Abstract
In this article, we propose a new model of dark matter. According to this new model, dark matter is a substance, that is a new physical element not constituted of classical particles, called dark substance and filling the Universe. Assuming some very simple physical properties to this dark substance, we theoretically justify the flat rotation curve of galaxies and the baryonic Tully-Fisher’s law. We then study according to our new theory of dark matter  the different possible distributions of dark matter in galaxies and in galaxy clusters, and the velocities of galaxies in galaxy clusters.
 
 Then using the new model of dark matter we are naturally led to propose a new geometrical model of Universe, finite, that is different from all geometrical models proposed by the Standard Cosmological Model (SCM). Despite that our Theory of dark matter is compatible with the SCM, we then expose a new Cosmological model based on this new geometrical form of the Universe and on the interpretation of the CMB Rest Frame (CRF), that has not physical interpretation on the SCM and that we will call local Cosmological frame. We then propose 2 possible mathematical models of expansion inside the new Cosmological model. The 1st mathematical model is based on General Relativity as the SCM and gives the same theoretical predictions of distances and of the Hubble’s constant as the SCM. The 2nd mathematical model of expansion of the Universe is mathematically much simpler than the mathematical model of expansion used in the SCM, but we will see that its theoretical predictions are in agreement with astronomical observations. Moreover, this 2nd mathematical model of expansion does not need to introduce the existence of a dark energy contrary to the mathematical model of expansion of the SCM. To end we study the evolution of the temperature of dark substance in the Universe and we make appear the existence of a dark energy, due to our model of dark matter.
Highlights
In the first part of the article, we expose a Theory of dark matter
We see that the new theory permits theoretical predictions of the dark radius of galaxies, in agreement with observations, and of the mean density of dark matter in the Universe, that is the origin of some anisotropies of the CMB
If we assume in our new Cosmological model that for this value of z, the temperature of the intergalactic dark substance was equal to the temperature of the CMB, we obtain that presently, the temperature of the intergalactic dark substance is 1500 times lower than the temperature of the CMB, which is an acceptable value, justifying our approximation in Section 2.3 expressing that the temperature of the intergalactic dark substance can be neglected in comparison with the temperature of spherical concentrations of dark substance corresponding to galaxies with flat rotation curve
Summary
In the preceding Part 2. we exposed a theory interpreting the whole of astronomical observations linked to dark matter. According to the 1st mathematical model of expansion, 1+z is obtained as it is obtained in the SCM, with a flat Universe: We apply locally the equations of General Relativity, assuming the same values as in the SCM for the densities of dark substance, baryonic matter and dark energy and assuming that those densities and that the Universe is flat. In this 1st mathematical model, the factor of expansion 1+z can be mathematically expressed the same way as in the SCM for a flat Universe. O would be the centre of the spherical Universe
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