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

€

dR

dt

⎛

⎝ ⎜

⎞

⎠ ⎟2

=8πG

3ρR2 − kc 2

€

dη = cdt

R(t)

€

dR

dη

⎛

⎝ ⎜

⎞

⎠ ⎟

2

=8πGρ 0R0

3

3c 2R − 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 0R0

3

3c 2=GM

c 2

€

d

dη

R

R*

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

2

= 2R

R*

⎛

⎝ ⎜

⎞

⎠ ⎟− 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*

c

η 3

6(1−

η 2

20)

€

R(t) ≈R*

2

6ct

R*

⎛

⎝ ⎜

⎞

⎠ ⎟

2 / 3

1−1

20

6ct

R*

⎛

⎝ ⎜

⎞

⎠ ⎟

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*

2

6ct

R*

⎛

⎝ ⎜

⎞

⎠ ⎟

2 / 3

=9GM

2

⎛

⎝ ⎜

⎞

⎠ ⎟

1/ 3

t 2 / 3

€

ρ(t ~ 0) =1

6πGt 2= ρ 0(t)

€

δρρ

=−3δR

R=

3

20

6ct

R*

⎛

⎝ ⎜

⎞

⎠ ⎟

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 =4π

3Δ ρ crit rvir

3

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