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The Universe owns the electronic radiation of blackbody at temperature 2.725 K, which we call the cosmic electron background. We calculate its radiation spectrum. The energy distribution of number density of electrons in the cosmic electron background becomes zero as energy goes to both zero and infinity. It has one maximum peak near the energy level of 10**(-23) J.

In the framework of the solvable model of cosmology constructed in the Earth-related coordinate system, we derive the modified Hubble law. This law carries the slowly time-varying Hubble parameter. The modified Hubble law eliminates the need for dark energy.

Relying on the obtained results in Rev.[1](physics/0505035), we derive the formula relating the red-shift of light signals coming from distant galaxies to the distance of these galaxies from us and the time of detecting of these light signals. The red-shift coefficient, instead of the Hubble parameter, is introduced. It varies with time and positive at all times. Its nowadays value equals the Hubble parameter. It increases forever as time is running from the past to the future. The derived formula enables us to estimate the nowadays increasing rate of the red-shift coefficient, which is nothing but the nowadays value of the "acceleration of the expansion of the Universe".

In the Earth-related coordinate system, we reconstruct the standard model of cosmology based on the assumption of the cosmological principle and the perfect gas (or fluid). We exactly solve Einstein's field equation involved. The solution consists of three parts respectively on the line element for space-time of the Universe, the value for the cosmological constant and the equation of state for the matter of the Universe.

Some problems relevant to the relativistic generalization of Maxwell's velocity distribution are discussed.

In solar interior, it is the equilibrium velocity distribution of few high-energy protons and nuclei that participates in determining nuclear fusion reaction rates. So, it is inappropriate to use the Maxwellian velocity distribution to calculate the rates of solar nuclear fusion reactions. We have to use the relativistic equilibrium velocity distribution for the purpose. The nuclear fusion reaction rate based on the relativistic equilibrium velocity distribution has a reduction factor with respect to that based on the Maxwellian distribution. The reduction factor depends on the temperature, reduced mass and atomic numbers of the studied nuclear fusion reactions, in other words, it varies with the sort of neutrinos. Substituting the relativistic equilibrium velocity distribution for the Maxwellian distribution is not important for the calculation of solar sound speeds. The relativistic equilibrium velocity distribution, if adopted in standard solar models, will lower solar neutrino fluxes and change solar neutrino energy spectra but maintain solar sound speeds. This velocity distribution is possibly a solution to the solar neutrino problem.

In the framework of special relativity, all particles are point-like or string-like. This nature of particles has caused the divergence difficulties in quantum field, string and superstring theories. In the framework of special relativity, due to the non-uniformity of the m-space and phase space in the usual inertial coordinate system, Boltzmann's hypothesis of the equality of the probability of equal volume element is no longer appropriate. That makes it very difficult to construct Lorentz-invariant statistical mechanics and thermodynamics for many-particle systems. Besides, some observations on special relativity itself and its experimental facts are also reported. The conclusions from these observations are: Special relativity is not an ultimate theory; Some modification is needed; Any modification must not violate the constancy of the light speed and the local Lorentz invariance; It seems that we have to change the assumption on local structures of gravity-free space and time in special relativity.

The generalized Finsler geometry, as well as Finsler geometry, is a generalization of Riemann geometry. The generalized Finsler geometry can be endowed with the Cartan connection. The generalized Finsler geometry and its Cartan connection are, as the necessary mathematical tools, involved in our modification of special relativity, which is made by assuming the generalized Finslerian structures of gravity-free space and time in the usual inertial coordinate system and combining this assumption with two fundamental postulates, (i) the principle of relativity and (ii) the constancy of the speed of light in all inertial frames of reference.

In the special theory of relativity it is assumed that gravity-free space and time possess flat metric structures in the usual inertial coordinate system. But there is an inconsistency in the theory formed by combining this assumption with the two fundamental postulates: (i) the principle of relativity, and (ii) the constancy of the one-way speed of light in all inertial frames of reference. We here propose assuming two generalized Finslerian structures of gravity-free space and time in the usual inertial coordinate system. The inconsistency no longer exists in the theory formed by combining the alternative assumption with the two postulates (i) and (ii), The new theory owns the localized Lorentz transformation between any two usual inertial coordinate systems and the validity of relativistic mechanics in the usual inertial coordinate system.

Keeping the two fundamental postulates of the theory of special relativity (the invariance of the velocity of light and the principle of relativity in inertial frames of reference), and assuming the generalized Finslerian structures of gravity-free space and time in the usual inertial coordinate system without loss of their flatness, we form a new test theory of special relativity. In the new test theory, the localized Lorentz transformation stands between any two usual inertial coordinate systems, and gravity-free space and time match the Fock velocity-space. We produce a new velocity distribution that is different from the Maxwell velocity distribution for free particles. This new distribution approaches zero as velocity goes to the velocity of light. It is claimed that the deviation of the new distribution from its previous formula will provide experimental means of testing the structures in gravity-free space and time of special relativity.

There is an inconsistency in the theory of special relativity. It is assumed in the theory of special relativity that space and time posses the flat metric structures, Sp² = S_rsdx^rdx^s, r,s = 1,2,3, and d1'2 = dt2, in the usual inertial coordinate system (x^r,t), r = 1,2,3. From this assumption one can find an equation, y² = S^rsy^ry^s, where Y = dp/dt is the velocity-length, 'y^r = dx^r/dt is the usual (Newtonian) velocity. The velocity-space embodied in the equation is boundless and subject to the Galilei addition law. On the other hand, the theory of special relativity owns the Lorentz transformation between any two usual inertial coordinate systems. That in fact indicates a finite velocity boundary-the speed of light c and the Einstein law governing velocity additions.

Experiments clearly support the finite velocity boundary c, the Einstein velocity addition law and the Lorentz or the local Lorentz invariance. The Fock velocity-space is characterized by the boundary c and the Einstein addition law. Recognition of the Fock velocity-space leads us, in resolving the inconsistency, to assume the non-flat generalized Finslerian structures of gravity-free space and time in the usual inertial coordinate system without loss of their flatness.