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