The effect of a perturbing mass on a homogeneous collisionless cloud of dark matter is considered in the linear approximation. It is shown that gravitational potential can have turning points, in sharp contrast with gravitating systems of finite extent. The model offers a reasonable explanation for the observed secondary maxima in the density distribution of rich clusters. The relevance of the model to the flatness of the rotation curves of galaxies is also discussed.

We argue that observations on Milky Way and dwarf spheroidals imply existence of individual haloes around dwarf spheroidals. If neutrinos (or any other ‘hot’ particle) provide the dark matter then we show that: (i) Embedding of visible matter inside large (∼ few Mpc) dark matter islands is observationally untenable. (ii) Dwarf spheroidals possess dark matter haloes of about 10 kpc radius around them, and have an (M/L) ratio of about 10⁴. (iii) The haloes of spiral galaxies (e.g. Milky Way) extend to about 100 kpc in radius. If ‘cold’ dark matter makes up the haloes, then no significant constraints are obtained. We discuss briefly the effect of these constraints on larger scales.

Cosmological scenarios with massive unstable neutrinos are discussed. Restrictions on the mass and the lifetime of the unstable neutrino are derived from (a) age and mass density of the universe and (b) the growth of primordial fluctuations. It will not be possible to accommodate unstable neutrinos with masses above ∼ 1 ke V in standard cosmology unless they have exceedingly small lifetime: Τ <5 × 10⁸ s.

We study the mass-radius relationship for aggregates of galaxies, viz. binaries, small groups and clusters. The data are subjected to a simple best-fit analysis similar to the one carried out earlier for individual field galaxies. The analysis shows that: (i) The data on binary galaxies are consistent with the assumption that binaries are just two galaxies, each with an individual isothermal (M ∫R) dark matter halo, moving under the mutual gravitational attraction, (ii) The data on the groups of galaxies are too scattered to obey a single power-law relation of the formM = kRⁿ with any degree of reliability, (iii) The data on groups and clusters fit better with a law of the formM = AR³ +BR. This form suggests the existence of two components in dark matter—one which is clustered around the galaxies (M ∫R) and another which is distributed smoothly (M ∫R³). The smooth distributions becomes significant only at scales ≥ 1 Mpc and hence does not affect binaries significantly. We briefly discuss the theoretical implications of this analysis