Cosmological Structure Formation A Short Course

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Cosmological Structure FormationA Short Course

• III. Structure Formation in the Non-Linear Regime
• Chris Power

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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

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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?

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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

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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

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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

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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

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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

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Spherical Collapse

• Turnaround occurs at t?R/c, when Rmax2R. 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 t2?R/c, when
  RvirR. In this case
• This is how simulators define the virial radius
  of a dark matter halo.

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Defining Dark Matter Haloes
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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

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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?

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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?

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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.

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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!

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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)

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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

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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