The Big Mistake in the Derivation of Friedmann Equation

The derivation of the Freedman equation requires the mathematical result known as the shell theorem which can easily be proved. The theorem states that in a spherical symmetric distribution of matter, a particle at a distance (r = R) from the centre (r=0) feels no force at all from the material in the outside region (r<R).Then the calculation is carried out to lead to Friedmann equation:

The full derivation can be found in any cosmology textbook and is usually followed by the discussion of the three possible geometries for the universe according to the value of (ĸ) being zero, positive or negative.

Unfortunately, this equation has been derived by erroneous method making use of the incorrect assumption that the shell theorem can be applied in the case of non-Euclidean geometry. Let us see the case of spherical geometry. Such geometry might well apply to our universe. It can easily be seen that in an isotropic universe with spherical geometry no gravitational force exists at all because every force in any direction is matched by a force of the same magnitude in the opposite direction because of total symmetry of space in this case. This brings out the interesting fact that the large scale geometry of an isotropic universe with spherical geometry is independent of its average density. This fact seems to contradict general theory of relativity at first glance but, interestingly, this contradiction will disappear  if we agree to define the cosmological constant in such a way that the average density of the universe is cancelled out from the  global application field equation. In fact, we are forced to do so because the application of Newtonian gravity in the large scale (not like the case of local application in which GR supersedes Newtonian gravity) should give the same results as that given by general theory of relativity. The independence of the large scale geometry of the universe from its average distance can be attained by assuming that the average density of the universe is a part of the cosmological constant and therefore it is cancelled out because it appears twice in the field equation  with opposite sign, first  as a part of the cosmological constant and then as the material part of the equation ( full details of this definition of the cosmological constant and  a proposed cosmological model associated  with it is found in (Alternative Cosmological Model without Ad Hoc Elements and without Modifications in GR or QM : http://vixra.org/abs/1706.0027).

Claims of alternative methods of derivation of the Friedmann equation such as Machian derivation (see for example: Machian derivation of the Friedmann equation by Herman Telkamp) also fail in front of the total symmetry of the isotropic spherical universe.