Engineering topics

Norman, Michael L.

Publications and source records attributed to Norman, Michael L..

At least 19 records

DOE OSTI2021

Structured Adaptive Mesh Refinement Adaptations to Retain Performance Portability With Increasing Heterogeneity

Adaptive mesh refinement (AMR) is an important method that enables many mesh-based applications to run at effectively higher resolution within limited computing resources by allowing high resolution only where really needed. This advantage comes at a cost, however: greater complexity in the mesh management machinery and challenges with load distribution. With the current trend of increasing heterogeneity in hardware architecture, AMR presents an orthogonal axis of complexity. Additionally, the usual techniques, such as asynchronous communication and hierarchy management for parallelism and memory that are necessary to obtain reasonable performance are very challenging to reason about with AMR. Different groups working with AMR are bringing different approaches to this challenge. Here, we examine the design choices of several AMR codes and also the degree to which demands placed on them by their users influence these choices.

42 ENGINEERING↗

DOE OSTI2017

First Light II: Emission Line Extinction, Population III Stars, and X-ray Binaries

Here, we produce synthetic spectra and observations for metal-free stellar populations and high-mass X-ray binaries in the Renaissance Simulations at a redshift of 15. We extend our methodology from the first paper in the series by modelling the production and extinction of emission lines throughout a dusty and metal-enriched interstellar and circum-galactic media extracted from the simulation, using a Monte Carlo calculation. To capture the impact of high-energy photons, we include all frequencies from hard X-ray to far-infrared with enough frequency resolution to discern line emission and absorption profiles. The most common lines in our sample in order of their rate of occurrence are Ly α, the C iv λλ1548, 1551 doublet, H α, and the Ca ii λλλ8498, 8542, 8662 triplet. The best scenario for a direct observation of a metal-free stellar population is a merger between two Population III Galaxies. In mergers between metal-enriched and metal-free stellar populations, some characteristics may be inferred indirectly. Single Population III galaxies are too dim to be observed photometrically at z = 15. Ly α emission is discernible by JWST as an increase in J200w – J277w colour off the intrinsic stellar tracks. Observations of metal-free stars will be difficult, though not impossible, with the next generation of space telescopes.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

Galaxy properties and UV escape fractions during the epoch of reionization: results from the renaissance simulations

Cosmic reionization is thought to be primarily fueled by the first generations of galaxies. We examine their stellar and gaseous properties, focusing on the star formation rates and the escape of ionizing photons, as a function of halo mass, redshift, and environment using the full suite of the Renaissance Simulations with an eye to provide better inputs to global reionization simulations. This suite probes overdense, average, and underdense regions of the universe of several hundred comoving Mpc{sup 3}, each yielding a sample of over 3000 halos in the mass range of 10{sup 7}–10{sup 9.5} M{sub ⊙} at their final redshifts of 15, 12.5, and 8, respectively. In the process, we simulate the effects of radiative and supernova feedback from 5000 to 10,000 Population III stars in each simulation. We find that halos as small as 10{sup 7} M{sub ⊙} are able to host bursty star formation due to metal-line cooling from earlier enrichment by massive Population III stars. Using our large sample, we find that the galaxy-halo occupation fraction drops from unity at virial masses above 10{sup 8.5} M{sub ⊙} to ∼50% at 10{sup 8} M{sub ⊙} and ∼10% at 10{sup 7} M{sub ⊙}, quite independent of redshift and region. Their average ionizing escape fraction is ∼5% in the mass range of 10{sup 8}–10{sup 9} M{sub ⊙} and increases with decreasing halo mass below this range, reaching 40%–60% at 10{sup 7} M{sub ⊙}. Interestingly, we find that the escape fraction varies between 10%–20% in halos with virial masses of ∼3 × 10{sup 9} M{sub ⊙}. Taken together, our results confirm the importance of the smallest galaxies as sources of ionizing radiation contributing to the reionization of the universe.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

X-ray background at high redshifts from pop III remnants: results from pop III star formation rates in the renaissance simulations

