Exponential Growth of Distance between Nearby Rays due to Multiple Gravitational Scatterings

We give an estimate of the relative error in the angular measurement of observations for high redshift objects induced by gravitational scatterings (lensing). Gunn (1967) concluded that the gravitational scatterings by galaxies induce the relative error of a few percent in the observations for objects at $z=1$. This estimate has been considered as a fundamental limitation of accuracy of the angular measurements in the observational cosmology. In multiple graviational scatterings, bending angle of single ray grows through the random work process. Gunn (1967) assumed that the difference of nearby rays also grows through the random walk process. However, distance between nearby photons grows exponentially because the two rays suffer coherent scatterings by the same scattering object. This exponential growth continues as long as the scattering is coherent. In the case of scattering by individual galaxies, the exponential growth continues until the angular distance reaches an arcminute or so. The relative error of the angular measurements under an arcminute due to the exponential growth is $\sim 30\%$ at $z=1$ and exceeds $100\%$ at $z=3$, in the case that the density parameter of galaxies is 0.2. The effects of clusters of galaxies or superclusters are more difficult to estimate accurately, but might be significant. In the case of supercluster the angular measurements up to a few degrees could be affected.Comment: compressed uuencoded postscript, 8 pages including 5 figures, APJL accepte

Mass-Loss Timescale of Star Clusters in an External Tidal Field. II. Effect of Mass Profile of Parent Galaxy

We investigate the long-term dynamical evolution of star clusters in a steady tidal field produced by its parent galaxy. In this paper, we focus on the influence of mass profile of the parent galaxy. The previous studies were done with the simplification where the parent galaxy was expressed by point mass. We express different mass profiles of the parent galaxy by the tidal fields in which the ratios of the epicyclic frequency to the angular velocity are different. We compare the mass-loss timescale of star clusters whose tidal radii are identical but in parent galaxies with different mass profile, by means of orbits calculations in fixed cluster potential and N-body simulations. In this situation, a cluster rotates around the parent galaxy more rapidly as the parent galaxy has shallower mass profile. We found that the mass-loss timescale increase 20% and 50% for the cases that the mass density profile of the parent galaxies are proportional to R^-2 and R^-1.5 where R is the distance from the galaxy center, compared to the point-mass case, in moderately strong tidal field. Counterintuitively, a cluster which rotates around the parent galaxy more rapidly has a longer lifetime. The increase of lifetime is due to the fact that the fraction occupied by regular-like orbit increases in shallower profile. Finally, we derive an evaluation formula for the mass-loss timescale of clusters. Our formula can explain a property of the population of the observed galactic globular clusters that their half-mass radii become smaller as their distances from the galactic center become smaller.Comment: Submitted to PAS

The Effect of Gravitational Scattering on the Anisotropy of the Cosmic Background Radiation

The homogeneity of the cosmic microwave background radiation (CBR) is one of the most severe constraint for theories of the structure formation in the universe. We investigated the effect of the gravitational scattering (lensing) of galaxies, clusters of galaxies, and superclusters on the anisotropy of the CBR by numerical simulations. Although this effect was thought to be unimportant, we found that the gravitational scatterings by superclusters can significantly reduce the anisotropy of the CBR. We took into account the exponential growth of the distance between two rays due to multiple scatterings. The bending angle of each ray grows through the random walk process. On the other hand, difference between two rays grows exponentially while it is small. This exponential growth is caused by coherent scatterings that two rays suffer, and was neglected in the previous studies. The gravitational scattering by superclusters reduces the observed temperature anisotropy of the CBR at present time approximately by 40--60 $\%$ from that at the recombination time for angular scale up to a few degrees if the supercluster were formed at $z=$2--4.Comment: compressed uuencoded postscript, 9 pages including 4 figures, APJL accepte

Environmental effect on the subhalo abundance -- a solution to the missing dwarf problem

Recent high-resolution simulations of the formation of dark-matter halos have shown that the distribution of subhalos is scale-free, in the sense that if scaled by the velocity dispersion of the parent halo, the velocity distribution function of galaxy-sized and cluster-sized halos are identical. For cluster-sized halos, simulation results agreed well with observations. Simulations, however, predicted far too many subhalos for galaxy-sized halos. Our galaxy has several tens of known dwarf galaxies. On the other hands, simulated dark-matter halos contain thousands of subhalos. We have performed simulation of a single large volume and measured the abundance of subhalos in all massive halos. We found that the variation of the subhalo abundance is very large, and those with largest number of subhalos correspond to simulated halos in previous studies. The subhalo abundance depends strongly on the local density of the background. Halos in high-density regions contain large number of subhalos. Our galaxy is in the low-density region. For our simulated halos in low-density regions, the number of subhalos is within a factor of three to that of our galaxy. We argue that the ``missing dwarf problem'' is not a real problem but caused by the biased selection of the initial conditions in previous studies, which were not appropriate for field galaxies.Comment: 8 pages, 5 figures, higher resolution run added, accepted by PAS

