cosmological structure formation a short course ii. the growth of cosmic structure chris power
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TRANSCRIPT
Cosmological Structure Formation
A Short Course
II. The Growth of Cosmic Structure
Chris Power
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
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.
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 €
d2a
dt 2> 0
€
d(H−1 /a)
dt< 0
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
€
N(t) = lna(tend )
a(t)
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.
The Seeds of Structure
Temperature Fluctuations in the Cosmic Microwave Background Credit: NASA/WMAP Science Team (http://map.gsfc.nasa.gov)
Temperature and Density Pertubations
• CMB corresponds to the last scattering surface of the radiation -- prior to recombination Universe was a hot plasma -- at z~1400, atoms could recombine.
• Temperature variations correspond to density perturbations present at this time -- the Sachs-Wolfe effect:
€
∂T
T=
Φ
c 2−
∂a
a=
Φ
3c 2
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.
€
(x, t) =ρ(x, t) − ρ (t)
ρ (t)
€
(x, t) = Σk
ˆ δ (k, t) e ikx
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
€
ξ(r) = δ(r)δ(r + x)
€
P(k) = |δk |2
€
P(k) = Ak nT(k)2
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.
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
• Poisson’s equation
€
Dρ
Dt= −ρ∇.v
€
Dv
Dt= −
∇p
ρ−∇Φ
€
∇Φ=4πGρ
Linear Perturbation Theory
• Find that• the continuity equation leads to
• the Euler equation leads to
• Poisson’s equation leads to
€
dδ
dt= −∇.δv
€
dδv
dt= −Hδv −
∇δp
ρ 0
−∇δΦ
€
∇Φ=4πGρ 0δ
Linear Perturbation Theory
• Combine these equations to obtain the growth equation
• Can take Fourier transform to investigate how different modes grow€
d2δ
dt 2+ 2H
dδ
dt= 4πGρ 0δ +
cs2∇ 2δ
a2
€
d2δk
dt 2+ 2H
dδk
dt= 4πGρ 0δk +
cs2k 2δk
a2
Linear Perturbation Theory
• Linear theory valid provided the size of perturbations is small -- <<1
• 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.
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
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