Local Approximations to the Gravitational Collapse of Cold Matter

We investigate three different local approximations for nonlinear gravitational instability in the framework of cosmological Lagrangian fluid dynamics of cold dust. They include the Zel'dovich approximation (ZA), the ``non-magnetic'' approximation of Bertschinger \& Jain (1994, NMA), and a new ``local tidal'' approximation (LTA). The LTA is exact for any perturbations whose gravitational and velocity equipotentials have the same constant shape with time, including spherical, cylindrical, and plane-parallel perturbations. We tested all three local approximations with the collapse of a homogeneous triaxial ellipsoid, for which an exact solution exists for an ellipsoid embedded in empty space and an excellent approximation is known in the cosmological context. We find that the LTA is significantly more accurate in general than the ZA and the NMA. Like the ZA, but unlike the NMA, the LTA generically leads to pancake collapse. For a randomly chosen mass element in an Einstein-de Sitter universe, assuming a Gaussian random field of initial density fluctuations, the LTA predicts that at least 78\% of initially underdense regions collapse owing to nonlinear effects of shear and tides.Comment: 29 pages of latex, uses aaspp4.sty (AASTeX v4.0), submitted to Ap

The Blackwell relation defines no lattice

Blackwell's theorem shows the equivalence of two preorders on the set of information channels. Here, we restate, and slightly generalize, his result in terms of random variables. Furthermore, we prove that the corresponding partial order is not a lattice; that is, least upper bounds and greatest lower bounds do not exist.Comment: 5 pages, 1 figur

Adding Long Wavelength Modes to an NN-Body Simulation

We present a new method to add long wavelength power to an evolved $N$-body simulation, making use of the Zel'dovich (1970) approximation to change positions and velocities of particles. We describe the theoretical framework of our technique and apply it to a P$^3$M cosmological simulation performed on a cube of $100$ Mpc on a side, obtaining a new ``simulation'' of $800$ Mpc on a side. We study the effect of the power added by long waves by mean of several statistics of the density and velocity field, and suggest possible applications of our method to the study of the large-scale structure of the universe.Comment: Revised version, shortened. 15 pages without figures. Accepted for publication in the Astrophysical Journal. Paper and 11 Figures available as .ps.gz files by anonymous ftp at ftp://ftp.mpa-garching.mpg.de/pub/bepi/MA

Velocity Structure of Self-Similar Spherically Collapsed Halos

Using a generalized self-similar secondary infall model, which accounts for tidal torques acting on the halo, we analyze the velocity profiles of halos in order to gain intuition for N-body simulation results. We analytically calculate the asymptotic behavior of the internal radial and tangential kinetic energy profiles in different radial regimes. We then numerically compute the velocity anisotropy and pseudo-phase-space density profiles and compare them to recent N-body simulations. For cosmological initial conditions, we find both numerically and analytically that the anisotropy profile asymptotes at small radii to a constant set by model parameters. It rises on intermediate scales as the velocity dispersion becomes more radially dominated and then drops off at radii larger than the virial radius where the radial velocity dispersion vanishes in our model. The pseudo-phase-space density is universal on intermediate and large scales. However, its asymptotic slope on small scales depends on the halo mass and on how mass shells are torqued after turnaround. The results largely confirm N-body simulations but show some differences that are likely due to our assumption of a one-dimensional phase space manifold.Comment: 11 pages, 4 figures. Accepted by PR

A Hot Spot Model for Black Hole QPOs

In at least two black hole binary systems, the Rossi X-Ray Timing Explorer has detected high frequency quasi-periodic oscillations (HFQPOs) with a 2:3 frequency commensurability. We propose a simple hot spot model to explain the positions, amplitudes, and widths of the HFQPO peaks. Using the exact geodesic equations for the Kerr metric, we calculate the trajectories of massive test particles, which are treated as isotropic, monochromatic emitters in their rest frames. By varying the hot spot parameters, we are able to explain the different features observed in ``Type A'' and ``Type B'' QPOs from XTE J1550-564. In the context of this model, the observed power spectra allow us to infer values for the black hole mass and angular momentum, and also constrain the parameters of the model.Comment: 4 pages, 2 figures, to be published in X-Ray Timing 2003: Rossi and Beyond, ed. P. Kaaret, F. K. Lamb, & J. H. Swank (Melville, NY: American Institute of Physics