Dark Matter is an unnecessary ad hoc fix to fill in the blanks in the Friedmann model under the FLRW metric. Galactic supermassive black-holes exist as true physical singularities according to the Kretschmann invariant and Schwartzchild's analysis. Therefore, as point masses, they must possess a hyperbolic (1/kr) gravitational field, NOT a field that falls off as 1/r2. Now, k = constant = 1m, S.I., for dimensional integrity. It is not true that GR cannot tolerate hyperbolic spacetime geometries. "The universe is hyperbolic." said Albert Einstein in his classic paper of 1915.
An hyperbolic field will give constant orbital acceleration to orbiting bodies as far from the center of a black-hole as we might like to measure. This means that bodies near the periphery of a galaxy should seem to move at constant velocity because rotational acceleration does not drop to near zero there as with a 1/r2 inverse square law. This constant velocity distribution effect has actually been measured and has given rise to the notion of Dark Matter.
Gravitation does not fall nearest to zero between galaxies in a cluster either. So they too can bend light and affect redshifts in ways that mimic Dark Matter. The rotation of galaxies in clusters is also influenced by the black-holes that they contain with their 1/kr gravitational potential profiles. The not quite counterbalanced redshift effects in the Sunyaev-Zeldovich phenomenon are influence by the hyperbolic galactic and galactic cluster gravitational fields that exist as light falls out of such clusters and super-clusters into a large void and as it climbs out of it again after the universe has expanded by another billion light years or more.
Scientists are mapping, not Dark Matter, but the huge extent of the network of hyperbolic galactic and super-galactic gravitational fields that behave like Dark Matter because of the mathematical properties of the hyperbolic gravitational field are similar to That expected for Dark Matter.
Primordial massive and supermassive black-holes with their 1/kr galactic gravitational fields can also mimic the “halos” of dark Matter that are postulated to have existed just after the big bang and before the emission of the cosmic microwave background. There is nothing that Dark Matter explains that cannot be accounted for just as well or better by the hyperbolic black hole gravitational field.
The conditions for validity of Birkhoff’s Theorem are not met for real black-holes. Therefore, Birkhoff’s Theorem does not apply. It sometimes may be used as a first approximation, but it cannot be depended upon as a rigid rule for precise calculations. “The physics near the extreme curvature of a black-hole singularity is not well defined”. This covers Birkhoff’s too.
By the way, any entity that possesses mass by virtue of its motion will be influenced by the gravitational fields that it encounters. It is not so much that a gas mimicking Dark Matter may be very much colder than other gases that such an entity might encounter, but whether such a gas may be much denser. But absolute zero is absolute. Only ground state vibration modes are allowed for gases at absolute zero, translational motion does not exist at K = 0 because it implies a temperature, T > 0 K. So, gases must not exist either. They must be solid crystals. Also, such ground state vibrational modes are only for multi-atom molecules. Intergalactic gas is almost non-existent, is not denser and is not a factor, so it cannot mimic Dark Matter.
