- Non-perturbative models and spatial averaging of the Einstein equations (Thesis defense talk, in French)
In the standard model of cosmology, the global dynamics of the Universe is modeled via a highly symmetric background space-time with homogeneous and isotropic spatial sections. However, the interpretation of observations within this model calls for an unexpectedly accelerated expansion requiring a poorly-understood Dark Energy component, in addition to Dark Matter.
Inhomogeneous cosmology aims at relaxing the restrictions of such models on the geometry and sources while staying within the framework of General Relativity. It can allow, in particular, for an improved modeling of the formation of structures accounting for strong deviations from homogeneity in the matter distribution and the geometry. It can also study the dynamical consequences, or backreaction effects, of the development of such inhomogeneities on the expansion of larger scales. Such a backreaction may then reproduce, at least partially, the behaviours attributed to Dark Energy or Dark Matter.
During my PhD under the direction of Thomas Buchert, I have been working on several analytical aspects of
general-relativistic inhomogeneous cosmology. I will briefly recall a first aspect: a relativistic Lagrangian approximation scheme for the description of the local dynamics of structures in irrotational perfect fluids. I will then focus on the other main approach considered in my PhD: the effective description of inhomogeneous fluids with vorticity and a general energy-momentum tensor in terms of spatial averaging. The averaging scheme I will present is applicable to any choice of spatial hypersurfaces of averaging. It provides for each choice a set of effective evolution equations, featuring several backreaction terms, for an averaging region comoving with the sources. I will also discuss the rewriting and extension of such an averaging scheme under a manifestly 4-covariant form.