Some solutions of the Maxwell equations with Dirac particles for the source in FRW spacetime are discussed. The Green's function of the equation for the radial component of the Maxwell fields, Frη and F θφ is solved. Green's function is found to reduce to that of Minkowskian spacetime in the appropriate limit. Also, the Lienard-Wiechert type solution is derived. Also, the solutions with the Dirac particle current is also presented. It is found that the Frηis composed of even angular momentum states while the odd states constitue F θφ .

Abstract:We have summarized the neutrino masses detected/predicted from different sources so far and calculated the Jeans masses of the neutrino structures for these values of neutrino masses. The size of the pure neutrino structures has come to be bigger than the superclusters and great attractor formed around 'recombination epoch'.

Sizes and masses of the Neutrino Structures calculated with different combination of parameters involved has shown interesting variation with remarkable shifting of position of peak value. The peak position of (ξRH0/Resc) shifts from ξ = x/x0 = 0.847 to 1.632, where x = m/T. Accordingly, the peak of the mass of neutrino structures also shifts.Key words: Neutrino; Large scale structures; Free streaming; Jeans mass; Momentum distributionDOI: 10.3126/sw.v5i5.2646 Scientific World, Vol. 5, No. 5, July 2007 1-5

Abstract:In the course of studying the role of massive neutrinos and other particles in the large scale structures formation in the universe, Virial moments of the neutrino structures have been calculated. The Jeans masses calculated this way have been compared with the ones calculated earlier in our previous paper. VIRIAL THEOREM AND MOMENTS:In a system of N particles, gravitational forces tend to pull the system together and the stellar velocities tend to make it fly apart. It is possible to relate kinetic and potential energy of a system through the change of its moment of inertia. In a steady-state system, these tendencies are balanced, which is expressed quantitatively through the Virial Theorem. A system that is not in balance will tend to move towards its virialized state. The Scalar Virial Theorem tells us that the average kinetic and potential energy must be in balance. The tensor Virial Theorem tells us that the kinetic and potential energy must be in balance in each separate direction. The scalar Virial Theorem is useful for estimating global average properties, such as total mass, escape velocity and relaxation time, while the tensor Virial Theorem is useful for relating shapes of systems to their kinematics, e.g. the flatness of elliptical galaxies to their rotational speed.The Virial Equations of the various orders are, in fact, no more than the moments of the relevant hydrodynamical equations. The scalar Virial Equation for a system is given by VIRIAL MOMENTS OF JEANS MASS:As stated earlier, in the virial method, we take the moments of the equation of motion. These equations obviously involve the moments of the distribution of density, pressure, velocity, gravitational potential, etc. Here we are taking the spatial moments of various orders of Jeans mass given by eq. CONCLUSIONS:From the above analysis, it is seen that the large-scale structures of neutrinos of different mass and random velocity distribution can form at different neutrino temperatures, corresponding to different time. The earliest peak that occurs at x = 1.9 corresponds to the time when the neutrino temperature was 1220 K for the 0.2 eV neutrino. Similarly, the latest peak occurs at x = 9.58 corresponding to a temperature of 242 K. In between these two values, it is found that the Jeans mass peaks at a number of different x. Thus it can be interpreted to mean that a distribution of neutrino structures of different masses and of different ages should be in existence. Typical masses of these structures range from 6 x 10 19 to 4.5 x 10 20 solar mass . The distribution in size and age of such neutrino structures, and the effect of cold dark matter will be studied in future work. REFERENCES:

Behaviour of the Dirac particle in Coulomb like field in FLRW space is investigated. Firstly, the Maxwell equations, in terms of the vector potentials are solved to identify the Lorentz and Coulomb like gauges.  The radial Coulomb like potential is solved in terms of Legendre functions. Then the Dirac equation is generalized to include this potential and the angular part is separated and solved. The radial and temporal parts of the mass less case is also separated and solved. But the massive case remains coupled. This is still reduced to the case where the Dirac particle can be represented as being in a combined gravitational and electric potential. This effective potential is found to develop an attractive well, which may require a revisit to the recombination era.