Distribution of dark matter in the galaxy a thesis submitted for the degree of Doctor of Philosophy in the faculty of science, Indian Institute of Science, Bangalore C. Ratnam [Ph.D Thesis]

Thesis Supervisor R. Cowsik

Doctor of Philosophy Indian Institute of Science, Bangalore 1998

Today, there is substantial observational evidence for the presence of dark matter (DM) on all scales, and in particular the galactic scales. The rotation curves of galaxies are flat at large distances beyond the optical radii indicating the presence of large amounts of "unseen matter". While the number of galaxies with such rotation curves are large, substantiating the above conclusion, yet there has been only a limited understanding of the nature and other properties of dark matter (DM). Since DM is seen to dominate the gravitational dynamics of galactic systems understanding its properties will be a crucial input to the study of the formation of structure in the •universe. The interest in the phase space structure of dark matter in general and the velocity dispersion in particular stems from the need to interpret data obtained from the laboratory experiments to detect dark matter particles. When the density distribution of all forms of matter in the Galaxy is modeled as that of an isothermal sphere the velocity dispersion is simply related to the asymptotic value of the rotational speed as (V2}1/2 ~ IfVdr = 00). Using the value of rotational speed at the Solar neighborhood of 220 kms-1yields (1)2)1/2 ~ 270 kms-1. It should be noted that in this commonly adopted procedure, the gravitational potential of the entire Galaxy without including the individual contribution of components like the spheroid and the disc etc is described as that of a single component isothermal sphere. In this thesis, we would like to investigate the dynamical effect of the visible matter on the distribution of dark matter and how it affects the estimate of the velocity dispersion.
In Chapter 1 an introduction to dark matter is given with particular stress on its significance in the Galaxy. The evidence for the presence of dark matter at different length scales is reviewed. A brief overview of different candidates that stake their claim do be the particles of dark matter is also presented.
In Chapter 2 we present the self consistent model for the halo of dark matter. In order to do this we start with an assumed form for the phase-space distribution for the dark matter particles and calculate how they will be distributed when subjected to their own gravity and that of the visible matter of the Galaxy. We have chosen two distribution functions, the Maxwellian and the King's, both consistent with stationarity to study the halo of the dark matter. The Maxwellian is essentially an isothermal distribution, with two free parameters the central density and the velocity dispersion. In the King's model the isothermal distribution is truncated at a finite value of the total energy and this translates to the so called tidal radius cut-off where the value of the density goes to zero. The emphasis ofthis chapter is on the fact that at small distances up to about 10 kpc the visible matter has also a very important role to play. The value of the velocity dispersion in the Solar neighborhood depends on both the visible and dark matter, and appears as one of free parameters of the problem. The value of the dark-matter central density is also independently constrained by its density 0.3 Ge Vcm- 3 in the local Solar neighborhood. We adopt well studied modf;lls to describe the density distribution of visible matter and proceed to solve the Poisson's equation for the dark matter, which because of the DF adopted involves both the potentials of visible matter and dark matter in a non-linear way. The self consistent dark matter potential calculated thus, yields in turn a rotation curve which is compared with the observed rotation curve of the Galaxy. The best fit between the theoretical and observed rotation curves yields a value for the velocity dispersion to be around 600 kms-1which is larger than the previously obtained value of 270kms-I.
In Chapter 3 the rotation curve of the Galaxy is well measured up to a galactocentric distance of ~ 8kpc with reasonable accuracy beyond which it becomes progressively uncertain and there is hardly any data beyond 20 kpc. In a recent analysis by Lynden-Bell and Lynden-Bell it was suggested that the motion of dwarf spheroidals may be used to derive the rotation curves at large distances. In this chapter we present an analysis following their suggestion. We show that the velocity distribution of dwarf spheroidals is skewed with (vi) = JL(v;}, with JL increasing beyond the isotropic value of JL = 2 to values of 5 or more because tidal disruption of dwarf spheroidals by the Galaxy has effectively removed orbits with high Vr• The data on radial velocity of the dwarf spheroidals then is used to determine the behavior of rotation curve at distances of ~ 100 kpc.
In Chapter 4 we explore the consequence of the large velocity dispersion of 600 kms-Iderived in this thesis. The detection of any non-baryonic matter like the WIMP's involves measuring the rate of energy deposited in a laboratory detector. However to model such a rate involves specific assumptions for the number density of WIMP's, the cross-section of interaction and the velocity dispersion of the dark matter particles. In other words the rate can be written as R = (p/mo}a(v2)1/2, where R is the rate; p, the mass density, a the cross-section of interaction, mo the DM mass and (v2 ) 1/2 the velocity dispersion of the dark matter particles. The mass density around the Solar neighborhood is taken to be 0.3 Ge V cm -3. Thus for any assumed mass of the WIMP particle (mo) the bound on the cross-section is obtained by fitting the theoretical rate to the observed event rate for a given value of velocity dispersion. We obtain exclusion plots in the ml')-a space, for the new velocity dispersion of 600 kms-l. It is cleat from this exercise that bounds on the scattering cross section is improved particularly for mo ~ 50 GeV, because of the larger dispersion we have obtained.
In Chapter 5 we apply the model for DM-halo developed in this thesis to 12 external spiral galaxies. We choose galaxies with well studied luminosity profiles and rotation curves. Luminosity profiles are used to estimate the density distribution of the visible matter and hence the potential due to it. In terms of the model we have developed, there are two parameters defining the dark halo, that are unknown i.e POM and (V2)OM' The observed rotation curves of these galaxies are used to place constraints on the velocity dispersions and density of DM in these galaxies and to search for possible correlations between the visible and dark matter contents of the galaxies. In Chapter 6 we summarize our work and discuss the results and suggest scope for further work.