Cosmological Structure Formation A Short Course

Cosmological Structure Formation A Short Course

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

Recently Viewed Presentations

  • Pipelining - faculty.utrgv.edu

    Pipelining - faculty.utrgv.edu

    Pipelining defined. Multiple instructions are overlapped in execution. Divide the work of the cpu as equally as possible. MIPS lends itself to pipelining because the only operations that affect memory are load and store and the instruction size is the...
  • Game Design Documents - WPI

    Game Design Documents - WPI

    IMGD 1001: Game Design Documents * * * * * * * IMGD 1001 * Types of Game Design Docs Concept Document Proposal Document Technical Specification Game Design Document Level Designs IMGD 1001 * Concept Document (1 of 2) Used...
  • Mythological Creatures Mythological Creatures Monsters and hybrids (human-animal

    Mythological Creatures Mythological Creatures Monsters and hybrids (human-animal

    Charybdis. Charybdis (Greek mythology) is one of several Greek monsters that appeared in multiple famous myths. She is often known only in her most vicious form - a swirling whirlpool of death that swallowed enormous amounts of water and anything...
  • 5-Minute Check on Activity 4-8 1. What point

    5-Minute Check on Activity 4-8 1. What point

    Activity 4 - 9 The Power of Power Functions Objectives Identify a direct variation function Determine the constant of variation Identify the properties of graphs of power functions defined by y = kxn, where n is a positive integer, k...
  • Unit 3 Review and Mystery Prize Thing

    Unit 3 Review and Mystery Prize Thing

    Carbon and hydrogen have similar electronegativities so they share electrons when they bond. Both the hydrogen and the carbon end up with filled outer energy levels. Objective #32. Predict what type of bond would form between 2 atoms using the...
  • 7th Grade Research Paper Everything you need to

    7th Grade Research Paper Everything you need to

    How do Search Engines Work? "Spiders" or "Robots (bots)" crawl the Web to find your information. Web masters must tell the search engine what their Webpage contains or the "spiders" will not find their page. Search engines do not search...
  • Chapter 1: Introduction to Chemistry

    Chapter 1: Introduction to Chemistry

    The responding variable (dependent variable) is the variable that you observe during an experiment. A good experiment only has one manipulated variable. Theory vs. Law. A theory is a well-tested explanation for a broad set of observations.
  • PowerPoint-Präsentation

    PowerPoint-Präsentation

    Jahresversammlung 2018 Herzlich willkommen! 23.11.2018 * Jahresversammlung Ausklang Die Heilbronner Bürgerstiftung bedankt sich bei allen Förderern und Freunden!