Twin Universes cosmology
Astronomy and Space Science
226 : 273-307, 1995
Observatory of Marseille, France
...Starting from the field equation S = c ( T - A(T) ), presented in a former paper, we present last results, based on numerical simulations, giving a new model applying to the very large structure of the Universe. A theory of inverse gravitational lensing is developped, in which the observed effects could be mainly due to the action of surrounding "antipodal matter". This is an alternative to the explanation based on dark matter existence. Then we develop a cosmological model. Because of the hypothesis of homogeneity, the metric must be solution of the equation S = 0, although the total mass of the Universe is non-zero. In order to avoid the trivial solution R = constant x t , we consider a model with "variable constants". Then we derive the laws linking the different constants of physics : G , c , h , m in order to keep the basic equations of physics invariant, so that the variation of these constants is not measurable in the laboratory : the only effect of this process is the red shift, due to the secular variation of these constants. All the energies are conserved, but not the masses. We find that all the characteristic lengths (Schwarzschild, Jeans, Compton, Planck) vary like the characteritic length R, whence all the characteristic times vary like the cosmic time t. As the energy of the photon hn is conserved over its flight, the decrease of its frequency n is due to the growth of the Planck constant h » t . In such conditions the field equations has a single solution, corresponding to a negative curvature and to an evolution law : R varies like t2/3.
The model is no longer isentropic and s » Log t. The cosmologic horizon varies like R, so that the homogeneity of the Universe is ensured at any time which constitues an alternative to the theory of inflation. We refind, for moderate distances, the Hubble's law. A new law : distance = f(z) is derived, very close to the classical one for moderate red shifts.
In a former
paper  a cosmological model was presented, based on a new field equation
S = c ( T - A(T) )
from the Lagragian ( R+ - R-)
The Einstein equation :
S = c T
is a local
equation, meaning that the local geometry of the universe
( tensor S ) is determined by the local content of energy-matter (tensor ). In the equation (1) we assumed that space-time hypersurface had a S3 x R1 topology and that the local geometry of the universe was determined both by the local content of energy-matter and by the content of energy-matter of the associated antipodal fold, through the antipodality relationship A.
Figure 1 : The coordinate-invariant antipodality relationship....
If s represents the space coordinates, two geodesics starting from M focus at the antipodal point M*, or A(M). A is an involutive mapping. We can give a didactic image in order to schematize the physical meaning of the equation (1).
...Consider a S2 hollow sphere made of some opaque material. We suppose that ,in this medium, the heat does not propagate, but causes dilatation. If we deposit thermal energy in in some places, the surface will be shaped by dilatation. In such a model, the heat represents the energy ( tensor T ). The dilatation materializes the impact of the local energy content on the local geometry. Light does not propagate in this medium, as assumed. But we can assume that sonic waves can propagate and may carry the information, from a point to another point.
...In classical General Relativity, light is not "contained" in the model, for the electromagnetic energy is not explicitly present in the energy tensor (although radiative pressure terms can be present in the tensor T), so that the propagation of light along null geodesics is nothing but an hypothesis, well-confirmed by the observations and experiences. The analogue of the sonic waves, in the classical RG model, are the gravitational waves, that we can build, perturbing the field equation. However, we cannot build electromagnetic waves from the equation (2) and we assume that they follow the null-geodesics of the manifold, as the gravitational waves do .
the equation (1) we assumed that light also follow the null-geodesic. Moreover,
we assumed that the local geometry S was determined both by the local
energy-matter content T and by the associated antipodal content A(T).
In our former paper , using the classical low field and small velocities
approximation, we have shown that the "antipodal matter"
( located in s*) acted on the matter ( located in s) as "a repulsive negative mass distribution", due to the presence of the minus sign of the field equation (1).
We can schematize that in the following 2d model. Take a plane and put masses on the two sides, symbolized by small disks.
...Two masses can collide, and exchange photons, if they are located in the same side. The cannot if they are located on different sides. Two masses located on the same side attract each other through Newtonian law. Two masses located on opposite sides repel each other, through a Newtonian law. Particules located on the same side can exchange photons, but not particules located on opposite sides (the plane is opaque). See figure 2.
Figure 2 : Two-dimensional image of the system of forces. If the particules are on the same side, they attract each other, according to the Newton law. If they belong to opposite sides they repel each other, according to the repulsive Newton law. Photons j can travel from A to B and from C to D and vice-versa, for they are located on the same side. The cannot travel from E to F, and vice-versa.
...In our former paper we have shown, through analytic solution, that this mechanism provided a "missing mass effect", for an observer located on one side, if he ignores the existence of the particles located on the other one. Some results of 2d numerical simulations were presented . They provided, ar large scale, a non-homogenous pattern. See reference , figure 7.
...But this does not look like the known Universe, which appears to be fairly spongy. In 1970 Zel'dovich proposed his well-known theory of the pancakes . The pancake effect was first demonstrated in numerical models for the evolution of the three-dimensional mass distribution by Doroshkevich and al.( 1980 ), Klypin and Shandarin ( 1983 ), and Centrella and Mellot (1983) [ 3, 4, 5]. Mellot and Shandarin (1990) gave an elegant demonstration of the effect by using two-dimensional computations that afforded considerably better resolution for given particule number, see reference . Shandarin (1988) and Kofma, Pogosyan and Shandarin (1990) presented a powerful semianalytic method for predicting the positions of pancakes from the initial conditions [7 and 8]. More recently (1992) Mellot used a 3d set of 643 particules, with periodic boundary conditions. From Mellot, the density fluctuations remains small. As pointed out by Peebles in 1993  : " This cannot be the whole story, for the pancakes found are a transient effect : with increasing time the mass in the pancakes drains into clumps that are concentrated in all the three dimensions. This means that if the local sheet of galaxies were a pancake, it must have been formed recently". Then Peebles asked : "could there be a second generation of pancakes that form by the collective collapse of the groups of the clumps that formed out of the first generation ? " But he concluded immediatly : "This does not follow from the analysis given, for it depends on the continuity of the velocity field that allows to write down a series expansion for the evolution of the relative positions. After the formation of the first generation of clumps, which might be the galaxies or their progenitors, the velocity field in general does not have the coherence length , and the analysis from the continuity does not apply".
...As a conclusion the pancake theory cannot describe, in its present state, the observed large scale structure.