Cosmological Structure Formation A Short Course III. Structure Formation in the NonLinear Regime Chris Power 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 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? 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 Spherical Collapse Friedmanns equation can be written as dR 2 8G 2 R kc 2 = dt 3 Introduce the conformal time to simplify the solution of Friedmanns equation dt d = c R(t) Friedmanns equation can be rewritten as
dR 2 8G 0 R03 2 R kR = 3c 2 d Spherical Collapse We can introduce the constant 4G 0 R03 GM R* = = 2 2 3c c which helps to further simplify our differential 2 2
equation R R d R = 2 k R* R* d R* For an overdensity, k=-1 and so we obtain the following parametric equations for R and t R( ) = R* (1 cos ), R* t( ) = ( sin ) c Spherical Collapse
Can expand the solutions for R and t as power series in R* R( ) = R* (1 cos ), t( ) = ( sin ) c Consider the limit where is small; we can ignore higher order terms and approximate R and t by 2 2 R* 3 2 R( ) R* (1 ), t( ) =
(1 ) 2 12 c 6 20 We can relate t and to obtain 2 / 3 2 / 3 R* 6ct 1 6ct R(t) 1 2 R* 20 R* Spherical Collapse
Expression for R(t) allows us to deduce the growth of the perturbation at early times. 2/3 R* 6ct R(t ~ 0) 2 R* This is the well known result for an Einstein de Sitter Universe 1 (t ~ 0) = = 0 (t) 2 6Gt 9GM 1/ 3 2 / 3
= t 2 Can also look at the higher order term to obtain linear theory result 2 / 3 R 3 6ct =3 = R 20 R* Spherical Collapse Turnaround occurs at t=R*/c, when Rmax=2R*. At this time, the density enhancment relative to the background is (R* /2) 3 (6ct max /R* ) 2 9 2
= = 3 0 16 Rmax Can define the collapse time -- or the point at which the halo virialises -- as t=2R*/c, when Rvir=R*. In this 3 2 case vir (R* /2) (6ct vir /R* ) 2 = = 18 178 3 0
Rvir This is how simulators define the virial radius of a dark matter halo. Defining Dark Matter Haloes What do FOF Groups Correspond to? Compute virial mass - for LCDM cosmology, use an overdensity 97 criterion of ,4i.e. M vir = 3
crit rvir3 Good agreement between virial mass and FOF mass 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: (r) c = crit r / rs (1+ r /rs ) Most hotly debated paper of the last decade? Dark Matter Halo Mass
Dark Matter Halo Mass Profiles 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? 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. Numerical Consideration
s 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! The SemiAnalytic 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) 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 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
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