Cosmological Structure Formation A Short Course

1. Cosmological Structure FormationA 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 magnitudeof 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 Birkhoff’s 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 • Friedmann’s equation can be written as • Introduce the conformal time to simplify the solution of Friedmann’s equation • Friedmann’s 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 Rmax=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 Rvir=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. Dark Matter Halo Mass Profiles Dark Matter Halo Mass Profiles • 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?

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. The First Stars • 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

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