1. Cosmological Structure Formation A Short Course III. Structure Formation in the Non-Linear Regime Chris Power

2. Recap Cosmological inflation provides mechanism for generating density perturbations… … which grow via gravitational instability Predictions of inflation consistent with temperature anisotropies in the Cosmic Microwave Background. Linear theory allows us to predict how small density perturbations grow, but breaks down when magnitude of perturbation approaches unity…

3. Key Questions What should we do when structure formation becomes non-linear? Simple physical model -- spherical or top-hat collapse Numerical (i.e. N-body) simulation What does the Cold Dark Matter model predict for the structure of dark matter haloes? When do the first stars from in the CDM model?

4. Spherical Collapse Consider a spherically symmetric overdensity in an expanding background. By Birkhoffs Theorem, can treat as an independent and scaled version of the Universe Can investigate initial expansion with Hubble flow, turnaround, collapse and virialisation

5. Spherical Collapse Friedmanns equation can be written as Introduce the conformal time to simplify the solution of Friedmanns equation Friedmanns equation can be rewritten as

6. Spherical Collapse We can introduce the constant which helps to further simplify our differential equation For an overdensity, k=-1 and so we obtain the following parametric equations for R and t

7. Spherical Collapse Can expand the solutions for R and t as power series in Consider the limit where is small; we can ignore higher order terms and approximate R and t by We can relate t and to obtain

8. Spherical Collapse Expression for R(t) allows us to deduce the growth of the perturbation at early times. This is the well known result for an Einstein de Sitter Universe Can also look at the higher order term to obtain linear theory result

9. Spherical Collapse Turnaround occurs at t= R * /c, when R max =2R *. At this time, the density enhancment relative to the background is Can define the collapse time -- or the point at which the halo virialises -- as t=2 R * /c, when R vir =R *. In this case This is how simulators define the virial radius of a dark matter halo.

10. Defining Dark Matter Haloes

11. What do FOF Groups Correspond to? Compute virial mass - for LCDM cosmology, use an overdensity criterion of, i.e. Good agreement between virial mass and FOF mass

12. Dark Matter Halo Mass Profiles Spherical averaged. Navarro, Frenk & White (1996) studied a large sample of dark matter haloes Found that average equilibrium structure could be approximated by the NFW profile: Most hotly debated paper of the last decade?

13. Most actively researched area in last decade! Now understand effect of numerics. Find that form of profile at small radii steeper than predicted by NFW. Is this consistent with observational data? Dark Matter Halo Mass Profiles

14. What about Substructure? High resolution simulations reveal that dark matter haloes (and CDM haloes in particular) contain a wealth of substructure. How can we identify this substructure in an automated way? Seek gravitationally bound groups of particles that are overdense relative to the background density of the host halo.

15. Numerical Considerations We expect the amount of substructure resolved in a simulation to be sensitive to the mass resolution of the simulation Efficient (parallel) algorithms becoming increasingly important. Still very much work in progress!

16. The Semi-Analytic Recipe Seminal papers by White & Frenk (1991) and Cole et al (2000) Track halo (and galaxy) growth via merger history Underpins most theoretical predictions Foundations of Mock Catalogues (e.g. 2dFGRS)

17. Dark matter haloes must have been massive enough to support molecular cooling This depends on the cosmology and in particular on the power spectrum normalisation First stars form earlier if structure forms earlier Consequences for Reionisation The First Stars

18. Some Useful Reading General Cosmology : The Origin and Structure of the Universe by Coles and Lucchin Physical Cosmology by John Peacock Cosmological Inflation Cosmological Inflation and Large Scale Structure by Liddle and Lyth Linear Perturbation Theory Large Scale Structure of the Universe by Peebles