**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/cm^{3 } (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/ cm^{3}. Cosmic ray is 10^{-32} kg /
cm^{3}. Dark sky brightness is 10^{-32} kg / cm^{3}; x
ray is l0^{-34} kg / cm^{3}. 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 R_{0} is the present value of the cosmic scale factor, H_{0}
and q_{0} is the present value of the Hubble constant and deceleration
parameter
respectively.

From (11-8) we know whether the spatial curvature k /R^{2} is
positive or negative that is determined by the factor of whether p_{0}
greater or less than the critical density

(11-10)

At present the observed value of the Hubble parameter is

H_{0} =50km · s^{-1} · Mpc^{-1
}(11-11)^{
}

The observed value of the deceleration parameter is

q_{0}
=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.

P_{0} << ρ_{0
}

Therefore, from (11-9) we have

k
/ R_{0}^{2}=(2q_{0} –1) H_{0}^{2
}(11-14)

Considering (11-8), we get the present value of the ratio of ρ_{0}
and p_{c
}

ρ_{0 }/ ρ _{c} = 2 q_{0}
(11-15)

However using (11-1) we get q_{0} = 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 m_{W}
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 = -h^{2 }·
^{ }·^{ }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 G^{2} N^{3} m_{WG}^{5} /
ħ^{2} 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 m_{W} is very small, putting (11.1-1) into (11.1-13) we
get

M_{MAX}
= 2.1 × 10^{36 }g ≈
10.5 × 10^{3 }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
m_{p}. 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 h_{ij} 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 m_{W
}– 0.058 N3 m_{W}^{5}
/ m_{pl}^{4
}(11.2-14)

Put m_{W}=3.6
x 10^{-35} kg into above equation we obtain

= (N – 4.3 x 10^{-155} N^{3}) m_{W}
(11.2-15)

In figure 4 I have plotted the distribution function of the average mass of WG star versus the particle number N.

**
figure 4**

Fig.
4

This is I believe a perfect universe!