Due to their long mean free paths, X-rays are expected to have global impacts on the properties of the intergalactic medium (IGM) by their large-scale heating and ionizing processes. At high redshifts, X-rays from Population (Pop) III binaries might have important effects on cosmic reionization and the Lyα forest. As a continuation of our previous work on Pop III binary X-rays, we use the Pop III distribution and evolution from the Renaissance Simulations, a suite of self-consistent cosmological radiation hydrodynamics simulations of the formation of the first galaxies, to calculate the X-ray luminosity density and background over the redshift range 20⩾z⩾7.6. As we find that Pop III star formation continues at a low, nearly constant rate to the end of reionization, X-rays are being continuously produced at significant rates compared to other possible X-ray sources, such as AGNs and normal X-ray binaries during the same period of time. We estimate that Pop III binaries produce approximately 6 eV of energy in the X-rays per hydrogen atom. We calculate the X-ray background for different monochromatic photon energies. KeV X-rays redshift and accumulate to produce a strong X-ray background spectrum extending to roughly 500 eV. The X-ray background is strong enough to heat the IGM to ∼1000 K and to ionize a few percent of the neutral hydrogen. These effects are important for an understanding of the neutral hydrogen hyperfine transition 21 cm line signatures, the Lyα forest, and the Thomson optical depth to the CMB.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2016

LATE POP III STAR FORMATION DURING THE EPOCH OF REIONIZATION: RESULTS FROM THE RENAISSANCE SIMULATIONS

We present results on the formation of Population III (Pop III) stars at redshift 7.6 from the Renaissance Simulations, a suite of extremely high-resolution and physics-rich radiation transport hydrodynamics cosmological adaptive-mesh refinement simulations of high-redshift galaxy formation performed on the Blue Waters supercomputer. In a survey volume of about 220 comoving Mpc{sup 3}, we found 14 Pop III galaxies with recent star formation. The surprisingly late formation of Pop III stars is possible due to two factors: (i) the metal enrichment process is local and slow, leaving plenty of pristine gas to exist in the vast volume; and (ii) strong Lyman–Werner radiation from vigorous metal-enriched star formation in early galaxies suppresses Pop III formation in (“not so”) small primordial halos with mass less than ∼3 × 10{sup 7} M {sub ⊙}. We quantify the properties of these Pop III galaxies and their Pop III star formation environments. We look for analogs to the recently discovered luminous Ly α emitter CR7, which has been interpreted as a Pop III star cluster within or near a metal-enriched star-forming galaxy. We find and discuss a system similar to this in some respects, however, the Pop III star cluster is far less massive and luminous than CR7 is inferred to be.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

SPATIALLY EXTENDED 21 cm SIGNAL FROM STRONGLY CLUSTERED UV AND X-RAY SOURCES IN THE EARLY UNIVERSE

We present our prediction for the local 21 cm differential brightness temperature (δT{sub b}) from a set of strongly clustered sources of Population III (Pop III) and II (Pop II) objects in the early universe, by a numerical simulation of their formation and radiative feedback. These objects are located inside a highly biased environment, which is a rare, high-density peak (“Rarepeak”) extending to ∼7 comoving Mpc. We study the impact of ultraviolet and X-ray photons on the intergalactic medium (IGM) and the resulting δT{sub b}, when Pop III stars are assumed to emit X-ray photons by forming X-ray binaries very efficiently. We parameterize the rest-frame spectral energy distribution of X-ray photons, which regulates X-ray photon-trapping, IGM-heating, secondary Lyα pumping and the resulting morphology of δT{sub b}. A combination of emission (δT{sub b} > 0) and absorption (δT{sub b} < 0) regions appears in varying amplitudes and angular scales. The boost of the signal by the high-density environment (δ ∼ 0.64) and on a relatively large scale combines to make Rarepeak a discernible, spatially extended (θ ∼ 10′) object for 21 cm observation at 13 ≲ z ≲ 17, which is found to be detectable as a single object by SKA with integration time of ∼1000 hr. Power spectrum analysis by some of the SKA precursors (Low Frequency Array, Murchison Widefield Array, Precision Array for Probing the Epoch of Reionization) of such rare peaks is found to be difficult due to the rarity of these peaks, and the contribution only by these rare peaks to the total power spectrum remains subdominant compared to that by all astrophysical sources.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2015

