1. The Theory/Observation connection lecture 3 the (non-linear) growth of structure Will Percival The University of Portsmouth

2. Lecture outline  Spherical collapse – standard model – dark energy  Virialisation  Press-Schechter theory – the mass function – halo creation rate  Extended Press-Schechter theory  Peaks and the halo model

3. Phases of perturbation evolution Inflation linearNon-linear Transfer function Matter/Dark energy domination

4. Linear vs Non-linear behaviour z=0 z=1 z=2 z=3 z=4 z=5 linear growth non-linear evolution z=0 z=1 z=2 z=3 z=4 z=5 large scale power is lost as fluctuations move to smaller scales P(k) calculated from Smith et al. 2003, MNRAS, 341,1311 fitting formulae

5. Spherical collapse  homogeneous, spherical region in isotropic background behaves as a mini-Universe (Birkhoff’s theorem)  If density high enough it behaves as a closed Universe and collapses (r  0)  Friedmann equation in a closed universe (no DE)  Symmetric in time  Starts at singularity (big bang), so ends in singularity  Two parameters: – density (  m ), constrains collapse time – scale (e.g. r 0 ), constrains perturbation size

6. The evolution of densities in the Universe Critical densities are parameteric equations for evolution of universe as a function of the scale factor a All cosmological models will evolve along one of the lines on this plot (away from the EdS solution)

7. Spherical collapse Contain equal mass collapsing perturbation Radius a p Background Radius a Set up two spheres, one containing background, and one with an enhanced density

8. Spherical collapse For collapsing Lambda Universe, we have Friedmann equation And the collapse requirement Can integrate numerically to find collapse time, but if no Lambda can do this analytically t coll apapapap  p is the curvature of the perturbation

9. Spherical collapse Problem: need to relate the collapse time t coll to the overdensity of the perturbation in the linear field (that we now think is collapsing).

10. Spherical collapse Problem: need to relate the collapse time t coll to the overdensity of the perturbation in the linear field (that we now think is collapsing). At early times (ignore DE), can write Friedmann equation as Obtain series solution for a For the background, Different for perturbation  p So that

11. Spherical collapse Can now linearly extrapolate the limiting behaviour of the perturbation at early times to present day Can use numerical solution for t coll, or can use analytic solution (if no Lambda) If  k =0,  m =1, then we get the solution, for perturbations that collapse at present day Problem: need to relate the collapse time t coll to the overdensity of the perturbation in the linear field (that we now think is collapsing).

12. Evolution of perturbations top-hat collapse limit for collapse evolution of scale factor W m =0.3, W v =0.7, h=0.7, w= - 1 virialisation

13. Spherical collapse Cosmological dependence of  c is small, so often ignored, and  c =1.686 is assumed

14. Spherical collapse: how to include DE? high sound speed means that DE perturbations are rapidly smoothed DE DM on large scales dark energy must follow Friedmann equation – this is what dark energy was postulated to fix! low sound speed means that large scale DE perturbations are important DE DM quintessence has ultra light scalar field so high sound speed The effect of the sound speed provides a potential test of gravity modifications vs stress-energy. If DE is not a cosmological constant, its sound speed controls how it behaves

15. Spherical collapse: general DE cosmology equation depends on the equation of state of dark energy p = w(a)  homogeneous dark energy means that this term depends on scale factor of background “perfectly” clustering dark energy – replace a with a p can solve differential equation and follow growth of perturbation directly from coupled cosmology equations For general DE, cannot write down a Friedmann equation for perturbations, because energy is not conserved. However, can work from cosmology equation

16. Evolution of perturbations top-hat collapse limit for collapse evolution of scale factor W m =0.3, W v =0.7, h=0.7, w= - 1 virialisation

17. Evolution of perturbations top-hat collapse limit for collapse evolution of scale factor W m =0.3, W v =0.7, h=0.7, w= - 2/3 virialisation

18. Evolution of perturbations top-hat collapse limit for collapse evolution of scale factor W m =0.3, W v =0.7, h=0.7, w= - 4/3 virialisation

19.  Real perturbations aren’t spherical or homogeneous  Collapse to a singularity must be replaced by virialisation  Virial theorem:  For matter and dark energy  If there’s only matter, then comparing total energy at maximum perturbation size and virialisation gives

20. virialisation The density contrast for a virialised perturbation at the time where collapse can be predicted for an Einstein-de Sitter cosmology This is often taken as the definition for how to find a collapsed object

