Scaling properties of the gravitational clustering in the nonlinear regime

The growth of density perturbations in an expanding universe in the non-linear regime is investigated. The underlying equations of motion are cast in a suggestive form, and motivate a conjecture that the scaled pair velocity, $h(a,x)\equiv -[v/(\dot{a}x)]$ depends on the expansion factor $a$ and comoving coordinate $x$ only through the density contrast $\sigma(a,x)$. This leads to the result that the true, non-linear, density contrast $^{1/2}=\sigma(a,x)$ is a universal function of the density contrast $\sigma_L(a,l)$, computed in the linear theory and evaluated at a scale $l$ where $l=x(1+\sigma^2)^{1/3}$. This universality is supported by existing numerical simulations with scale-invariant initial conditions having different power laws. We discuss a physically motivated ansatz $h(a,x)=h[\sigma^2(a,x)]$ and use it to compute the non-linear density contrast at any given scale analytically. This provides a promising method for analysing the non-linear evolution of density perturbations in the universe and for interpreting numerical simulations.Comment: 14 pages 2 figures available on request, TeX, IUCAA-12/9

Constrained Violent Relaxation to a Spherical Halo

Violent relaxation during the collapse of a galaxy halo is known to be incomplete in realistic cases such as cosmological infall or mergers. We adopt a physical picture of strong but short lived interactions between potential fluctuations and particle orbits, using the broad framework outlined by Tremaine (1987) for incorporating incompleteness of the relaxation. We are guided by results from plasma physics, viz. the quasilinear theory of Landau damping, but allow for significant differences in our case. Crucially, wave particle scattering does not drive the system to an equilibrium distribution function of the exponential type, even in regions of phase space allowed by the constraints. The physical process is mixing without friction in ``action'' space, for which the simplest possible model is a constant phase space density modulated by the constraints. Our distribution function does not use the exponential functions of the energy prevalent in other work, which we regard as inappropriate to collisionless systems. The halo of the self-consistent, parameter-free solutions show an r^(-4) behavior in density at large r, an r^(1/4) surface brightness profile in the region 0.1-8 r_e, and a radially anisotropic velocity dispersion profile outside an isotropic core. The energy distribution seen in simulations N(E) singles out the pericenter cutoff model as the most realistic among the variants we have explored.Comment: 25 pages, 12 figures; scheduled to appear in ApJ, vol 524, #2 (oct. 99). Figures in gif format. Preprints are also available on request from [email protected]

A critique of scaling behaviour in non-linear structure formation scenarios

Moments of the BBGKY equations for spatial correlation functions of cosmological density perturbations are used to obtain a differential equation for the evolution of the dimensionless function, $h = - ({v/{\dot{a}x}})$, where $v$ is the mean relative pair velocity. The BBGKY equations are closed using a hierarchical scaling ansatz for the 3-point correlation function. Scale-invariant solutions derived earlier by Davis and Peebles are then used in the non-linear regime, along with the generalised stable clustering hypothesis ($h \to$ const.), to obtain an expression for the asymptotic value of $h$, in terms of the power law index of clustering, $\gamma$,and the tangential and radial velocity dispersions. The Davis-Peebles solution is found to require that tangential dispersions are larger than radial ones, in the non-linear regime; this can be understood on physical grounds. Finally, stability analysis of the solution demonstrates that the allowed asymptotic values of $h$, consistent with the stable clustering hypothesis, lie in the range $0 \leq h \leq 1/2$. Thus, if the Davis-Peebles scale-invariant solution (and the hierarchical model for the 3-pt function) is correct, the standard stable clustering picture ($h \to 1$ as $\bar\xi \to \infty$) is not allowed in the non-linear regime of structure formation.Comment: 14 pages, no figures. Scheduled to appear in ApJ, Mar 1 issue. Final version, contains added discussion to match the accepted versio

The gravitational dynamics of galaxies

The broad area of galactic dynamics is presented for a physics audience, with the requisite astronomy background in outline, and focusing on gravitational effects. The basic underlying model is a large number of particles (which could be stars or dark matter) moving in their self-consistent gravitational potential. The effects of two-particle correlations/scattering, although weak, can be cumulative and hence important for a class of systems such as star clusters which are hence termed collisional. On the larger scale of galaxies, we have collisionless behaviour which is different and in some ways richer. The basic ideas and applications in both these regimes are described, and some issues highlighted in conclusion

The importance of being ignorant using entropy for interpretation and inference

In many real life situations, we have to draw conclusions from data which are not complete and have been affected by measurement errors. Such problems have been addressed from the time of Bayes and Laplace (late 1700's) using concepts which parallel Boltzmann's use of entropy in thermal physics. The idea is to assign probabilities to different possible conclusions from a given set of data. A critical - and sometimes controversial - input is a 'prior probability', which represents our knowledge before any data are given or taken! This body of ideas is introduced in this article with simple examples

Thermal ionistion and the Saha equation!

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