Do attraction between two galaxies count for dark matter? We know that the uniform rotation curve of individual spiral galaxies does require presence of some form of dark entity (matter). Does the attraction between two (or more) spiral galaxies also require same dark entity? Have such calculations been done? If there is dark matter, it should be required in same amount by both phenomena - 1) uniform rotation curve of individual galaxies, and 2) mutual interaction of two (or more) such galaxies.
This is not necessarily about clusters because the dark matter cloud can span galaxies in a cluster. I am trying to see whether local (rotation curve) effects of dark matter match the inter galactic effects where the dark clouds of two galaxies are disjoint. 
In the same manner as above two clusters can be considered for comparing local and inter cluster effects as long as the dark clouds of two clusters are disjoint.
 A: Observations imply that there is a considerable amount of dark matter 
or unseen matter on astronomical scales ranging from clusters of galaxies to 
individual galaxies themselves. For instance the masses of galaxies in a 
cluster as estimated from the virial theorem to account for their observed 
velocity dispersion (v^2
) giving the dynamical mass, Md ~ (v^2
 )R/G turns out 
to be at least a factor of ten higher than what one would except from the 
luminosity(Faber and Gallagher 1979). Even groups of galaxies seem to 
have inadequate luminous mass by a similar factor. To account for their 
dynamical dispersion, the proportion of unseen non luminous mass should 
increase with increasing scales. Again studies of the dynamics and structure 
of large spiral galaxies suggest that a universal feature of all the rotation 
curves is that at large galacto centric distances they are either flat or slowly 
rising, there being no large spiral galaxy whose rotation curve falls (Rubin et 
al. 1982). The rotational velocities for a point mass (keplerian) are given by 
v^2
 being proportional to GMr
 /r, Mr
 being the mass contained within a radius 
r. These observations of flat , v = constant, rotation curves imply Mr
increasing linearly with r indicating the presence of much unseen dark 
matter up to large distances from the centre of the spiral galaxies. 
The progressive increase in the dynamical mass with radius is a characteristic 
feature of all these galaxies, i.e. individual galaxies are surrounded by 
massive dark halos, which have as much as ten times the mass of the visible 
matter. It is now known that x-ray emitting hot gas (e.g. from clusters and 
galactic coronae) would account for only a small fraction of the required 
missing mass. Other propositions for DM ranging from black holes to very 
low mass stars have met with various difficulties, So finally the presence of 
dark matter in halos and beyond halos (in clusters) imply a large ratio of 
dynamical mass to luminous mass. This non baryonic mass is present for 
large distances from the galaxy. The orbital velocity remains constant at 
larger distance from the galactic core. 
The galaxy as has long been suspected has at its centre a massive 
black hole, with estimated mass of around 3million suns. If the galaxy was 
held together by the attraction of that mass, and the motion around it was 
circular.
Thus the effective value of M increases with distance, and the 
rotation velocity v in the denser parts of the galaxy may falls off less steeply 
than like 1/r. However, beyond densest part, v should drop off, and this fall 
off should be close to 1/r. But in practice the velocity of the objects beyond 
the halos become constant. The matter implies still present but its not 
radiating. One can give some models.
Conclusion: Several observations imply that the rotational curves of the galaxies 
for long distances are flat. It indicates the presence of DM or unseen matter.Considering suitable models for these DM halos, one can plot the flat 
rotation curves. If we consider the effects of cosmological constant on large 
scales, the flat curves take a dip at very large distances. From this we can get 
the formula for the distance beyond which the dark energy dominates. By 
applying different values for ω in the generalized metric, we conclude that 
the ω value should be always< -1/3.
Sorry for such long answer, I haven't mentioned the mathematical formulae as I thought this is sufficient.
