Transcript Cosmological Structure Formation A Short Course

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
dR2 8G 2
R  kc 2
  
 dt 
3
•
Introduce the conformal time to simplify the solution of Friedmann’s
equation
dt
d  c
R(t)

•
Friedmann’s equation can be rewritten as

dR2 8G0R03
2
R

kR
  
3c 2
d 
Spherical Collapse
•
We can introduce the constant
4G0 R03 GM
R* 
 2
2
3c
c
which helps to further simplify our differential equation
2
 d  R 
 R   R 
    2  k 
R*  R* 
d R* 
2

•
For an overdensity, k=-1 and so we obtain the following parametric
equations for R and t
 R()  R (1 cos),
*
R*
t()  (  sin)
c
Spherical Collapse
•
Can expand the solutions for R and t as power series in 
R*
R()  R* (1 cos), t()  (  sin)
c
•
Consider the limit where  is small; we can ignore higher order terms and
approximate R and t by
2

•
2
R*  3
2
R()  R* (1 ), t() 
(1 )
2
12
c 6
20
We can relate t and  to obtain
R* 6ct 
R(t)   
2  R* 
2/3

2 / 3 



1 1 6ct  
 20  R  
*


Spherical Collapse
•
Expression for R(t) allows us to deduce the growth of the perturbation at early
times.
2/3
1/ 3




R* 6ct
9GM
2/3
R(t ~ 0)     
 t
 2 
2  R* 
•
This is the well known result for an Einstein de Sitter Universe
1
(t ~ 0) 
 0 (t)
2
6Gt

• Can also look at the higher order term to obtain linear theory result

2/3



R 3 6ct
 3   

R 20  R* 
Spherical Collapse
•
Turnaround occurs at t=R*/c, when Rmax=2R*. At this time, the density
enhancment relative to the background is
 (R* /2)3 (6ctmax /R* )2 9 2


3
0
16
Rmax
•
Can define the collapse time -- or the point at which the halo virialises -as t=2R*/c, when Rvir=R*. In this case

•
vir (R* /2) 3 (6ctvir /R* ) 2
2


18

 178
3
0
Rvir
This is how simulators define the virial radius of a dark matter halo.

Defining Dark Matter Haloes
What do FOF Groups Correspond to?
• Compute virial mass for LCDM cosmology,
use an overdensity
criterion of
 , i.e.
97
4
• Good agreement
3
M



r
vir virial mass
crit vir
between
3
and FOF mass



Dark Matter Halo Mass Profiles

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:

(r)
c

crit r /rs (1 r /rs )

Most hotly debated paper of
the last decade?
Dark
HaloMass
MassProfiles
Profiles
Dark Matter
Matter Halo
• 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?
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
Considerations
• 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)
The First Stars
• 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
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