# A Lattice Universe with Matter-Antimatter Repulsion

Paul R. Gerber

Gerber Molecular Design, Forten 649, CH-8873 Amden, Switzerland

Assuming repulsion between matter and antimatter
(see evidence) the universe will be built
up by islands of matter and antimatter. A simple model would be to assume an
arrangement of matter and antimatter islands of equal size and mass in a
lattice of NaCl type. A cell would represent an average galaxy cluster.
## Confinment Potential

In a cell, the rest of the lattice generates a gravitational potential, in
which matter (or antimatter) of the cell moves. It can be calculated as a
Taylor series about the center of the cell. The distance from the center
is r and the distance to the next (antimatter) cell-center is R.
One obtains, with the usual non-chalance, the potential:

V(r) = M_{1}GuR^{2} + 2M_{3}Gur^{2} + ...,

where G is the gravitational constant, u the average mass density of the universe,
M_{1} = 1.748 is the Madelung constant of the NaCl lattice, and
M_{3} the analog sum with the minus third power of the distance
and a cos^{2} in the direction with respect to any nearest-neibor
lattice bond (or any fixed direction). M_{3} is also of order one.
The harmonic part of this confinement potential is attractive and independent
of the lattice constant R.
## Galaxy rotation

A galaxy in our cell moves in this potential and experiences corresponding
tidal forces. For the harmonic part, the tidal potential is the same harmonic
potential, which provides an additional centripetal force. One can ask at which
distance, r_{e}, from the galactic center the galactic and the tidal
centripetal forces are equal. One obtains:

m(r_{e}) = 4M_{3}ur_{e}^{3}.

The mass of the galaxy within this distance, m(r_{e}), is a couple of
times the mass with a filling of the average density of the Universe.
In addition, the tidal contribution to the centripetal force keeps growing
with increasing distance from the galactic center. This is quite comparable
to measured galaxy rotation curves without any need for postulating
the presence of dark matter.

Furthermore, to keep larger galaxy clusters together, increasing amounts of
dark matter appear to be necessary, whereas with the tidal potential it is
obvious that confinment becomes increasingly efficient for more extended
matter regions.

Amden, February 4, 2011