Cosmological Structure Formation

A Short Course

III. Structure Formation in the Non-Linear Regime

Chris Power

Recap• Cosmological inflation provides mechanism

for generating density perturbations…• … which grow via gravitational instability• Predictions of inflation consistent with

temperature anisotropies in the Cosmic Microwave Background.

• Linear theory allows us to predict how small density perturbations grow, but breaks down when magnitude of perturbation approaches unity…

Key Questions• What should we do when structure formation

becomes non-linear?• Simple physical model -- spherical or “top-hat” collapse

• Numerical (i.e. N-body) simulation

• What does the Cold Dark Matter model predict for the structure of dark matter haloes?

• When do the first stars from in the CDM model?

Spherical Collapse• Consider a

spherically symmetric overdensity in an expanding background.

• By Birkhoff’s Theorem, can treat as an independent and scaled version of the Universe

• Can investigate initial expansion with Hubble flow, turnaround, collapse and virialisation

Spherical Collapse• Friedmann’s equation can be written as

• Introduce the conformal time to simplify the solution of Friedmann’s equation

• Friedmann’s equation can be rewritten as

€

dRdt ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

= 8πG3ρR2 − kc 2

€

dη = c dtR(t)

€

dRdη ⎛ ⎝ ⎜

⎞ ⎠ ⎟2

= 8πGρ 0R03

3c 2 R − kR2

Spherical Collapse• We can introduce the constant

which helps to further simplify our differential

equation

• For an overdensity, k=-1 and so we obtain the following parametric equations for R and t

€

R* = 4πGρ 0R03

3c 2 = GMc 2

€

ddη

RR*

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥2

= 2 RR*

⎛ ⎝ ⎜

⎞ ⎠ ⎟− k R

R*

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

€

R(η ) = R*(1− cosη ), t(η ) = R*

c(η − sinη )

Spherical Collapse• Can expand the solutions for R and t as power series in

• Consider the limit where is small; we can ignore higher order terms and approximate R and t by

• We can relate t and to obtain

€

R(η ) = R*(1− cosη ), t(η ) = R*

c(η − sinη )

€

R(η ) ≈ R*η 2

2(1− η

2

12), t(η ) = R*

cη 3

6(1− η

2

20)

€

R(t) ≈ R*

26ctR*

⎛ ⎝ ⎜

⎞ ⎠ ⎟2 / 3

1− 120

6ctR*

⎛ ⎝ ⎜

⎞ ⎠ ⎟2 / 3 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Spherical Collapse• Expression for R(t) allows us to deduce the growth of the

perturbation at early times.

• This is the well known result for an Einstein de Sitter Universe

• Can also look at the higher order term to obtain linear theory result

€

R(t ~ 0) ≈ R*

26ctR*

⎛ ⎝ ⎜

⎞ ⎠ ⎟2 / 3

= 9GM2

⎛ ⎝ ⎜

⎞ ⎠ ⎟

1/ 3

t 2 / 3

€

ρ(t ~ 0) = 16πGt 2 = ρ 0(t)

€

δρρ

=−3δRR

= 320

6ctR*

⎛ ⎝ ⎜

⎞ ⎠ ⎟2 / 3

Spherical Collapse• Turnaround occurs at t=R*/c, when Rmax=2R*. At this time,

the density enhancment relative to the background is

• Can define the collapse time -- or the point at which the halo virialises -- as t=2R*/c, when Rvir=R*. In this case

• This is how simulators define the virial radius of a dark matter halo.

€

ρρ0

= (R* /2)3(6ctmax /R*)2

Rmax3 = 9π 2

16

€

ρvirρ 0

= (R* /2)3(6ctvir /R*)2

Rvir3 =18π 2 ≈178

Defining Dark Matter Haloes

What do FOF Groups Correspond to?

• Compute virial mass - for LCDM cosmology, use an overdensity criterion of , i.e.

• Good agreement between virial mass and FOF mass

€

Δ ≈97

€

Mvir = 4π3

Δ ρ crit rvir3

Dark Matter Halo Mass Dark Matter Halo Mass ProfilesProfiles

Spherical averaged.

Navarro, Frenk & White (1996) studied a large sample of dark matter haloes

Found that average equilibrium structure could be approximated by the NFW profile:

Most hotly debated paper of the last decade?

€

ρ(r)ρ crit

= δcr /rs(1+ r /rs)

• Most actively researched area in last decade!

• Now understand effect of numerics.

• Find that form of profile at small radii steeper than predicted by NFW.

• Is this consistent with observational data?

Dark Matter Halo Mass ProfilesDark Matter Halo Mass ProfilesDark Matter Halo Mass Profiles

What about Substructure?

• High resolution simulations reveal that dark matter haloes (and CDM haloes in particular) contain a wealth of substructure.

• How can we identify this substructure in an automated way?

• Seek gravitationally bound groups of particles that are overdense relative to the background density of the host halo.

Numerical Consideration

s

• We expect the amount of substructure resolved in a simulation to be sensitive to the mass resolution of the simulation

• Efficient (parallel) algorithms becoming increasingly important.

• Still very much work in progress!

The Semi-

Analytic Recipe

• Seminal papers by White & Frenk (1991) and Cole et al (2000)

• Track halo (and galaxy) growth via merger history

• Underpins most theoretical predictions

• Foundations of Mock Catalogues (e.g. 2dFGRS)

• Dark matter haloes must have been massive enough to support molecular cooling

• This depends on the cosmology and in particular on the power spectrum normalisation

• First stars form earlier if structure forms earlier

• Consequences for Reionisation

The First Stars

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