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 The fundamental gravitino and its physical dark world

Chapter 11 

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                           (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 << 老0

      Therefore, from (11-9) we have

    k / R02=(2q0 每1) H02                                       (11-14)

      Considering (11-8), we get the present value of the ratio of 老0 and pc

        老/ 老 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 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 mF 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﹞    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 G2 N3 mFG5 / ħ2 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 mW is very small, putting (11.1-1) into (11.1-13) we get

   MMAX  = 2.1 ℅ 1036 g  >  10.5 ℅ 10          (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 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 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 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 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 mF  每 0.058 N3 mF5  / mpl4          (11.2-14)

      Put mF=3.6 x 10-35 kg into above equation we obtain

       =(N每4.3x10-155 N3)mF                       (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!