scholarly journals Total Collisions in the N-Body Shape Space

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1712
Author(s):  
Flavio Mercati ◽  
Paula Reichert
Keyword(s):  
Configuration Space ◽  
Equations Of Motion ◽  
Shape Space ◽  
Big Bang ◽  
Matter Field ◽  
Stiff Matter ◽  
Simple Modification ◽  
N Body Problem ◽  
The Big Bang ◽  
Total Collision

We discuss the total collision singularities of the gravitational N-body problem on shape space. Shape space is the relational configuration space of the system obtained by quotienting ordinary configuration space with respect to the similarity group of total translations, rotations, and scalings. For the zero-energy gravitating N-body system, the dynamics on shape space can be constructed explicitly and the points of total collision, which are the points of central configuration and zero shape momenta, can be analyzed in detail. It turns out that, even on shape space where scale is not part of the description, the equations of motion diverge at (and only at) the points of total collision. We construct and study the stratified total-collision manifold and show that, at the points of total collision on shape space, the singularity is essential. There is, thus, no way to evolve solutions through these points. This mirrors closely the big bang singularity of general relativity, where the homogeneous-but-not-isotropic cosmological model of Bianchi IX shows an essential singularity at the big bang. A simple modification of the general-relativistic model (the addition of a stiff matter field) changes the system into one whose shape-dynamical description allows for a deterministic evolution through the singularity. We suspect that, similarly, some modification of the dynamics would be required in order to regularize the total collision singularity of the N-body model.

scholarly journals Investigating Total Collisions of the Newtonian N-Body Problem on Shape Space

2021 ◽  
Vol 51 (2) ◽  
Author(s):  
Paula Reichert
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Shape Space ◽  
Absolute Space ◽  
Gravitational System ◽  
N Body Problem ◽  
All Solutions ◽  
Dynamical Description ◽  
Total Collision ◽  
Newtonian Gravitational System

AbstractWe analyze the points of total collision of the Newtonian gravitational system on shape space (the relational configuration space of the system). While the Newtonian equations of motion, formulated with respect to absolute space and time, are singular at the point of total collision due to the singularity of the Newton potential at that point, this need not be the case on shape space where absolute scale doesn’t exist. We investigate whether, adopting a relational description of the system, the shape degrees of freedom, which are merely angles and their conjugate momenta, can be evolved through the points of total collision. Unfortunately, this is not the case. Even without scale, the equations of motion are singular at the points of total collision (and only there). This follows from the special behavior of the shape momenta. While this behavior induces the singularity, it at the same time provides a purely shape-dynamical description of total collisions. By help of this, we are able to discern total-collision solutions from non-collision solutions on shape space, that is, without reference to (external) scale. We can further use the shape-dynamical description to show that total-collision solutions form a set of measure zero among all solutions.

scholarly journals Gauss–Bonnet gravity in D = 4 without Gauss–Bonnet coupling to matter: Cosmological implications

2019 ◽  
Vol 34 (07n08) ◽  
pp. 1950051
Author(s):  
Eduardo Guendelman ◽  
Emil Nissimov ◽  
Svetlana Pacheva
Keyword(s):  
Functional Dependence ◽  
Equations Of Motion ◽  
Scalar Potential ◽  
Big Bang ◽  
Gravity Models ◽  
Back Reaction ◽  
Field Equations ◽  
Dark Energy Density ◽  
Bonnet Gravity ◽  
The Big Bang

We propose a new model of D = 4 Gauss–Bonnet gravity. To avoid the usual property of the integral over the standard D = 4 Gauss–Bonnet scalar becoming a total derivative term, we employ the formalism of metric-independent non-Riemannian spacetime volume elements which makes the D = 4 Gauss–Bonnet action term nontrivial without the need to couple it to matter fields unlike the case of ordinary D = 4 Gauss–Bonnet gravity models. The non-Riemannian volume element dynamically triggers the Gauss–Bonnet scalar to be an arbitrary integration constant M on-shell, which in turn has several interesting cosmological implications. (i) It yields specific solutions for the Hubble parameter and the Friedmann scale factor as functions of time, which are completely independent of the matter dynamics, i.e. there is no back reaction by matter on the cosmological metric. (ii) For M[Formula: see text]0, it predicts a “coasting”-like evolution immediately after the Big Bang, and it yields a late universe with dynamically produced dark energy density given through M. (iii) For the special value M = 0, we obtain exactly a linear “coasting” cosmology. (iv) For M[Formula: see text]0, we have in addition to the Big Bang also a Big Crunch with “coasting”-like evolution around both. (v) It allows for an explicit analytic solution of the pertinent Friedmann and [Formula: see text] scalar field equations of motion, while dynamically fixing uniquely the functional dependence on [Formula: see text] of the scalar potential.

