This blog is based on a recent talk on the Horizon supercomputer simulation for galaxy formation. The talk (in English) was given at the Ecole Normale Superieure by Julien Devrient, of the University of Oxford, available on YouTube here:

The background for the simulation of galaxy formation on supercomputers is the standard Lambda-Cold Dark Matter cosmology with 4.8% ordinary matter, 26.8% dark matter and 68.4% dark energy, which are the measured values from the Planck satellite and other observations. These are the proportions at present, but until the last few billion years, dark matter was dominant over dark energy. The ratio of dark matter to ordinary matter has stayed essentially fixed since the universe was 1 second old, with about 5 times or so as much dark matter as ordinary matter.

The collisionless components to consider are cold dark matter (CDM) and stars, as the stars form inside the simulation.

Then there is a collisional fluid composed of gas, in both atomic (neutral and ionized) and molecular forms and consisting primarily of hydrogen, helium and a small amount, up to around 1% by mass, of heavy elements including carbon, nitrogen, oxygen, silicon, iron and so forth. This fraction increases during the history of the universe as star formation and evolution proceeds. This ‘primordial’ gas is heated by falling into the gravitational potential determined primarily by the CDM (but also by the ordinary matter) and it cools via various radiative processes that depend on density, temperature and composition.

There are many complicating factors and feedback processes. This is an extremely messy problem to address. Dust, supernovae, turbulent gas dynamics, magnetic fields, and black holes that merge and grow into supermassive black holes (SMBH) are all things to consider. The SMBH are surrounded by accretion disks and also may emit jets and these components are visible as highly luminous AGN (active galactic nuclei). Not all of these can be included in simulations at present, or they are treated empirically.

Although the physics is well understood for the collisionless component behavior and for the atomic and molecular gas, including the cooling (radiative) functions, the modeling must occur over many, many orders of magnitude, since scales range from less than 1 parsec to 100s of Megaparsecs (a million parsecs, where 1 parsec = 3.26 light-years). This huge range in scale, plus complex physics, makes the calculation extremely computationally expensive.

The Horizon simulation had 7 billion grid cells and 1 billion dark matter particles. The highest resolution is down to 1 kiloparsec. Gas cooling, star formation, stellar winds, two types of supernovae are included and the abundances of C, N, O, Si, Mg, and Fe tracked. Black hole formation was included. Two million CPU core hours were required for the simulation.

Figure 1. The multi-scale problem

Many scales are involved in simulating galaxy formation – 11 or 12 orders of magnitude. Each tick mark in the above Figure 1 is 3 orders of magnitude (a factor of 1000) in linear scale. From the largest to the smallest objects (moving from right to left) we have LSS = large-scale structure: the universe has evolved into a web-like structure with filaments and sheets of galaxies and high-density and low-density regions. The scale is 100s of Megaparsecs to more than a Gigaparsec. Below this are the galaxy clusters, which are the largest gravitationally bound structures, at around 1 Megaparsec, and then galaxies which are found primarily in the 1 kiloparsec to 100 kiloparsec range.

Then within galaxies, star formation happens within molecular clouds and the scales are parsecs to 100s of parsecs. At the smallest scale, we have highly energetic active galactic nuclei (AGN), that are powered by SMBH (supermassive black holes), with millions to billions of solar masses, and have surrounding accretion disks, confined within a very small region of order 1/1000 of a parsec, reaching down towards the scale of our solar system.

Figure 2. The Dark Matter Halo mass function and the galaxy mass function

It is impossible with current supercomputers and techniques to directly model across all these scales, but the Horizon-AGN Simulation, one of the largest galaxy formation simulations today, spans around 5 orders of magnitude by using adaptive mesh refinement strategies. When and where the density of matter is high and the physics is interesting, an increasingly finer mesh is employed for the calculations. Without this method, it would be impossible to make progress.

Galaxies are formed within the gravitational potentials of dark matter halos (DMH). There is about 5 times as much mass in dark matter as in ordinary matter (baryons, e.g. protons and neutrons). So the ordinary matter falls into the gravitational potentials of DMH, is heated up, and cools by radiation which allows for further collapse, and so on until galaxies are formed.

The interesting scales for DMH are from about 100 billion to 1000 trillion solar masses. The size distribution for the density perturbations that self-collapse under their own gravity follows a power law (with an index of close to -1 in the inverse linear scale). This comes from the cosmic microwave background measurements and inflationary Big Bang theory. How these density perturbations evolve and collapse to DMH is now a well-studied problem in cosmology.

One might assume that each DMH results in a single galaxy, and in the mid-range, this matches observations fairly well. But at the low-end and the high-end, this simple model breaks down, when comparison is made to the observed galaxy mass function (which is simply a measurement of how many galaxies we see per unit volume with a given mass).

At the low end we see fewer galaxies than expected. These are very faint however more and more dwarf galaxies with low luminosity yet with significant mass dominated by dark matter are being detected, and this is helping to resolve this issue. An important factor is most likely feedback from supernovae. As supernovae explode they produce blast waves which drive gas out and prevent molecular cloud formation and star formation.

Supernova physics is tricky as it can result in gas compression which enhances the star formation rate but also can drive gas out of a galaxy, partcularly if it is smaller and has a lower gravitational field, and this suppresses star formation.

In the left panel of Figure 2 above, the first black line is the DMH mass function, and the second black line is just shifted to the left by the baryon to dark matter ratio. What is being plotted is the frequency of galaxies expected for a given mass.  The actual observed curve for galaxy stellar masses is in red, and one sees fewer galaxies at the low end and especially at the very high end. The right panel shows the observational data which is replotted as the red line in the left panel.

At the high end of the mass function there are fewer galaxies with a rapid cutoff around 1 to 10 trillion solar masses for baryon content, which is about an order of magnitude lower than the DMH  mass function would suggest. At the high end it is believed that feedback from AGN (SMBH) is the cause of inhibited star formation, placing a limit on the maximum size of an individual galaxy. Of course multiple galaxies may form out of a single halo as well.

Figure 3. The Horizon simulation without and with Active Galactic Nuclei included

The upper panel on the right in Figure 3 is the simulation without AGN, the lower one with AGN. The simulation including AGN is a better fit to observed galaxy properties.

The simulation had 7 billion grid cells and 1 billion dark matter particles. The highest resolution is down to 1 kiloparsec. Gas cooling, star formation, stellar winds, two types of supernovae are included and the abundances of C, N, O, Si, Mg, and Fe tracked. Black hole formation was included. Two million CPU core hours were required for the simulation.

Including modeling of AGN, the larger galaxies in the simulation are less massive and dimmer, and are more likely to be ellipticals than spiral galaxies. The high mass galaxies in the center of clusters are generally observed to be ellipticals, so this is a desired result.

There is much room for refining and improving galaxy simulation work, including adding additional physics and more small-scale resolution to the models. I encourage you to look at the YouTube video, there are many other interesting results discussed by Prof. Devrient from the Horizon-AGN simulation work.

References:

https://www.youtube.com/watch?v=ZRDITkkqqUg – Prof. Devrient’s talk

http://www.horizon-simulation.org/about.html – Horizon simulation home page