The speed at which stars in the spiral arms of galaxies revolve around the galactic center remains more or less constant with increasing radius because the scale factor within the galaxy is inversely proportional to the radius. This is because black holes at the centers of galaxies produce new space, which is gradually dispersed into surrounding space.
Phenomena explained: It is observed that the rotational speeds with which stars in spiral arms of galaxies revolve around the galactic center of mass does not continue to attenuate after a given distance from the center, but remains almost constant. 
We posit that local expansion of space results in more space, without directly affecting the surrounding space. For example, if space expands within a closed sphere, but not outside of it, distances, areas and volumes within the sphere will expand, whilst the surface area of the sphere will remain unchanged. The space within the sphere is ‘richer’ than the surrounding space. We define space to be ‘rich’ to the degree in which the distances within the volume, as measured by the passage of light, are larger than our trigonometric calculations of those distances. In other words, the scale factor (a) in rich space is higher than that of the observer. The scale factor a is a numerical value, without units. Its value at any point in spacetime is the limit as line length tends to 0 of the ratio between the actual length of a line through that point and perpendicular to our line of sight and the length we would assign to the line on the basis of trigonometric calculations. By definition, a = 1 on Earth, now.
The behavior of gravity in space in which the scale factor is variable can best be understood by applying the Gaussian formulation of gravity. The gravitational flux exerted by a mass on any closed surface containing that mass is, as Gauss observed, a constant depending only on the mass, regardless of distance. In the Gaussian formulation of Newtonian gravity, the gravitational flux due to a massive object of mass M on any closed surface surrounding that object is a constant, given by the formula

(1) 
If we take an isodistant surface at distance r from a mass M, g is uniform, so we have

(2) 
where a is the scale factor at the surface, relative to the observer. If the scale factor is 1, this formula reduces to the conventional theory. But if it is smaller than 1,  g will be larger than conventional theory would predict. Therefore, if the scale factor in the spiral arms of a galaxy is inversely proportional to r, g will be constant with respect to r, presuming M does not substantially increase with r. Constant g results in a constant orbital speed.
Let us assume that the center of the galaxy is a fountain of space, generating new space at a constant number of cubic meters per second, and that this new space diffuses into the surrounding volumes in proportion to the surface area over which the diffusion takes place. Note that it is not necessarily the case that all the additional space is passed on immediately. The outgoing flow for a particular surface may be limited by the speed at which the neighboring poorer space can absorb new space, and the incoming flow could in principle exceed that. In such a case, the space in the vicinity of the source becomes more rich. As long as the center of the galaxy is producing new space, a scale factor differential may be maintained. Whenever the production of new space ceases, the scale factor within the galaxy is expected to tend to uniformity.
If the rate at which poorer space can accept new space is the determining factor in the diffusion of space, it follows that an initial state in which the scale factor is more or less directly proportional to the inverse of the distance from the galactic center r will be perpetuated as long as the center of the galaxy produces enough new space. For then the speed of diffusion across the isodistant surface at radius r within the spiral arms is proportional to the effective area of that surface, which is constant.
On the face of it, the most plausible sources of local space production are black holes, given that the centers of galaxies are known to contain supermassive black holes. We find black holes at the scene of the crime, so to speak. Further, black holes are perhaps the only things of which our understanding is insufficiently complete to justify a priori rejection of the assertion that they produce space. Admittedly, this explanation is, in multiple senses, a leap in the dark. But it also presents an opportunity: Perhaps a theory of black holes which includes space production will be able to do without singularities.
Credibility This explanation requires only the mass which we actually observe. It has a simple gap for its hypothesis that black holes produce new space, giving it an Occam Score of 0100. That compares favorably with the explanations proffered by mainstream cosmology, which require dark matter that just happens to be located in the right distribution to produce flat galactic rotation curves but cannot account for the variation they exhibit. These explanations at at best complex gaps, with Occam Scores of 0300, two notches worse than the QO explanation. However, we have not shown that there are no other explanations which are equally or more credible. Therefor the QO explanation is tentatively credible. 