Chapter 11 



      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

   image081.gif (381 bytes)                          (11-2)

  image082.gif (1219 bytes)               (11-3)

      Where the R (t) is the cosmic scale factor, k = -1, 0 1 corresponding to open, flat and closed universe respectively. Eliminating image083.gif (862 bytes) from (11-2) and (11-3) we can get a differential equation of first order

   image084.gif (1111 bytes)                                   (11-4)

      The definition of the Hubbell parameter, which is a measure of the universe, is

    image085.gif (969 bytes)                                                   (11-5)      

      Using this expression (11-4) can be rewritten as





      The present value of the cosmic energy density and pressure can be obtained from (11-2) and (11-3)



      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 image0810.gif (953 bytes) 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



      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


      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


      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





        r = -h2 ·  · G-1 · N-1 · u                            (11.1-8)

      image0818.gif (1456 bytes)           (11.1-9)

       image0819.gif (1171 bytes)                          (11.1-10)

      image0820.gif (1174 bytes)                          (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

image0821.gif (1636 bytes) (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



      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

   image0824.gif (1185 bytes)                                     (11.2-1)

      The total Hamiltonian of the system is



     image0826.gif (1581 bytes)           (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

      image0828.gif (1254 bytes)                     (11.2-4)

      Thus I have got the lower limit of ground state energy of N WG original particles system of self- drawing:

     image0829.gif (650 bytes)         (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

     image0830.gif (871 bytes)    (11.2-6)

      The Hamiltonian for relative motion of N WG particles in coordinate of the center of mass is



    image0832.gif (627 bytes)                           (11.2-8)


      The definition of the conjugate momentum of ( image0833.gif (223 bytes)) is image0834.gif (199 bytes)= (image0835.gif (239 bytes))/2, which satisfies the canonical transformation. The (11.2-8) can be rewritten as

      image0836.gif (546 bytes)                                 (11.2-9)  

      The lower limit of the expected value of hij is

     image0837.gif (464 bytes)                                     (11.2-10)

      Therefore the lower limit of ground state is

      image0836.gif (546 bytes)              (11.2-11)

      On the other hand, if I was using trial wave function

     image0839.gif (567 bytes)                    (11.2-12)

and standard variation approach I would get the upper limit of ground state energy

      image0840.gif (578 bytes)               (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.

                                                     figure 4

       image0843.gif (3171 bytes)                                           


















Fig. 4





This is I believe a perfect universe!