Authors: Manuel Abarca Hernandez
The hypothesis of this work is that D.M is generated by the own gravitational field, according an Universal mechanism based on a quantum gravitation phenomenon. In order to study DM as a Universal law is studied rotation curve of M31. The theory developed for galaxies is extended successfully to Coma cluster . In this paper is published by first time the called The equivalence theorem, which is a crucial tool to extend the theory to cluster scale. Another crucial finding made in this paper is that a characteristic parameter of the theory has been calculated by data coming from Coma cluster and by data from M31. Both results match with a high precision. Briefly will be explained method followed to develop this paper. Rotation curve data come from  Sofue,Y.2015. Thanks this remarkable rotation curve, the regression curve of velocity depending on radius has a correlation coefficient bigger than 0.96 and data range from 40 kpc up to 300 kpc. In fourth chapter it is got the function of DM density depending on radius, called direct DM density. In fifth chapter it is demonstrated that function direct DM density is mathematically equivalent to the function DM density depending on E. Namely a power of E whose exponent is B= 1.6682. In sixth chapter it is got that for radius bigger than 40 kpc the ratio baryonic density versus DM density is under 1% . So it is reasonable to consider negligible baryonic matter density in order to simplify calculus. In seventh chapter it is got a Bernoulli differential equation for field and is solved. In eighth chapter it is made dimensional analysis for magnitudes Density, field E and universal constants G, h and c. It is demonstrated that it is needed a formula with two Pi monomials. It is found that B = 5/3 is the value coherent with Buckingham theorem and differs only two thousandth regarding B=1.6682 which was got by regression analysis.In the chapter 9, considering B= 5/3 are recalculated the others parameters a&b and A and formulas of theory are rewriting. In the tenth chapter are introduced the called reduced formulas for field, density and total mass. The variable of these functions is dimensionless, this way the formulas gain simplicity. In the eleventh chapter, is calculated masses in M31 at different radius by the reduced formula of mass. In the chapter 12 is introduced The equivalence theorem which is crucial in order to extend the dark gravitation theory to cluster scale. As mass and density are additive magnitudes it is found a new parameter D associated to the cluster, considering as start point the parameter D associated to one single galaxy. In the thirteenth chapter are developed two methods to calculate parameter D in clusters and it is demonstrated that both are mathematically equivalents. Also it is calculated three different parameter D for Coma cluster using recent measures of Coma cluster mass. In the fourteenth chapter is used the formula of total mass in Coma cluster by the three different parameter D calculated in previous chapter. Also it is discussed its result and compared with data measured of Coma cluster mass. In the fifteenth chapter is introduced the average density formula, it is applied to Coma cluster and it is shown that at virial radius Coma cluster has an average density equal to 100 times the critical density of universe, which match very well with measures. Also it is calculated the radius of the sphere where the average density is equal to the Universal average D.M. density, In the sixteenth chapter it is proposed a method to calculate parameters C and D, rightly from mass formula. This method is totally independent to the equivalence theorem, so it is highly interesting in order to compare both methods.
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