**Structure Formation**

It has already been known for some time that the Universe we live in, a Universe which is today full of very distinct objects separated by huge voids, used to be remarkably homogeneous. The cosmic microwave background (CMB) shows that some 400 000 years after the Big Bang the density fluctuations were about one part in 100 000. Certainly, one of the biggest challenges of modern cosmology is to explain how the transition from homogeneous to inhomogeneous conditions has been made. The current, observationally favored cosmological scenario postulates that the main constituent of the total mass in the Universe is non-relativistic (or cold) dark matter, whose physical nature still has to be determined. Within this framework, modern structures form from initial small fluctuations by gravitational instability, and they form hierarchically. Very small objects form first, and through the process of hierarchical merging these small objects then form bigger ones. This bottom-up scenario is strongly supported by the observations, in which the most distant objects we can see are quasars, then galaxies, and finally clusters of galaxies as the largest and most recently virialized objects. We also have evidence of clustering which is still ongoing on even bigger scales.

**Seeds of Structures**

The story of structure formation starts with an epoch of rapid and accelerated expansion of the universe called inflation. It does not matter what was the density field before inflation, because during inflation, the universe expanded for a such a large factor that the observable universe (since information propagates with finite speed, no faster than the speed of light it is plausible to interact only with a finite part of the universe) will appear as geometrically flat, and with a homogeneous density distribution. In a classical theory, the inflation outcome will be a universe with a perfectly uniform density, making it dynamically stable and thus unevolving. However, due to quantum physical effects, namely Heisenberg uncertainty principle, not whole universe exited inflation in the same moment, but rather some parts of it inflated shorter, ending up denser, while some parts inflated longer becoming more rarefied than the average value. This provides the universe that appears homogeneous on a large scale, but still has a spectar of perturbations that serves as a source for gravitational instability. The inflationary inhomogeneities are the seeds of future structures (galaxies, clusters…). While inflation is an ad hoc solution to some inconsistencies within Big Bang theory – horizon problem, monopoles problem, and a flatness problem, it is very predictive as a theory of cosmological structures, and therefore testable. It predicts that perturbations will have Gaussian distribution of amplitudes, and a power spectrum of the form: P(k) = A kn The most of today favored models predict n = 1 – ε, i.e. close to scale invariant power spectrum. The biggest support for inflation is coming from observations of the CMB done by WMAP satellite. On this nice image made by Prof. Tegmark, one can see fluctuations in temperature (which are related to fluctuations in the density) of the background radiation where blue spots correspond to colder regions, and red to hotter. This is a very high contrast map, the fluctuations are only around one part in 100 000. The WMAP measurements show that the fluctuations have Gaussian distribution of amplitudes, and n = 0.951 ± 0.019.

**Linear Growth**

As long as the density of perturbation on a certain length scale λ is small enough with respect to background density (ρ0) :

one can use linear theory to describe its evolution. Since mean free path of a particle is smaller then the horizon size, the ideal fluid approximation can be used. Also, for scales much smaller then the horizon (λ << H-1) Newtonian approach is good; otherwise full general relativity should be used. Thus one can take fluid equations (Continuity and Euler) and Poisson equation for gravitational potential, linearize them in perturbations, and look for solutions in terms of plane waves. The resulting equation which governs evolution of a particular Fourier mode k (which corresponds to scale λ) is:

It is a standard equation of fluid instabilities, with competing gravitational and pressure terms; their equality defines the Jeans scale or Jeans mass. However, here we have an additional term called Hubble drag because the fluid in this case is embedded in a space which itself expands. It enters the equation as a friction term, effectively slowing down gravitational collapse. Here we can also see that in cosmological case, growth of structures can be suppressed via pressure of the fluid, but also if the space expands fast enough. Since above equation is a second order differential equation, we have two solutions, and for instable modes (larger than the Jeans) they are usually called growing (d+) and decaying mode (d-). Dark matter behaves as pressureless fluid, thus all modes are instable, and in classical fluid dynamics, these modes would exponentially grow (or decay), but here Hubble drag reduce their growth to a power law. The rate of growth depends on cosmic epoch (through the rate of expansion), and on the size of perturbations, so modes that are inside the horizon grow as:

Treatment of modes larger than the horizon can be done in general relativity, and the result is:

We see that in epoch of matter domination, all modes grow the same factor (~ a), while in radiation dominated epoch there is a big difference between modes that are inside the horizon – they do not grow at all, and modes which are larger than the horizon size, that grow rapidly. Also, horizon is expanding, thus modes are entering it, and as soon as they enter their growth gets suppressed. Therefore, we expect to measure a power spectrum which on large scales still have its primordial shape coming from the inflation (~ k, for n=1), but then it falls down as we look at modes that were inside horizon during radiation domination. Indeed, that is what is observed.

We see that during radiation epoch structures couldn’t form, but when the matter starts to dominate the content of the universe, the expansion slows down and structures will grow on all scales. At some point the density contrast will become of order unity, violating validity of linear theory, set at the beginning of this section. Even though there are analytical approaches to nonlinearity, they have to use simplifications, thus making their quantitative predictions unreliable. The only qualitatively correct and quantitatively accurate way to follow nonlinear evolution of structures is through numerical simulations.

**Nonlinear Regime**

The nonlinear evolution up to the current epoch, results in a universe full of structures – filaments, voids, and halos. In the pioneering work, Gregory and Thompson have observed such structures in 1978, and since then accuracy and sample size have been increasing, with now more than 140 000 galaxies mapped by the Sloan Digital Sky Survey. All these observations show that universe exhibits structure even on the very big scale (100 Mpc). In early times of numerical simulations (1980s), one would check if, for a given cosmology, simulations reproduce structures which by eye look the same as observations. If they do, it would be a proof that certain cosmological parameters are really describing our universe. That was important step in qualitative understanding of structure formation. Nowadays, simulations have considerably grown in size, and while they are still not able to reproduce formation of galaxies in ab initio large scale simulations (due to large computational resources required – but it should happened in next decade), making morphological studies impossible, they do result in a clumps of matter called halos which can be associated with real astronomical objects. One can perform a rigorous statistical study of all these objects, and check if their spatial distribution (correlation functions), distribution of masses (mass function), or density profiles match observations. Since the overall picture is understood well and the goals are to quantitatively prove our understandings, it is often said that we are in “precision cosmology” era. Through the collaboration between University of Illinois and Los Alamos National Laboratory, Katrin Heitmann, Salman Habib, Paul Ricker, and me, are doing numerical study of structure formation from very early times till today. For that purpose we use a particle-mesh code MC2, where matter is discretized in form of simulation particles which evolve self-gravitationally in the expanding universe. Large part of our effort is going into understanding and quantifying errors of simulations, domains of their result’s validity, and mutual agreement of different codes. If one’s goal is accuracy, these are all important issues, and they have been dangerously overlooked until recent years. For now, in our simulations we consider only the most dominant matter component, responsible for the overall dynamics – the dark matter. We examined “concordance” cosmology (n=1, h=0.7, 25.2% dark matter, 70% dark energy, baryons 4.8%), and we run 62 simulations with different box sizes (from 4 Mpc on a side to 256 Mpc), and with numerous number of realizations for each box size. The fact we have simulations spanning from small regions of the universe (43 Mpc3) up to a reasonably big ones (2563 Mpc3) enables us to probe very large range of masses and redshifts, and since we have a lot of statistical samples for each box size, we have excellent statistics. We have compared mass function starting at very high redshift (z=20) with analytical predictions and numerical fits, and our finding is that Warren et al. (2005) fit works the best over the whole redshift and mass range.