Fully coupled simulation of cosmic reionization. I. numerical methods and tests

Here, we describe an extension of the Enzo code to enable fully coupled radiation hydrodynamical simulation of inhomogeneous reionization in large similar to(100 Mpc)(3) cosmological volumes with thousands to millions of point sources. We solve all dynamical, radiative transfer, thermal, and ionization processes self-consistently on the same mesh, as opposed to a postprocessing approach which coarse-grains the radiative transfer. But, we employ a simple subgrid model for star formation which we calibrate to observations. The numerical method presented is a modification of an earlier method presented in Reynolds et al. differing principally in the operator splitting algorithm we use to advance the system of equations. Radiation transport is done in the gray flux-limited diffusion (FLD) approximation, which is solved by implicit time integration split off from the gas energy and ionization equations, which are solved separately. This results in a faster and more robust scheme for cosmological applications compared to the earlier method. The FLD equation is solved using the hypre optimally scalable geometric multigrid solver from LLNL. By treating the ionizing radiation as a grid field as opposed to rays, our method is scalable with respect to the number of ionizing sources, limited only by the parallel scaling properties of the radiation solver. We test the speed and accuracy of our approach on a number of standard verification and validation tests. We show by direct comparison with Enzo's adaptive ray tracing method Moray that the well-known inability of FLD to cast a shadow behind opaque clouds has a minor effect on the evolution of ionized volume and mass fractions in a reionization simulation validation test. Finally, we illustrate an application of our method to the problem of inhomogeneous reionization in a 80 Mpc comoving box resolved with 3200(3) Eulerian grid cells and dark matter particles.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

Scaling relations for galaxies prior to reionization

The first galaxies in the universe are the building blocks of all observed galaxies. We present scaling relations for galaxies forming at redshifts z ≥ 15 when reionization is just beginning. We utilize the 'Rarepeak' cosmological radiation hydrodynamics simulation that captures the complete star formation history in over 3300 galaxies, starting with massive Population III stars that form in dark matter halos as small as ∼10{sup 6} M {sub ☉}. We make various correlations between the bulk halo quantities, such as virial, gas, and stellar masses and metallicities and their respective accretion rates, quantifying a variety of properties of the first galaxies up to halo masses of 10{sup 9} M {sub ☉}. Galaxy formation is not solely relegated to atomic cooling halos with virial temperatures greater than 10{sup 4} K, where we find a dichotomy in galaxy properties between halos above and below this critical mass scale. Halos below the atomic cooling limit have a stellar mass-halo mass relationship log M {sub *} ≅ 3.5 + 1.3log (M {sub vir}/10{sup 7} M {sub ☉}). We find a non-monotonic relationship between metallicity and halo mass for the smallest galaxies. Their initial star formation events enrich the interstellar medium and subsequent star formation to a median of 10{sup –2} Z {sub ☉} and 10{sup –1.5} Z {sub ☉}, respectively, in halos of total mass 10{sup 7} M {sub ☉}, which is then diluted by metal-poor inflows well beyond Population III pre-enrichment levels of 10{sup –3.5} Z {sub ☉}. The scaling relations presented here can be employed in models of reionization, galaxy formation, and chemical evolution in order to consider these galaxies forming prior to reionization.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

Heating the intergalactic medium by X-rays from population III binaries in high-redshift galaxies

Due to their long mean free path, X-rays are expected to have an important impact on cosmic reionization by heating and ionizing the intergalactic medium (IGM) on large scales, especially after simulations have suggested that Population III (Pop III) stars may form in pairs at redshifts as high as 20-30. We use the Pop III distribution and evolution from a self-consistent cosmological radiation hydrodynamic simulation of the formation of the first galaxies and a simple Pop III X-ray binary model to estimate their X-ray output in a high-density region larger than 100 comoving (Mpc){sup 3}. We then combine three different methods—ray tracing, a one-zone model, and X-ray background modeling—to investigate the X-ray propagation, intensity distribution, and long-term effects on the IGM thermal and ionization state. The efficiency and morphology of photoheating and photoionization are dependent on the photon energies. The sub-kiloelectronvolt X-rays only impact the IGM near the sources, while the kiloelectronvolt photons contribute significantly to the X-ray background and heat and ionize the IGM smoothly. The X-rays just below 1 keV are most efficient in heating and ionizing the IGM. We find that the IGM might be heated to over 100 K by z = 10 and the high-density source region might reach 10{sup 4} K, limited by atomic hydrogen cooling. This may be important for predicting the 21 cm neutral hydrogen signals. On the other hand, the free electrons from X-ray ionizations are not enough to contribute significantly to the optical depth of the cosmic microwave background to the Thomson scattering.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

