The Universe on Small Scales

During the last 10 years new extensive observations of the Universe were made using both ground-based telescopes and space instruments. These measurements have provided new insights into the structure of the Universe on various scales. A wide range of the electromagnetic spectrum emitted by cosmic objects has been studied. The wavelengths extend from very long radio wavelengths to energetic gamma rays. This observational progress has been accompanied by considerable effort in our theoretical understanding of the formation of different components of the observed structure of the Universe: galaxies and their satellites, clusters of galaxies, and superclusters. A substantial part of this theoretical progress is due to the increasing possibilities of using ever improving numerical models, which mimic the structure formation on different scales using the new generation of massively parallel supercomputers. Looking back in recent history (20 years or so), it is interesting to note that the resolution of the numerical simulations roughly followed Moore‘s Law. In the eighties the best simulations handled about 32³ particles, whereas now we can reach 1024³ particles, a factor of 2¹⁵ increase during 20 years. Moore‘s Law predicts a factor of 2¹³. The remaining factor of 2² in speedup is due to more sophisticated numerical algorithms. The dramatic increase in the accuracy and quality of modeling is astonishing.

The effort of observers and theorists brought about the so called concordance or standard cosmological model. This model is based on the idea that some kind of dark energy contributes about 70% of the total energy density of the spatially flat Universe. The simplest form of the dark energy is the cosmological constant, which was introduced in 1917 by Albert Einstein in his paper treating the cosmological solutions of the field equations of general relativity. The remaining 30% of energy density consists of matter. About 85% of this matter is made of unknown dark matter particles, the remaining 15% is the contribution of “normal” baryonic particles well known to particle physicists. This means that the nature of more than 95% of the matter in the Universe is not yet understood.

According to the standard cosmological model, the main process responsible for the formation of observed structures is gravitational instability. The initial seeds, which eventually became galaxies and superclusters and all other structures, resulted from the quantum fluctuations generated during the early inflationary phase: ~ 10^-35 sec or so from the beginning of the Big Bang. The power spectrum of these primordial fluctuations has been confirmed by measuring the temperature fluctuations of the cosmic microwave background radiation. These temperature fluctuations tell us the magnitude of the small fluctuations in the Universe about 300,000 years after the Big Bang.

One of the key features of the standard model is its simplicity. The expansion rate and the clustering properties are described by only few parameters which can be measured with quite high accuracy. Moreover, there are no preferred length scales in the model. Thus, structure formation is predicted to be essentially scale invariant: in a statistical sense the structures on scales of galaxy clusters are repeated on scales of galaxies. Typically, small objects merge together and form more and more massive objects. However, the small objects do not disappear within those larger objects but rather form a complex hierarchy of substructures. This hierarchical scenario predicts that our Milky Way Galaxy is expected to have as many satellites (many hundreds) as a cluster of galaxies has galaxies. However, the Milky Way has only a dozen satellite galaxies: a far cry from what is predicted.

Altogether, we arrive at a picture in which dark matter particles form the backbone structure for all objects in the Universe from clusters of galaxies to dwarf satellite galaxies. Normal matter (baryons) falls into the potential wells formed by the dark matter particles and forms the luminous objects. The details of this formation process must be followed using hydrodynamical simulations. However, many features can already be studied by semi-analytical methods which are based on the evolution of the dark matter halos as measured in the dark matter imulations.

The nonlinear evolution of cosmological fluctuations can be studied only numerically. The requirements for modern cosmological simulations are extreme: a very large dynamical range for force resolution and many millions of particles are needed. These requirements are just a reflection of the vast range of masses and spatial scales in real astronomical objects. For example, from dwarf galaxies to galaxy clusters the mass spans about seven orders of magnitude. The range of scales is also enormous: from the inner structure of galaxies (sub-kiloparsec scales) to cosmological distances of many megaparsecs (1 pc = 3.26 light years).

