Cosmological Structure Formation A Short Course

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Cosmological Structure FormationA Short Course

• II. The Growth of Cosmic Structure
• Chris Power

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Recap

• The Cold Dark Matter model is the standard
  paradigm for cosmological structure formation.
• Structure grows in a hierarchical manner -- from
  the bottom-up -- from small density
  perturbations via gravitational instability
• Cold Dark Matter particles assumed to be
  non-thermal relics of the Big Bang

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Key Questions

• Where do the initial density perturbations come
  from?
• Quantum fluctuations imprinted prior to
  cosmological inflation.
• What is the observational evidence for this?
• Angular scales greater than 1 in the Cosmic
  Microwave Background radiation.
• How do these density perturbations grow in to the
  structures we observe in the present-day
  Universe?
• Gravitational instability in the linear- and
  non-linear regimes.

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Cosmological Inflation

• Occurs very early in the history of the Universe
  -- a period of exponential expansion, during
  which expansion rate was accelerating
• or alternatively, during which comoving
  Hubble length is a decreasing function of time

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Cosmological Inflation

• Prior to inflation, thought that the Universe was
  in a chaotic state -- inflation wipes out this
  initial state.
• Small scale quantum fluctuations in the vacuum
  stretched out by exponential expansion -- form
  the seeds of the primordial density
  perturbations.
• Can quantify the amount of inflation in terms
  of the number of e-foldings it leads to

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Cosmological Inflation

• Turns out the 70 e-foldings are required to
  solve the so-called classical cosmological
  problems
• Flatness
• Horizon
• Abundance of relics -- such as magnetic monopoles
• Homogeneity and Isotropy
• Inflation thought to be driven by a scalar field,
  the inflaton -- could it also be responsible for
  the accelerated expansion (i.e. dark energy) we
  see today?
• Turns out that angular scales larger than 1º in
  the CMB are relevant for testing inflation --
  also expect perturbations to be Gaussian.

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The Seeds of Structure

• Temperature Fluctuations in the Cosmic Microwave
  Background
• Credit NASA/WMAP Science
  Team (http//map.gsfc.nasa.gov)

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Temperature and Density Pertubations

• CMB corresponds to the last scattering surface of
  the radiation -- prior to recombination Universe
  was a hot plasma -- at z1400, atoms could
  recombine.
• Temperature variations correspond to density
  perturbations present at this time -- the
  Sachs-Wolfe effect

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Characterising Density Perturbations

• We define the density at location x at time t by
• This can be expressed in terms of its Fourier
  components
• Inflation predicts that ? can be characterised as
  a Gaussian random field.

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Gaussian Random Fields

• The properties of a Gaussian Random Field can be
  completely specified by the correlation function
• Common to use its Fourier transform, the Power
  Spectrum
• Expressible as

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Aside Setting up Cosmlogical Simulations

• Generate a power spectrum -- this fixes the dark
  matter model.
• Generate a Gaussian Random density field using
  power spectrum.
• Impose density field d(x,y,z) on particle
  distribution -- i.e. assignment displacements and
  velocities to particles.

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Linear Perturbation Theory

• Assume a smooth background -- how do small
  perturbations to this background evolve in time?
• Can write down
• the continuity equation
• the Euler equation
• Poissons equation

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Linear Perturbation Theory

• Find that
• the continuity equation leads to
• the Euler equation leads to
• Poissons equation leads to

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Linear Perturbation Theory

• Combine these equations to obtain the growth
  equation
• Can take Fourier transform to investigate how
  different modes grow

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Linear Perturbation Theory

• Linear theory valid provided the size of
  perturbations is small -- ?ltlt1
• When ?1, can no longer trust linear theory
  predictions -- problem becomes non-linear and we
  enter the non-linear regime
• Possible to deduce the approximate behaviour of
  perturbations in this regime by using a simple
  model for the evolution of perturbations -- the
  spherical collapse model
• However, require cosmological simulations to
  fully treat gravitational instability.

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Next Lecture

• The Spherical Collapse Model
• Defining a dark matter halo
• The Structure of Dark Matter Haloes
• The mass density profile -- the Navarro, Frenk
  White universal profile
• The Formation of the First Stars
• First Light and Cosmological 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