1. 2dF Galaxy Redshift Survey ¼ M galaxies 2003
1/4 of the horizon
CFA Survey 1980

2. The Initial Fluctuations
At Inflation: Gaussian, adiabatic
a realization of an ensemble
fluctuation field
ensembe average ~ volume average
Fourier
Power Spectrum
rms M δ k Pk

3. Scale-Invariant Spectrum (Harrison-Zel’dovich)
mass M t
time  M

4. Cosmological Scales
mass
teq
zeq~104 t0
time

5.  growth when matter is self-gravitating
CDM Power Spectrum
mass t
time
teq
 growth when matter is self-gravitating 
Meq
mass
CDM Pk
kpeak  m h k
CDM
HDM
HDM free streeming

6. Formation of Large-Scale Structure
Fluctuation growth in the linear regime:
rms fluctuation at mass scale M:
Typical objects forming at t:
example M 1
CDM: bottom-up M 1
free streaming
HDM: top-down

7. Power Spectrum

8. ΛCDM Power Spectrum
normalization:

9. Non-linear Growth of Structure
Lecture
Non-linear Growth of Structure
Spherical Collapse,
Virial Theorem,
Zel’dovich Approximation,
N-body Simulations

12. Formation of Large-Scale Structure: comoving

13. היווצרות הגלקסיות המוקדמות מהמארג הקוסמי

14. Formation of Large-Scale Structure: comoving
30 Mpc

15. Filamentary Structure: Zel’dovich Approximation
Approximate the displacement from initial position
Velocity & acceleration along displacement → trajectories straight lines
In physical coordinates
as in linear
central force → potential flow
Density (Lagrangian):
deformation tensor eigenvalues
continuity
Jacobian
→ caustics
pancake 42%
cluster 8%
filament 42%

16. Zel’dovich Approximation cont’d
linear
→ D is the growing mode of GI obeying
plug density in Poisson eq.
error small in linear regime
or pancakes
error big in spherical collapse
Error:
→ error is 2nd +3rd terms

17. Non-dissipative Pancakes: why flat?R
oscilation time << expansion time
adiabatic invariant
pancake becomes flatter in time

18. N-body simulation
CDM
N-body simulation

19. N-body simulation

20. N-body simulation

21. spherical collapse or mergers
N-body simulation
spherical collapse or mergers

22. spherical collapse or mergers
N-body simulation
spherical collapse or mergers

23. spherical collapse or mergers
N-body simulation
spherical collapse or mergers

24. xxx

25. N-body simulation of Halo Formation

26. N-body simulation of Halo Formation

27. The Friedman Equation Einstein’s equations For the isotropic RW metric
Newton’s gravity: space fixed, external force determining motions
Einstein’s equations
left side of E’s eq. is the most general function of g and its 1st and 2nd time derivatives that reduces to Newton’s equation
Gravity is an intrinsic property of space-time. geometry <-> energy density. Particles move on geodesics (local straight lines) determined by the local curvature.
For the isotropic RW metric
Stress-energy tensor
Einstein’s tensor
energy conservation
eq. of motion
conservation of number of photons
mass conservation
A differential equation for a(t)

28. Solutions of Friedman eq. (matter era)
radiation era
acceleration
expansion
forever
conformal time
collapse
critical density
big bang
here & now

29. Light travel in a closed universe
A photon is emitted at the origin (ue=0) right after the big-bang (te=0)
photon:
conformal time
π/ π
north pole u south pole 2π
3π/2 π
π/2 η
big bang
big crunch t

30. Top-Hat Model (Λ=0, matter era)
a bound sphere (k=1) in EdS universe (k=0)
conformal time
tmax tvir
universe
perturbation
overdensity:
5.55
linear perturbation
Taylor
turnaround
linear equivalent to collapse

31. 8
universe
200 4
5.55
perturbation
tmax tvir t

32. Spherical Collapse universe radius perturbation virial equilibrium
time
radius
virial equilibrium
virial equilibrium:
perturbation

33. Virial Scaling Relations
Virial equilibrium:
Spherical collapse:
Weak dependence on time of formation:
Practical formulae:

34. Hierarchical Clustering
Lecture 6
Hierarchical Clustering
Press Schechter Formalism

35. Press Schechter Formalism
halo mass function n(M,a)
Gaussian random field
random spheres of mass M
linear-extrapolated δrms at a:
a0 =1
a(t)
nonlinear σ
linear
fraction of spheres with δ>δc =1.68:
PS ansaz: F is the mass fraction in halos >M (at a) δc x δ
derivative of F with respect to M:
Mo & White 2002

36. Press Schechter Formalism cont.
self-similar evolution, scaled with M*
log M
log n(M)
M*(a)
Example:
time Pk
approximate
Top Hat R x k
kR=π
in a flat universe

37. Press Schechter cont. Comparison of PS to N-body simulations
Better fit using ellipsoidal collapse (Sheth & Tormen 2002)
log M
log M2n(M)
Comparison of PS to N-body simulations
factor 2

38. Press-Schechter in ΛCDM14 13 12 11 10 9
log M/Mʘ 1σ 2σ
Mo & White 2002

39. Press-Schechter
Mo & White 2002

40. Merger Tree t

41. Mass versus Light Distribution
halo mass
galaxy
stellar mass
40% of baryons