We have developed a highly efficient parallel Adaptive Refinement Tree (ART) code, which tracks the evolution of small-amplitude perturbations from the early Universe until the present time. Our code covers a dynamical range of up to 500,000 and handles millions of particles of different masses. The code also includes hydrodynamics. At present, for very large simulations we use a hybrid MPI-OpenMP mode of parallelization. In this mode the code can handle up to 1 billion particles.

Figure 1:
Decomposition of the simulation box into 64 domains with each domain handled by one MPI task. The code decreases the volume of domains in regions with many particles and high force resolution (the center of the box) to achieve load balance. The colored 24 smallest domains cover the central region.

The first step of running cosmological simulations is to set up the initial conditions: amplitudes and phases of small perturbations at very high redshifts. Having in mind that the largest structures in the Universe - superclusters and voids - have sizes of 10-50 Mpc, the simulated volume should be significantly larger than a supercluster or a void. However, we may be interested in the structure of a much smaller object such as our Milky Way Galaxy or its satellites. In N-body simulations each mass element is represented by a point-like particle. The mass resolution is limited by the total number of particles computers can handle at a given time. Thus, increasing the representative cosmological volume decreases the mass resolution. To overcome this problem we developed a mass refinement technique: To construct suitable initial conditions, we first create an unconstrained random realization at highest possible resolution, i.e. 2048³ (~ 8.6 billion) particles. The initial displacements and velocities of N particles are calculated using all waves ranging from the fundamental mode k=2π/L_box to the Nyquist frequency k_Ny=2π/L_box x N^1/3/2. We then merge (lump together) particles, assigning to merged particles a velocity and a displacement equal to the average values of the original small-mass particles. Using the smaller number of more massive merged particles we first run low-resolution simulations until the present epoch. In this simulation we select the regions of interest. Then we repeat the process by starting from refined initial conditions. This time we preserve the original, very small-mass particles but only inside the region, which later collapses to produce our object of interest. Outside of this region we progressively merge more and more of the small particles creating shells of larger and larger particles. This procedure ensures that our object evolves in the proper cosmological environment and with the right gravitational tidal fields.

We have used the technique described above to simulate the evolution of a region similar to our cosmological neighborhood. In this simulation we use more than 150 million particles within the high resolution region. The mass resolution is 5 x 106 Mø and hence a Milky Way Galaxy is represented by 200,000 particles. The force resolution reaches 300 pc. The total CPU time used for this simulation was about 300,000 CPU hours. The initial conditions (they need 105 GByte of shared memory) have been calculated on the SP4 at NIC Jülich. A substantial part of the simulation has been done using 512 CPUs of the Hitachi SR8000 at Leibniz Rechenzentrum Munich, the rest at NASA‘s SGI Altix 3000.

Figure 2:
Matter distribution in a cosmological simulation. The colors represent the logarithm of the local density. Galaxies with stars are formed in high density (white) regions.

Figure 2 shows a region of 35 Mpc size. It is a small part (3%) of a larger simulation with a total volume of (115 Mpc)³. One can clearly see the filamentary structure. The first blow-up shows a region of 10 Mpc and the second one of 2 Mpc. The object in the second blow-up has a mass comparable to our Milky Way.

Even though high resolution simulations already allow the comparison of many theoretical predictions with observed properties of galaxies in various environments they are still far from providing us with a complete understanding of the processes of galaxy formation. In the near future observations will deliver detailed information about the inner structure of galaxies and the formation and evolution of low mass objects, in particular about such objects at high redshift, which are the progenitors of the present-day galaxies. These improved observational data sets need to be accompanied by an equally sophisticated theoretical and numerical modeling of structure formation. To this end one also needs to follow the evolution of the gas and the formation of stars in cold clouds; to know their impact on the interstellar medium and on the formation of the observed stellar disks and bulges. These models become more and more complex and span an even wider dynamical range. To cope with such tasks the largest available supercomputers will be required by astrophysicists working in numerical cosmology.

• Stefan Gottlöber
• Arman Khalatyan
Astrophysikalisches Institut Potsdam

• Anatoly Klypin
New Mexico State University, Las Cruces, USA

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