Fully coupled simulation of cosmic reionization. II. Recombinations, clumping factors, and the photon budget for reionization

We use a fully coupled cosmological simulation including dark matter dynamics, multispecies hydrodynamics, nonequilibrium chemical ionization, flux-limited diffusion radiation transport, and a parameterized model of star formation and feedback (thermal and radiative) to investigate the epoch of hydrogen reionization in detail. In this paper, the first of several application papers, we investigate the mechanics of reionization from stellar sources forming in high-z galaxies, the utility of various formulations for the gas clumping factor on accurately estimating the effective recombination time in the intergalactic medium (IGM), and the photon budget required to achieve reionization. We also test the accuracy of the static and time-dependent models of Madau et al. as predictors of reionization completion/maintenance. We simulate a WMAP7 ΛCDM cosmological model in a 20 comoving Mpc cube, resolved with 800{sup 3} uniform fluid cells and dark matter particles. By tuning our star formation recipe to approximately match the observed high-redshift star formation rate density and galaxy luminosity function, we have created a fully coupled radiation hydrodynamical realization of hydrogen reionization, which begins to ionize at z ≈ 10 and is completed at z ≈ 5.8 without further tuning. We find that roughly two ionizing photons per H atom are required to convert the neutral IGM to a highly ionized state. After reionization concludes, we find that the quantity n-dot{sub ion}×(1 Gyr)/n{sub H} is ∼9 at z = 5, in rough agreement with measurements of the ionizing emissivity by Becker and Bolton. The complicated events during reionization that lead to this number can be generally described as inside-out, but in reality, the narrative depends on the level of ionization of the gas one attributes as being ionized. We find that the formula for the ionizing photon production rate needed to maintain the IGM in an ionized state derived by Madau et al. should not be used to predict the epoch of reionization completion because it ignores history-dependent terms in the global ionization balance which are not ignorable. We find that the time-dependent model for the ionized volume fraction Q{sub HII} is more predictive, but overestimates the redshift of reionization completion z{sub reion} by Δz ≈ 1. We propose a revised formulation of the time-dependent model that agrees with our simulation to a few percent accuracy. Finally, we use our simulation data to measure the absorption of ionizing radiation due to circumgalactic gas resolved on our mesh and find f-bar{sub esc}(CGM)≈0.7.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2014

ENZO: AN ADAPTIVE MESH REFINEMENT CODE FOR ASTROPHYSICS

This paper describes the open-source code Enzo, which uses block-structured adaptive mesh refinement to provide high spatial and temporal resolution for modeling astrophysical fluid flows. The code is Cartesian, can be run in one, two, and three dimensions, and supports a wide variety of physics including hydrodynamics, ideal and non-ideal magnetohydrodynamics, N-body dynamics (and, more broadly, self-gravity of fluids and particles), primordial gas chemistry, optically thin radiative cooling of primordial and metal-enriched plasmas (as well as some optically-thick cooling models), radiation transport, cosmological expansion, and models for star formation and feedback in a cosmological context. In addition to explaining the algorithms implemented, we present solutions for a wide range of test problems, demonstrate the code's parallel performance, and discuss the Enzo collaboration's code development methodology.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2013

POPULATION III STARS AND REMNANTS IN HIGH-REDSHIFT GALAXIES

Recent simulations of Population III star formation have suggested that some fraction form in binary systems, in addition to having a characteristic mass of tens of solar masses. The deaths of metal-free stars result in the initial chemical enrichment of the universe and the production of the first stellar-mass black holes. Here we present a cosmological adaptive mesh refinement simulation of an overdense region that forms a few 10{sup 9} M{sub Sun} dark matter halos and over 13,000 Population III stars by redshift 15. We find that most halos do not form Population III stars until they reach M{sub vir} {approx} 10{sup 7} M{sub Sun} because this biased region is quickly enriched from both Population III and galaxies, which also produce high levels of ultraviolet radiation that suppress H{sub 2} formation. Nevertheless, Population III stars continue to form, albeit in more massive halos, at a rate of {approx}10{sup -4} M{sub Sun} yr{sup -1} Mpc{sup -3} at redshift 15. The most massive starless halo has a mass of 7 Multiplication-Sign 10{sup 7} M{sub Sun }, which could host massive black hole formation through the direct gaseous collapse scenario. We show that the multiplicity of the Population III remnants grows with halo mass above 10{sup 8} M{sub Sun }, culminating in 50 remnants located in 10{sup 9} M{sub Sun} halos on average. This has implications that high-mass X-ray binaries and intermediate-mass black holes that originate from metal-free stars may be abundant in high-redshift galaxies.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1997