Nonminimal derivative coupling: Some cosmological behaviors of John and George models

2014 ◽  
Vol 23 (08) ◽  
pp. 1450067
Author(s):  
A. Kanfon ◽  
G. Edah ◽  
E. Baloïtcha
Keyword(s):  
Exponential Function ◽  
Coupling Parameter ◽  
Big Bang ◽  
De Sitter ◽  
Stiff Matter ◽  
Derivative Coupling ◽  
Sitter Model ◽  
The Big Bang ◽  
The Universe ◽  
De Sitter Model

We examine some nonminimal derivative coupling models with a term of potential in front of the Ricci scalar–tensor. We limited ourselves to three models of this family: — the potential proportional to the square of the field — the potential proportional to the inverse of the field — the potential proportional to the exponential function of the field. The first one leads to an universe which closes a few moment after its creation. The two other models show an accelerated expanding universe after inflation. The model with a potential proportional to the exponential function of the field, pointed out, just after the big bang primordial, the predominance of dark energy guiding inflation. At the end of inflation, in its expansion, the universe tends to de Sitter model dominated by the stiff matter. These results are those obtained by using the potential which is a linear function of the field. What is interesting about this model is that these results are not very sensitive to variations of the coupling parameter and the initial velocity of the field.

scholarly journals REPARAMETRIZATION-INVARIANT PATH INTEGRAL IN GR AND "BIG BANG" OF QUANTUM UNIVERSE

2001 ◽  
Vol 16 (10) ◽  
pp. 1715-1742 ◽  
Author(s):  
M. PAWLOWSKI ◽  
V. N. PERVUSHIN
Keyword(s):  
General Relativity ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Big Bang ◽  
Constrained System ◽  
Generating Functional ◽  
Finite Space ◽  
Creation Of Matter ◽  
The Big Bang ◽  
Friedmann Robertson Walker

The reparametrization-invariant generating functional for the unitary and causal perturbation theory in general relativity in a finite space–time is obtained. The region of validity of the Faddeev–Popov–DeWitt functional is studied. It is shown that the invariant content of general relativity as a constrained system can be covered by two "equivalent" unconstrained systems: the "dynamic" (with "dynamic" evolution parameter as the metric scale factor) and "geometric" (given by the Levi–Civita type canonical transformation to the action-angle variables where the energy constraint converts into a new momentum). "Big Bang," the Hubble evolution, and creation of matter fields by the "geometric" vacuum are described by the inverted Levi–Civita transformation of the geomeric system into the dynamic one. The particular case of the Levi–Civita transformations are the Bogoliubov ones of the particle variables (diagonalizing the dynamic Hamiltonian) to the quasiparticles (diagonalizing the equations of motion). The choice of initial conditions for the "Big Bang" in the form of the Bogoliubov (squeezed) vacuum reproduces all stages of the evolution of the Friedmann–Robertson–Walker universe in their conformal (Hoyle–Narlikar) versions.

A view of the universe before the big bang

2006 ◽  
Vol 190 ◽  
pp. 15-15
Author(s):  
D CASTELVECCHI
Keyword(s):  
Big Bang ◽  
The Big Bang ◽  
The Universe

The First Galaxies in the Universe

2013 ◽  
Author(s):  
Abraham Loeb ◽  
Steven R. Furlanetto
Keyword(s):  
Galaxy Evolution ◽  
Big Bang ◽  
First Stars ◽  
Distant Galaxies ◽  
Formation And Evolution ◽  
Early Galaxies ◽  
First Galaxies ◽  
Large Telescopes ◽  
New Generation ◽  
The Big Bang

This book provides a comprehensive, self-contained introduction to one of the most exciting frontiers in astrophysics today: the quest to understand how the oldest and most distant galaxies in our universe first formed. Until now, most research on this question has been theoretical, but the next few years will bring about a new generation of large telescopes that promise to supply a flood of data about the infant universe during its first billion years after the big bang. This book bridges the gap between theory and observation. It is an invaluable reference for students and researchers on early galaxies. The book starts from basic physical principles before moving on to more advanced material. Topics include the gravitational growth of structure, the intergalactic medium, the formation and evolution of the first stars and black holes, feedback and galaxy evolution, reionization, 21-cm cosmology, and more.

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