(11-2)THE MATH-PHYSICAL MODEL FOR THE UNIVERSE
CONSISTING OF DARK MATTER WG
In all of the observed cosmic substances, the contribution from the galaxy mass to the average density of the universe is decisive.
P
= 3.1x10-28 kg/cm3 (11-1)
The Contribution from other types of matter is several orders smaller
than it. For example, the density of the cosmic microwave background radiation
is 4 x 10-31 kg/ cm3. Cosmic ray is 10-32 kg /
cm3. Dark sky brightness is 10-32 kg / cm3; x
ray is l0-34 kg / cm3. Therefore the density of (11-1) can
be viewed as the total average density of cosmic substance.
On the other hand, in the so-called Big Bang theory of cosmology, the
basic equation of the universe is
(11-2)
(11-3)
Where the R (t) is the cosmic scale factor, k = -1, 0 1 corresponding to
open, flat and closed universe respectively. Eliminating
from (11-2)
and (11-3) we can get a differential equation of first order
(11-4)
The definition of the Hubbell parameter, which is a measure of the
universe, is
(11-5)
Using this expression (11-4) can be rewritten as
(11-6)
Where
(11-7)
The present value of the cosmic energy density and
pressure can be obtained from (11-2) and (11-3)
(11-8)
(11-9)
Where R0 is the present value of the cosmic scale factor, H0
and q0 is the present value of the Hubble constant and deceleration
parameter
respectively.
From (11-8) we know whether the spatial curvature k /R2 is
positive or negative that is determined by the factor of whether p0
greater or less than the critical density
(11-10)
At present the observed value of the Hubble parameter is
H0 =50km · s-1 · Mpc-1
(11-11)
The observed value of the deceleration parameter is
q0
=1.0
± 0.8
(11-12)
There is adequate evidence to confirm that mainly the non-relativistic
matter determines the present value of cosmic energy.
P0 << ρ0
Therefore, from (11-9) we have
k
/ R02=(2q0 –1) H02
(11-14)
Considering (11-8), we get the present value of the ratio of ρ0
and pc
ρ0 / ρ c = 2 q0
(11-15)
However using (11-1) we get q0 = 0.02. Which is much different from the observed value
(11-12). This implies that inevitably there exists an invisible matter in the
cosmos, and at least 90% of cosmic matters are made up by non-baryon,
furthermore the electromagnetic interaction of such substance must be very weak,
otherwise it could not be so dark as to be observed. In the previous section I
have stated that an individual WG original particle cannot drag “WG ether”
strongly (g=0), which implies that a very weak electromagnetic interaction of WG
original particle. So WG original particle can be a candidate for dark matter.
11.1.
The WG star
A mathematical study about WG composing the entire universe
Since
the distribution density of WG matter is p(r) in Newtonian mechanics frame, WG
matter satisfies the Poisson equation
∆V
= 4πGρ
(11.1-1)
Where V is the gravity potential of WG matter, G is gravitational
constant. On the other hand, under non-relativistic approximation, WG matter
must satisfy the Schrodiger equation
(11.1-2)
The density distribution of WG original particle in the same quantum state is
ρ
= N mW
ψ*ψ
(11.1-3)
Where N is the number of particles, and ψ is the wave function of a
single particle this satisfies the normalization condition
(11.1-4)
I am now discussing the spherically symmetry WG star, so as to examine
only the ground state wave function, i.e. the state with quantum number n = 1, l
= 0. The spherical symmetry radial function of ground state under dimensionless
unit satisfies
(11.1-5)
(11.1-6)
(11.1-7)
Where
r = -h2 ·
· G-1 · N-1 · u
(11.1-8)
(11.1-9)
(11.1-10)
(11.1-11)
The boundary condition of equation is Φ(u) → 0, for
u→∞. Since I am only discussing ground state of system, there are no
nodes in the wave function Φ(u). Using the Runge-Kutta method to integrate
the differential equation numerically, the value of binding energy of ground
state is E = -0.054 G2 N3 mWG5 /
ħ2 Therefore the total energy of WG star is
(11.1-12)
From (11.1-12) the upper limit of the total energy of WG star, the
maximum value of total energy takes place at
.
(11.1-13)
On the other hand because the value of mW is very small, putting (11.1-1) into (11.1-13) we
get
MMAX
= 2.1 × 1036 g ≈
10.5 × 103 M
(11.1-14)
I believe that using WG theory I can solve the cosmic dark matter
problem.
11.2 The
analytical study for the mass of WG star.
In Newtonian approximation I will analytically study the ground state
energy of the N WG original particle system further. The Newtonian potential
between two of WG particles is
(11.2-1)
The total Hamiltonian of the system is
(11.2-2)
Where
(12.2-3)
Comparing with two bodies Hamiltonian of hydrogen atom, I have found that
the only difference is that
replaces
mp. Therefore I can use the results of hydrogen atom Schrodiger
equation with appropriate replacement. For example the expected value of the
ground state must satisfy an inequality P
(11.2-4)
Thus I have got the lower limit of ground state energy of N WG original
particles system of self- drawing:
(11.2-5)
This is a preliminary analytical result. Separating the kinetic energy
and that of center of mass, I can get a better analytical result.
Using the mathematical identity
(11.2-6)
The Hamiltonian for relative motion of N WG particles in coordinate of
the center of mass is
(11.2-7)
Where
(11.2-8)
The definition of the conjugate momentum of (
) is 
= (
)/2,
which satisfies the canonical transformation. The (11.2-8) can be rewritten as
(11.2-9)
The lower limit of the expected value of hij is
(11.2-10)
Therefore the lower limit of ground state is
(11.2-11)
On the other hand, if I was using trial wave function
(11.2-12)
and standard variation approach I would get the upper
limit of ground state energy
(11.2-13)
Considering that WG star consists of a lot of WG original particles, I
now find the difference between the upper and lower limits is only 15%. I
suggest that the average mass of WG star is
=
N mW
– 0.058 N3 mW5
/ mpl4
(11.2-14)
Put mW=3.6
x 10-35 kg into above equation we obtain
= (N – 4.3 x 10-155 N3) mW
(11.2-15)
In figure 4 I have plotted the distribution function of the average mass
of WG star
versus the particle number N.
Fig.
4
This is I believe a perfect universe!