1. Cosmological Structure Formation A Short Course
II. The Growth of Cosmic Structure
Chris Power

2. 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

3. 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.

4. 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

5. 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

6. 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.

7. The Seeds of Structure
Temperature Fluctuations in the Cosmic Microwave Background
Credit: NASA/WMAP Science Team (

8. 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:

9. 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.

10. 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

11. 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.

12. 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

13. Linear Perturbation Theory
Find that
the continuity equation leads to
the Euler equation leads to
Poisson’s equation leads to

14. Linear Perturbation Theory
Combine these equations to obtain the growth equation
Can take Fourier transform to investigate how different modes grow

15. 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.

16. 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

17. Some Useful Reading General Cosmological Inflation
“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