**
The
fundamental gravitino and its physical dark world**

**
THE MATH-PHYSICAL MODEL FOR THE UNIVERSE **

**
CONSISTING OF DARK MATTER FG**

In all of the observed cosmic substances, the contribution from the galaxy mass to the average density of the universe is decisive.

曲=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} <<
老_{0}

Therefore, from (11-9) we have

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

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

老_{0 }/
老 _{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
FG original particle cannot drag ※FG ether§ strongly (g=0), which implies that a
very weak electromagnetic interaction of FG original particle. So FG original
particle can be a candidate for dark matter.

**
11.1. The FG star**

**
A mathematical study about FG composing the entire universe**

Since the distribution density of FG matter is p(r) in Newtonian mechanics frame, FG matter satisfies the Poisson equation

∆V = 4羽G老 (11.1-1)

Where V is the gravity potential of FG matter, G is gravitational constant. On the other hand, under non-relativistic approximation, FG matter must satisfy the Schrodiger equation

(11.1-2)

The density distribution of FG original particle in the same quantum state is

老 = N m_{F} 肉^{*}肉 (11.1-3)

Where N is the number of particles, and 肉 is the wave function of a single particle this satisfies the normalization condition

(11.1-4)

I am now discussing the spherically symmetry FG 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 朴(u) ↙ 0, for u↙﹢. Since I am only
discussing ground state of system, there are no nodes in the wave function 朴(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_{FG}^{5} /
ħ^{2} Therefore
the total energy of FG star is

(11.1-12)

From (11.1-12) the upper limit of the total energy of FG 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 FG theory I can solve the cosmic dark matter problem.

**
11.2 The analytical study for the mass of FG star.**

In Newtonian approximation I will analytically study the ground state energy of the N FG original particle system further. The Newtonian potential between two of FG 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 FG 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 FG particles in coordinate of the center of mass is

(11.2-7)

Where

(11.2-8)

The definition of the conjugate momentum of is = , 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 FG star consists of a lot of FG original particles, I now find the difference between the upper and lower limits is only 15%. I suggest that the average mass of FG star is

= N
m_{F }每
0.058 N^{3} m_{F}^{5} /
m_{pl}^{4
}(11.2-14)

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

=(N每4.3x10^{-155} N^{3})m_{F}
(11.2-15)

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

**
figure 4**

This is I believe a perfect universe!