cosmological structure formation a short course
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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 - PowerPoint PPT PresentationTRANSCRIPT
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