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So I have recently watched a simulation of dark matter density and gas temperature evolution in a universe. However I couldn't find description of under what assumptions it was was made and what it is telling us.

Simulation is here: http://www.illustris-project.org/movies/illustris_movie_cube_sub_frame.mp4

The description next to the video:

Time evolution of a 10Mpc (comoving) cubic region within Illustris, rendered from outside. The movies shows on the left the dark matter density field, and on the right the gas temperature (blue: cold, green: warm: white: hot). The rapid temperature fluctuations around massive haloes are due to radiative AGN feedback that is most active during quasar phases. The larger 'explosions' are due to radio-mode feedback.

Maybe some of you had spent some time thinking about these things and could answer some of my questions about simulation?

1) It seems where we have large dark matter density, we also have large temperature, meaning large matter density, is these are the regions new galaxies or stars forms?

2) It seems that dark matter density just builds up slowly in this net but do not move anywhere, does it mean that dark matter is slow? (non relativistic)

3) Stellar mass counter is increasing meaning that new suns are forming. Does this means that initially universe were more uniform in matter density, however some regions attracted more matter and started forming galaxies?

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Standard cosmology assumes homogeneity (as well as isotropy) of the universe. Though we know that this is only an approximation, we also know that for cosmic scales it is a good one. This can be seen in the cosmic microwave background radiation (CMB). However, the initially small inhomogeneities are responsible for galaxy and star formation through gravitational interaction. (This should answer your third question).

Careful: the next paragraph is full of speculations!
Whether these irregularities are (quantum) fluctuations in an initial system, say after inflation, or origin in clumps of dark matter is not known. If the latter is the case, this still raises the question why dark matter (DM) clustered in the first place. This question however is impossible to answer right now, since we do not know what DM actually is. To answer your first question, yes these are the regions where galaxies form, but we cannot say why.
However, it is unlikely that stars formed in a region of higher DM density, since the density is assumed to be rather homogeneous in the halo of a galaxy, especially in a small part, i.e. a solar system. Since DM does not interact electromagnetically, it doesn't scatter to build stars and planets like luminous matter.

The last sentence already kind of addresses your second question. Furthermore, there is a distinction between cold dark matter (CDM) and hot dark matter. While most theories favor CDM (Axions, WIMPs, MACHOs), which is nonrelativistic, there are also hot dark matter models with ultrarelativistic neutrinos.

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