PGPG: An Automatic Generator of Pipeline Design for Programmable GRAPE Systems

We have developed PGPG (Pipeline Generator for Programmable GRAPE), a software which generates the low-level design of the pipeline processor and communication software for FPGA-based computing engines (FBCEs). An FBCE typically consists of one or multiple FPGA (Field-Programmable Gate Array) chips and local memory. Here, the term "Field-Programmable" means that one can rewrite the logic implemented to the chip after the hardware is completed, and therefore a single FBCE can be used for calculation of various functions, for example pipeline processors for gravity, SPH interaction, or image processing. The main problem with FBCEs is that the user need to develop the detailed hardware design for the processor to be implemented to FPGA chips. In addition, she or he has to write the control logic for the processor, communication and data conversion library on the host processor, and application program which uses the developed processor. These require detailed knowledge of hardware design, a hardware description language such as VHDL, the operating system and the application, and amount of human work is huge. A relatively simple design would require 1 person-year or more. The PGPG software generates all necessary design descriptions, except for the application software itself, from a high-level design description of the pipeline processor in the PGPG language. The PGPG language is a simple language, specialized to the description of pipeline processors. Thus, the design of pipeline processor in PGPG language is much easier than the traditional design. For real applications such as the pipeline for gravitational interaction, the pipeline processor generated by PGPG achieved the performance similar to that of hand-written code. In this paper we present a detailed description of PGPG version 1.0.Comment: 24 pages, 6 figures, accepted PASJ 2005 July 2

GreeM : Massively Parallel TreePM Code for Large Cosmological N-body Simulations

In this paper, we describe the implementation and performance of GreeM, a massively parallel TreePM code for large-scale cosmological N-body simulations. GreeM uses a recursive multi-section algorithm for domain decomposition. The size of the domains are adjusted so that the total calculation time of the force becomes the same for all processes. The loss of performance due to non-optimal load balancing is around 4%, even for more than 10^3 CPU cores. GreeM runs efficiently on PC clusters and massively-parallel computers such as a Cray XT4. The measured calculation speed on Cray XT4 is 5 \times 10^4 particles per second per CPU core, for the case of an opening angle of \theta=0.5, if the number of particles per CPU core is larger than 10^6.Comment: 13 pages, 11 figures, accepted by PAS

GRAPE-6: The massively-parallel special-purpose computer for astrophysical particle simulation

In this paper, we describe the architecture and performance of the GRAPE-6 system, a massively-parallel special-purpose computer for astrophysical $N$-body simulations. GRAPE-6 is the successor of GRAPE-4, which was completed in 1995 and achieved the theoretical peak speed of 1.08 Tflops. As was the case with GRAPE-4, the primary application of GRAPE-6 is simulation of collisional systems, though it can be used for collisionless systems. The main differences between GRAPE-4 and GRAPE-6 are (a) The processor chip of GRAPE-6 integrates 6 force-calculation pipelines, compared to one pipeline of GRAPE-4 (which needed 3 clock cycles to calculate one interaction), (b) the clock speed is increased from 32 to 90 MHz, and (c) the total number of processor chips is increased from 1728 to 2048. These improvements resulted in the peak speed of 64 Tflops. We also discuss the design of the successor of GRAPE-6.Comment: Accepted for publication in PASJ, scheduled to appear in Vol. 55, No.

PROGRAPE-1: A Programmable, Multi-Purpose Computer for Many-Body Simulations

We have developed PROGRAPE-1 (PROgrammable GRAPE-1), a programmable multi-purpose computer for many-body simulations. The main difference between PROGRAPE-1 and "traditional" GRAPE systems is that the former uses FPGA (Field Programmable Gate Array) chips as the processing elements, while the latter rely on the hardwired pipeline processor specialized to gravitational interactions. Since the logic implemented in FPGA chips can be reconfigured, we can use PROGRAPE-1 to calculate not only gravitational interactions but also other forms of interactions such as van der Waals force, hydrodynamical interactions in SPH calculation and so on. PROGRAPE-1 comprises two Altera EPF10K100 FPGA chips, each of which contains nominally 100,000 gates. To evaluate the programmability and performance of PROGRAPE-1, we implemented a pipeline for gravitational interaction similar to that of GRAPE-3. One pipeline fitted into a single FPGA chip, which operated at 16 MHz clock. Thus, for gravitational interaction, PROGRAPE-1 provided the speed of 0.96 Gflops-equivalent. PROGRAPE will prove to be useful for wide-range of particle-based simulations in which the calculation cost of interactions other than gravity is high, such as the evaluation of SPH interactions.Comment: 20 pages with 9 figures; submitted to PAS