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.