21. Aside: energy evolution in a perturbation in a standard cosmological constant cosmology, we can write down a Friedmann equation for a perturbation for dark energy “fluid” with a high sound speed, this is not true – energy can be lost or gained by a perturbation the potential energy due to the matter U G and due to the dark energy U X

22. Press-Schechter theory  Builds on idea of spherical collapse and the overdensity field to create statistical theory for structure formation – take critical density for collapse. Assume any pertubations with greater density (at an earlier time) have collapsed – Filter the density field to find Lagrangian size of perturbations. If collapse on more than one scale, take largest size  Can be used to give – mass function of collapsed objects (halos) – creation time distribution of halos – information about the build-up of structure (extended PS theory)

23. The mass function in PS theory Smooth density field on a mass scale M, with a filter Result is a set of Gaussian random fields with variance  2 (M). For each location in space we have an overdensity for each smoothing scale: this forms a “trajectory”: a line of  as a function of  2 (M).

24. The mass function in PS theory For sharp k-space filtering, the overdensity of the field at any location as a function of filter radius (through  2 (M) ), forms a Brownian random walk We wish to know the probability that we should associate a point with a collapsed region of mass >M At any mass it is equally likely that a trajectory is now below or below a barrier given that it previously crossed it, so Where

25. The mass function in PS theory Differentiate in M to find fraction in range dM and multiply by  /M to find the number density of all halos. PS theory assumes (predicts) that all mass is in halos of some (possibly small) mass High Mass: exponential cut-off for M>M *, where Low Mass: divergence

26. The mass function The PS mass function is not a great match to simulation results (too high at low masses and low at high masses), but can be used as a basis for fitting functions Sheth & Tormen (1999) Jenkins et al. (2001) PS theory - dotted Sheth & Toren - dashed

27. Halo creation rate in PS theory Can also use trajectories in PS theory to calculate when halos of a particular mass collapse This is the distribution of first upcrossings, for trajectories that have an upcrossing for mass M For an Eistein-de Sitter cosmology,

28. Creation vs existence Formation rate of galaxies per comoving volume Redshift distribution of halo number per comoving volume

29. Extended Press-Schechter theory Extended PS theory gives the conditional mass function, useful for merger histories Given a halo of mass M 1 at z 1, what is the distribution of masses at z 2 ? Can simply translate origin - same formulae as before but with  c and  m shifted

30. Problems with PS theory   Mass function doesn’t match N-body simulations   Conditional probability is lop-sided f(M 1,M 2 |M) ≠ f(M 2,M 1 |M)   Is it just too simplistic? MM1MM1 MM2MM2 M

31. Halo bias  If halos form without regard to the underlying density fluctuation and move under the gravitational field then their number density is an unbiased tracer of the dark matter density fluctuation  This is not expected to be the case in practice: spherical collapse shows that time depends on overdensity field  A high background enhances the formation of structure  Hence peak-background split

32. Peak-background split Split density field into peak and background components Collapse overdensity altered Alters mass function through

33. Peak-background split Get biased formation of objects Need to distinguish Lagrangian and Eulerian bias: densities related by a factor (1+  b ), and can take limit of small  b For PS theory For Sheth & Tormen (1999) fitting function

34. Halo clustering strength on large scales

35. Bias on small scales comes from halo profile N-body gives halo profile: r = [ y(1+y) 2 ] -1 ; y = r/r c (NFW) r = [ y 3/2 (1+y 3/2 ) ] -1 ; y = r/r c (Moore) (cf. Isothermal sphere r = 1/y 2 )

36. The halo model M=10 15 M=10 10 linear non-linear bound objects galaxies large scale clustering small scale clustering Predicts power spectrum of the form Simple model that splits matter clustering into 2 components small scale clustering of galaxies within a single halo large scale clustering of galaxies in different halos

37. Further reading  Peacock, “Cosmological Physics”, Cambridge University Press  Coles & Lucchin, “Cosmology: the origin and evolution of cosmic structure”, Wiley  Spherical collapse in dark energy background – Percival 2005, A&A 443, 819  Press-Schechter theory – Press & Schechter 1974, ApJ 187, 425 – Lacey & Cole 1993, MNRAS 262, 627 – Percival & Miller 1999, MNRAS 309, 823  Peaks – Bardeen et al (BBKS) 1986, ApJ 304, 15  halo model papers – Seljak 2000, MNRAS 318, 203 – Peacock & Smith 2000, MNRAS 318, 1144 – Cooray & Sheth 2002, Physics reports, 372, 1