Authors: Manuel Abarca Hernandez
The main target this paper is to check a theory about non baryonic dark matter nature, which was published by the author in  Abarca,M.2014.  Abarca,M.2015.  Abarca,M.2016. [ 11] Abarca,M.2016 and others papers. It was postulated that non baryonic dark matter density depend on E, gravitational field, through a power of E as a Universal law. In order to check this theory in this paper will be studied DM density in halo region of MW and M31. Throughout the paper, DM refers to Non Baryonic DM. Briefly will be described paper procedure. In third chapter are introduced rotation curves of MW,  Huang,Y.2016, and M31,  Sofue, Y.2015. It is fitted a power regression for velocity depending on radius in halo of both galaxies. Formula is v = a• R^b. In fourth chapter it is deduced mathematically a DM density profile for MW & M31 in halo region. This new profile is called Direct DM density because it is got directly from power regression velocity depending on radius. In fifth chapter is found a new DM density as power of E, which is mathematically equivalent to Direct DM density. Its formula is DM density = A• E^B. Where A & B are coefficients which depend on a & b, coefficients of velocity power regression in rotation curve. Also are calculated A&B for MW and M31. Finally it is explained that hypothesis of DM depending on gravitational field as Universal law allows enlarge dominion DM as power E for radius inside disc and radius bigger than dominion measures. In sixth chapter, according theory of DM generated by gravitational field, is defined galactic halo as region where own gravitational field dominates over neighbour gravitational field. Through this criterion is found halo MW = 310 kpc and halo M31 = 460 kpc. The seventh is a crucial chapter, because it is successfully compared DM as power E in MW & M31. Both formulas are compared and relative differences throughout dominion of E are below 14 %. Agreement of both functions is the main evidence about DM power E as Universal law. In eighth chapter is compared DM density as power E with NFW profile given by author of rotation curve in his paper.  Huang,Y.2016. NFW density is bigger than DM power E throughout all dominion, and relative differences oscillate between 34% and 13%. Such remarkable differences will be properly justified. In ninth chapter is calculated total mass through NFW profile and dynamical method. Mass through NFW profile is bigger than through dynamical method. Difference might be explained by the same reason given in previous chapter. In tenth chapter is calculated Local DM density through DM power E profile, which gives 0,16 GeV/cm^3 and is compared with same magnitude given by  Huang,Y.2016, which is 0,32 GeV/cm^3. It is defended that there is not contradiction between both data because the first profile gives Non Baryonic DM only, whereas experimental data measure Baryonic and non Baryonic DM. In eleventh chapter is developed a mathematical procedure to change physics magnitudes as power into dimensionless power variables. This change has been possible thanks a new parameter Rs called scale radius. So it is defined x = radius/ Rs being x dimensionless. This way velocity v = c• x^d. As a consequence Direct DM is changed into a new dimensionless power and similarly happen to DM as power of E, where E is changed by a new dimensionless variable. In twelfth chapter is made use of new formulas to Milky Way data. New results have being checked and it has been shown that they are identical to got in previous chapters. It is concluded three main ideas which will be properly justified in following pages. The first one is that there are strong evidences about that fraction of Baryonic DM vs Non Baryonic DM inside bulge and galactic disc is bigger than the same fraction inside halo. The second one is DM generated by gravitational field theory leads rightly a new definition for halo radius. According this theory are calculated halo radius MW equal to 310 kpc and halo radius M31 equal to 460 kpc. The third one is that there are strong evidences that non baryonic DM density is generated by gravitational field as a Universal law. Main reason to support this hypothesis is got in chapter seventh.
Comments: 27 Pages. V2-Two new chapters with DM density formulas through dimensionless variables power.
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