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
<V*e> the mean thermal
velocity of a unity ghost matter particle, outside the clumps, and <V*c>
the averaged velocity in the clump.
(16)
<V*e> »
<V*c>
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�s 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 len. A ghost matter clump
would act as a concave len. 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 cbr will be analyzed in details
in the next paper,including negative curvature effect (k = - 1).