Parallel Multiscale Algorithms for Astrophysical Fluid Dynamics Simulations

Our goal is to develop software libraries and applications for astrophysical fluid dynamics simulations in multidimensions that will enable us to resolve the large spatial and temporal variations that inevitably arise due to gravity, fronts and microphysical phenomena. The software must run efficiently on parallel computers and be general enough to allow the incorporation of a wide variety of physics. Cosmological structure formation with realistic gas physics is the primary application driver in this work. Accurate simulations of e.g. galaxy formation require a spatial dynamic range (i.e., ratio of system scale to smallest resolved feature) of 104 or more in three dimensions in arbitrary topologies. We take this as our technical requirement. We have achieved, and in fact, surpassed these goals.

Norman, Michael L.↗

NASA NTRS1994

The evolution of X-ray clusters in a cold plus hot dark matter universe

We present the first self-consistently computed results on the evolution of X-ray properties of galaxy clusters in a cold + hot dark matter (CHDM) model. We have performed a hydrodynamic plus N-body simulation for the COBE-compatible CHDM model with standard mass components: Omega(sub hot) = 0.3, Omega (sub cold) = 0.6 and Omega(sub baryon) = 0.1 (h = 0.5). In contrast with the CDM model, which fails to reproduce the observed temperature distribution function dN/dT (Bryan et al. 1994b), the CHDM model fits the observational dN/dT quite well. Our results on X-ray luminosity are less firm but even more intriguing. We find that the resulting X-ray luminosity functions at redshifts z = 0.0, 0.2, 0.4, 0.7 are well fit by observations, where they overlap. The fact that both temperatures and luminosities provide a reasonable fit to the available observational data indicates that, unless we are missing some essential physics, there is neither room nor need for a large fraction of gas in rich clusters: 10% (or less) in baryons is sufficient to explain their X-ray properties. We also see a tight correlation between X-ray luminosity and gas temperature.

Bryan, Greg L.↗

NASA NTRS1994

Shock interactions with magnetized interstellar clouds. 1: Steady shocks hitting nonradiative clouds

We study the interaction of a steady, planar shock with a nonradiative, spherical, interstellar cloud threaded by a uniform magnetic field. For strong shocks, the sonic Mach number scales out, so two parameters determine the evolution: the ratio of cloud to intercloud density, and the Alfven Mach number. We focus on the case with initial field parallel to the shock velocity, though we also present one model with field perpendicular to the velocity. Even with 100 zones per cloud radius, we find that the magnetic field structure converges only at early times. However, we can draw three conclusions from our work. First, our results suggest that the inclusion of a field in equipartition with the preshock medium can prevent the complete destruction of the cloud found in the field-free case recently considered by Klein, McKee, & Colella. Second, the interaction of the shock with the cloud can amplify the magnetic field in some regions up to equipartition with the post-shock thermal pressure. In the parallel-field case, the shock preferentially amplifies the parallel component of the field, creating a 'flux rope,' a linear structure of concentrated magnetic field. The flux rope dominates the volume of amplified field, so that laminar, rather than turbulent, amplification is dominant in this case. Third, the presence of the cloud enhances the production of X-ray and synchrotron emission. The X-ray emission peaks early, during the initial passage of the shock over the cloud, while the synchrotron emission peaks later, when the flow sweeps magnetic field onto the axis between the cloud and the main shock.

