twin universe astrophysics and cosmology Matter ghost matter astrophysics. 5 : Results of numerical 2D simulations.
VLS. About a possible schema for galaxies' formation. (p6)
On figure 17, call d the diameter of a cell and f the diameter of a clump. For different given initial conditions, and randomized initial positions of mass-points, the number of clumps nc (and cells on the screen) does not change so much. The standard deviation obeys :
(7) snc << nc
Same thing for the masses and diameters of the clumps.
(8)
smc << < mc >
(9) sf << < f > Of course, these are only 2D simulations. Nothing says that such a system, with three dimensions, would behave in the same way, but we may presume it would. This model is certainly not comparable to observations, but is an exploration of our qualitative ideas. However, these structures are very stable in time and space.
Although it comes from 2D simulation, we can examine some features, for this peculiar numerical computation. Matter forms a cellular structure. Call rs the mean mass density of matter in that structure. We use the subscript s for, in 3D, one could expect to get some "spongy structure". The mass density, in the clumps, obeys :
(10)
Outside the clumps, the ghost matter has a constant density (subscript e, for "external"), corresponding to
(11)
which gives (12)
The mean diameter of the clumps, compared to the mean diameter of the cells, obeys :
(13)
which gives (14)
which means that there is the same amount of ghost matter inside and outside the clumps. As these results correspond to 2D it is difficult to define temperatures and Jeans' lengths. Perhaps can we define some sort of "pseudo-temperature", as a measure of the mean kinetic energy in these 2D gazes.
(15)
T » < Vx2 + Vy2 > = < V2 >
Call <Ve> the mean thermal velocity of a unity ghost matter particle, outside the clumps, and <Vc> the averaged velocity in the clump.
(16)
<Ve> » <Vc>
Outside the clumps, the ghost matter density and the mean random velocity (the thermal velocity) are constant in space. In addition :
(17)
If we consider that the diameter f of the clump is close to some two-dimensional Jeans' length we find that the order of magnitude of that length, in the interclumps space, for ghost matter, is close to the distance d, between clumps, which suggests that, between the clumps, the ghosty matter is gravitationnaly stable. Where the matter is (from that definition of "2D temperature") :
(18)
Before the galaxies' formation (this comes from the paper [3]) the temperature of the ghost matter is higher than the temperature of matter (T* » 16 T) .
Can we estimate the effect of these hypothetic ghost matter clumps on the light coming from distant sources ? A photon, located in our fold of the universe, cannot be captured by a ghost matter particle, on pure geometric grounds [3]. But ghost matter clumps act on the photons' paths by negative gravitational lensing ( [6] and [8] ).
Can the presence of ghost matter clumps be evidenced by some cosmological test ? We can build a rough evaluation, taking a non-realistic situation where the universe is described as euclidean and steady, that would fit moderate distances.
The diameters f of the ghost matter clumps are very similar. As seen before, the standard deviations (5) and (9) are weak so that we can figure space, over large distances, as a regular distribution of cells, with a spheroidal clump nested at the center of each cell, and we can take the same diameter f for all clumps. Call n the number of density of the clumps, assumed to be constant over space.
(19)
A photon travels with the velocity c. The cross-section of a clump is :
(20)
The encounter frequency is (remember that the photon cannot be absorbed by the clumps) :
(21)
The mean free path is :
(22)
What about the reduction of the number of observed galaxies, located at a given distance r ? From kinetic theory we know how to compute the probability to observe a free path of a given length r. It is :
(23)
Let :
(24)
then :
(25)
p strongly depends on the value of a .The probability h to get a gravitational lensing effect is 1 - p , which correspond to the curves :
** ** Fig. 18 :** probability to observe antilensing effect** versus distance, for different values of f/d
The computational results, presented in the paper, correspond to the value f/d » 0.14. But dissipative processes may then occur in the clumps, that could drastically reduce their diameter, transforming these objects, for example, into some giant galaxies. From [3] the today's averaged ratio (ghost matter density / normal matter density) r*/r is » 65. Rough calculation gives the mass of a clump : 105 MG, where MG is the mass of a galaxy. If clumps transform into relatively small objects we could expect to get unaltered images from distant sources (quasars, galaxies). A cluster of galaxies, roughly speaking, acts as a biconvex lens. A ghost matter clump would act as a concave lens. The images of distant galaxies, through such gravitational lens, should appear smaller, fainter and more numerous. As pointed out by Peebles (ref. [13], page 311) they are too much large redshift faint galaxies for an Einstein-de-Sitter model.
The effect of antilensing on the background objects (galaxies, QSO), and on CMB will be analyzed in details in the next paper, including negative curvature effect (k = - 1).