# User:Helge Rosé

Helge Rosé - Physicist - System Analysis and Simulation - Fraunhofer Institut für Offene Kommunikationssysteme

## Geometric Matter

[1] – On the geometrization of matter by exotic smoothness

## Calculation of the Cosmological Constant by Unifying Matter and Dark Energy

Paper available arXiv.org 0710.1562

During the late 1990s, observations of supernovae suggested that the expansion of the cosmos is accelerating by an unknown kind of energy. Observations of the cosmic microwave background indicate that this dark energy accounts for 70% of the total energy in the cosmos. In the last years, great effort has been invested to understand this unknown kind of energy.

In our paper "Calculation of the Cosmological Constant by Unifying Matter and Dark Energy" arXiv.org gr-qc/0609004 we propose a new approach to explain the origin of dark energy. We are able to calculate Einsteins cosmological constant which coincides very well with the observations: We obtain Omega_D = 0.734 and current observations report Omega_D in the range 0.65 ... 0.85

What do we assume?

We following Einsteins way. Einstein introduced a great idea to physics: Think geometrically. He assumes that a physically thing - gravity - is nothing else but a geometrically property of spacetime - metric. We use this geometrically thinking for our approach. There are three properties of spacetime: topology, differential structure and metric. We use the differential structure to explain the global structure of the cosmos. Knowing this global appearance of the cosmos we can interpret dark energy and calculate the cosmological constant.

What mathematics says

File:Akbulutcork.jpg Spacetime is a 4-dimensional space - a 4-manifold. The differential structure (see this post) of spacetime M determines a special subset of M. This subset is called the "Akbulut cork" A. The 4-dimensional Akbulut cork has a 3-dimensional boundary Sigma (like the 3D earth has a 2D surface). This boundary is also a special one: the mathematicians call it a homology 3-sphere. There are some important examples: First of all the Poincaré sphere - it is a Dodecahedron with opposite faces glued together (You can't do that in 3D but in 4D spacetime it is easy. A forward driving spacecraft leaving through one face will enter again through the opposite one).

For fun, read: The fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used for arranging the constellations on the whole heaven". Aristotle added a fifth element, aithêr (aether in Latin, "ether" in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato's fifth solid. The second example are the spheres Sigma(p,q,r) of the mathematician Brieskorn. The Poincaré sphere is a special case with Sigma(2,3,5) and we use the Brieskorn sphere Sigma(2,5,7) as model for the cosmos.

The mathematicians have made a theory of 3-dimensional spaces like the Brieskorn spheres. They found the astonishing fact that there are only three kinds of components that can form a such a 3-sphere: negatively curved pieces Ki, positively curved 3-spheres S3 and positively curved Poincaré spheres.

Our conjecture

We assume that the boundary of the Akbulut cork - this Brieskorn sphere Sigma(2,5,7) - is the cosmos. We was very amazed about the fact that such a sphere is build by only three types of components. There are also three types of matter in the cosmos - could this be an accident? Nobody knows, but we could make a conjecture. If it leads to a fruitful calculation it could be a promising new approach.

So we conjecture that the three kinds of 3-dimensional spaces forming a Brieskorn sphere correspond to the three kinds of matter: baryonic matter, dark matter and dark energy. In this approach matter and energy are interpreted as an expression of the differential-geometric state of spacetime. Matter is understood purely geometrically, as negatively curved pieces Ki. Dark matter, being the linking pieces S3 between the particles Ki, forms a sort of "shadow matter". The third component - two Poincaré spheres - is the origin of dark energy.

The calculation

If you have a curved piece of space and you know is curvature, then you can also calculate the energy representing it by using the Einstein equation. Now we have this two Poincaré spheres representing the dark energy. We can calculate its energy by calculating its curvature.

This is not an easy task. But with the help of a result of Edward Witten one can reduce this to the calculation of a special mathematical number - the Chern-Simons invariant. This number characterize a 3-sphere like Pi characterize a circle - it is a so-called topological invariant.

We can not calculate the absolute energy value. But we are able to calculate the fraction of dark energy with regard to the total energy of the cosmos. This calculated energy fraction - sqrt(14/27) - is a purely mathematical number. The astronomer measure the dark energy by the density parameter Omega. The observation of the WMAP project yields Omega = 1.02. With this value we obtain for the cosmological constant and the dark energy

Lambda = sqrt(14/27) 3H0^2/c^4 Omega = 1.4 10^-52 m^-2. Omega_D = sqrt(14/27) Omega = 0.734

The current observations report Omega_D = 0.65 ... 0.85, which coincides very well with our calculation.