Cosmological Structure Formation A Short Course

1. Cosmological Structure FormationA 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 (http://map.gsfc.nasa.gov)

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 • “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