Low, Mordecai-Mark Mac↗

NASA NTRS1994

A comparison of cosmological hydrodynamic codes

We present a detailed comparison of the simulation results of various hydrodynamic codes. Starting with identical initial conditions based on the cold dark matter scenario for the growth of structure, with parameters h = 0.5 Omega = Omega(sub b) = 1, and sigma(sub 8) = 1, we integrate from redshift z = 20 to z = O to determine the physical state within a representative volume of size L(exp 3) where L = 64 h(exp -1) Mpc. Five indenpendent codes are compared: three of them Eulerian mesh-based and two variants of the smooth particle hydrodynamics 'SPH' Lagrangian approach. The Eulerian codes were run at N(exp 3) = (32(exp 3), 64(exp 3), 128(exp 3), and 256(exp 3)) cells, the SPH codes at N(exp 3) = 32(exp 3) and 64(exp 3) particles. Results were then rebinned to a 16(exp 3) grid with the exception that the rebinned data should converge, by all techniques, to a common and correct result as N approaches infinity. We find that global averages of various physical quantities do, as expected, tend to converge in the rebinned model, but that uncertainites in even primitive quantities such as (T), (rho(exp 2))(exp 1/2) persists at the 3%-17% level achieve comparable and satisfactory accuracy for comparable computer time in their treatment of the high-density, high-temeprature regions as measured in the rebinned data; the variance among the five codes (at highest resolution) for the mean temperature (as weighted by rho(exp 2) is only 4.5%. Examined at high resolution we suspect that the density resolution is better in the SPH codes and the thermal accuracy in low-density regions better in the Eulerian codes. In the low-density, low-temperature regions the SPH codes have poor accuracy due to statiscal effects, and the Jameson code gives the temperatures which are too high, due to overuse of artificial viscosity in these high Mach number regions. Overall the comparison allows us to better estimate errors; it points to ways of improving this current generation ofhydrodynamic codes and of suiting their use to problems which exploit their best individual features.

Kang, Hyesung↗

NASA NTRS1994

The evolution of cosmic-ray-mediated magnetohydrodynamic shocks: A two-fluid approach

We study the shock structure and acceleration efficiency of cosmic-ray mediated Magnetohydrodynamic (MHD) shocks both analytically and numerically by using a two-fluid model. Our model includes the dynamical effect of magnetic fields and cosmic rays on a background thermal fluid. The steady state solution is derived by following the technique of Drury & Voelk (1981) and compared to numerical results. We explore the time evolution of plane-perpendicular, piston-driven shocks. From the results of analytical and numerical studies, we conclude that the mean magnetic field plays an important role in the structure and acceleration efficiency of cosmic-ray mediated MHD shocks. The acceleration of cosmic-ray particles becomes less efficient in the presence of strong magnetic pressure since the field makes the shock less compressive. This feature is more prominent at low Mach numbers than at high Mach numbers.

Jun, Byung-Il↗

NASA NTRS1994

X-ray clusters from a high-resolution hydrodynamic PPM simulation of the cold dark matter universe

A new three-dimensional hydrodynamic code based on the piecewise parabolic method (PPM) is utilized to compute the distribution of hot gas in the standard Cosmic Background Explorer (COBE)-normalized cold dark matter (CDM) universe. Utilizing periodic boundary conditions, a box with size 85 h(exp-1) Mpc, having cell size 0.31 h(exp-1) Mpc, is followed in a simulation with 270(exp 3)=10(exp 7.3) cells. Adopting standard parameters determined from COBE and light-element nucleosynthesis, Sigma(sub 8)=1.05, Omega(sub b)=0.06, we find the X-ray-emitting clusters, compute the luminosity function at several wavelengths, the temperature distribution, and estimated sizes, as well as the evolution of these quantities with redshift. The results, which are compared with those obtained in the preceding paper (Kang et al. 1994a), may be used in conjuction with ROSAT and other observational data sets. Overall, the results of the two computations are qualitatively very similar with regard to the trends of cluster properties, i.e., how the number density, radius, and temeprature depend on luminosity and redshift. The total luminosity from clusters is approximately a factor of 2 higher using the PPM code (as compared to the 'total variation diminishing' (TVD) code used in the previous paper) with the number of bright clusters higher by a similar factor. The primary conclusions of the prior paper, with regard to the power spectrum of the primeval density perturbations, are strengthened: the standard CDM model, normalized to the COBE microwave detection, predicts too many bright X-ray emitting clusters, by a factor probably in excess of 5. The comparison between observations and theoretical predictions for the evolution of cluster properties, luminosity functions, and size and temperature distributions should provide an important discriminator among competing scenarios for the development of structure in the universe.

Bryan, Greg L.↗