2dF Galaxy Redshift Survey ¼ M galaxies 2003

1. 2dF Galaxy Redshift Survey ¼ M galaxies 2003 CFA Survey 1980

2. The Initial Fluctuations At Inflation: Gaussian, adiabatic a realization of an ensemble ensembe average ~ volume average   ( ) x     fluctuation field  ( ) x        k i x     ~   ( ) Fourier x ke  Power Spectrum 2| ) n ( ) | ( P k k k      k   ~ 2 / K K K rms      2 3 3 3 ~ exp[ ( ) ' ] ' ~ i k k  x d k d k d k   k 0   0          ' k k k k  ' 0 k k   ( ) ' k k  Dirac   2 /  2 3 ( 3 / ) 3  n     2| P d k M |           k k k k  0 k δ Pk  3 / 2  1 n M    / 1  2 0  n M    3 . n const   M k

3. Scale-Invariant Spectrum (Harrison-Zel’dovich) R  t mass  MH t Horizon 3 2   a t  H t m M time t 2 / 3 t   2 / 3 2 / 3     ( , ) M t M t        H ( ) t M H  . const H  1 n P k  k 2 / 3   M M

4. Cosmological Scales mass ~ R  ct MH t Horizon 3 2   a t  m  3  a matter  4  a rad t0 teq time zeq~104

5. CDM Power Spectrum MH t  mass H  growth when matter is self- gravitating  H  H   rad mass t time teq CDM 1  k Pk  2 / 3  3 M  k CDM HDM HDM free streeming mass k kpeak m h Meq

6. Formation of Large-Scale Structure Fluctuation growth in the linear regime: rms fluctuation at mass scale M: Typical objects forming at t:   ~ 1    / 2 t 6 3 1 a  / ) 3   M M   n  0  a 2    ( n    2 / 3     / 1   M a * example 6 M a   * HDM: top-down CDM: bottom-up / 1 2 / 1 2 2 2                     1 1 free streaming 0 0 M M

7. Power Spectrum

8. ΛCDM Power Spectrum 2k ( ) ( ) P k k T  ln( 1 . 2 34 q ) q k   1  / 1  4 2 3 4 ( ) . 3 89 16 ( 1 . ) . 5 (  46 ) . 6 (  71 ) T k q q q q q     2 1 . 2 34  h Mpc  m normalization:    1 ( 8 )  R h Mpc  8 tophat

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

12. Formation of Large-Scale Structure: comoving

13. ימסוקה גראמהמ תומדקומה תויסקלגה תורצוויה

14. Filamentary Structure: Zel’dovich Approximation Approximate the displacement from initial position    ( , ) ( ) ( ), x q t q D t q     Velocity & acceleration along displacement → trajectories straight lines D  D   as in linear  ,  x  x   x     central force → potential flow , r ax v r  a  x a x  Hr v       In physical coordinates pec Density (Lagrangian):   3 3 continuity ( , ) x t d x d q  q   2  q  q  q      ( , ) x t ,         1 2 3 i    2 || / || 1 ( ( ) )( 1 ( ) )( 1 ( ) ) x q D t D t D t      i 1 2 3 deformation tensor eigenvalues Jacobian → caustics      1 2 3    cluster 8%   1 2 3    pancake 42%   1 2 3

15. Zel’dovich Approximation cont’d  2  q  q      ( , ) x t ,     1 2 3 i    2 1 ( ( ) )( 1 ( ) )( 1 ( ) ) D t D t D t     i 1 2 3           2 3 1 ( ) ( ...) ( ) ... D D D           1 2 3 1 2 1 2 3  q 1 x  D linear       ( ) D D D v v                    1 2 3 D  D  ( ) Hf  D   D  → D is the growing mode of GI obeying  4  2 H G D   Error: plug density in Poisson eq.        2 ( )          1 2 3 Poisson grav linear           → error is 2nd +3rd terms   2 3 3 2 3 2 3 ( ) ( 2 ) 2   D D          1 1 2 2      1 1 1 1 error small in linear regime  1 D  1    or pancakes ,  1 2 3 error big in spherical collapse    ~ ~ 1 2 3

16. Non-dissipative Pancakes: why flat?  h v R oscilation time << expansion time  1   H  adiabatic invariant  2 2 . ~ ~ ~ const x dt v h v 0  2 3 / 1 / 2  3 ~  v f fv v f R     2  f R    1 / 2 3  h v R   h / 1  3 / 1  3 R a   R pancake becomes flatter in time

17. N-body simulation CDM N-body simulation

18. N-body simulation

19. N-body simulation

20. N-body simulation spherical collapse or mergers

21. N-body simulation spherical collapse or mergers

22. N-body simulation spherical collapse or mergers

23. xxx

24. N-body simulation of Halo Formation

25. N-body simulation of Halo Formation

26. Top-Hat Model (Λ=0, matter era) * 2 a a bound sphere (k=1) in EdS universe (k=0)   2 * 3 4 ( ) 3 / . a  k a G a const     a conformal time  cos  * 2 * 3 ( a ) 2 / 1 ( (  ) 6 / ( p a a a t ) a dt   t d   * * (t ) a    sin ) a    p p p p p 3 / 2   * *   6 a 6 a 1  p p        3 ( ) ( sin ) a ( ) ( sin ) t t       p p p p p p p    * * 2 a a universe overdensity: 3 *    2 ( 9 sin ) a    a p p p p   * 3     * 3 a a a a       5.55   * 3 1 ( 2 cos ) a a p p p  p p    / 1 1   linear perturbation Taylor perturbation p 1 1 1 1        2 4 3 5 cos 1 sin        2 24 6 120      a  2 p  3 / 2 t . 0 15 a      p   2  0 p p  2 9 0 tmax tvir   . 5 55   turnaround p  16 3 / 2 / 2 3  (  ) 1   ( 2 ) t  3    2 2   p linear equivalent to collapse        ( ) 1 . 0 15  . 1 68      2 p p c           t / 6  p p

27. universe 4 5.55 200 8 perturbation   2 0 t 0 tmax tvir

28. Spherical Collapse  universe max  5 . 5 vir radius 200  perturbation  virial equilibrium time 1 GM GM virial equilibrium: E     2 R max R vir

29. Virial Scaling Relations GM Virial equilibrium: 2 V  R M Spherical collapse:   3  a 200       0 u u  3 4 ( ) 3 / R 3 / 3 2 3 3  M V a R a    Weak dependence on time of formation:    ( ) ( ) 1 ( / ) 3  6 1 . 0  2 . 0  D a M a M n     0 4 for 2 M V n    Practical formulae:  30 3 2 3 . 2 76 10    g cm 7 . 0 h a    3 . 0 m u 3 / 3  2 3 Mpc 3  M V A R A   11 100 2 / 1  3 ( ) A a 7 . 0 h    200 3 . 0 m  2 2 ( ) 18 [ 82 ( ) 39 ( ) / ] ( ) ( ) 1 178  340  a a a a a           0 m   3  a  ( ) ( ) ( ) 1 m a a a       m m  3  a    m 

30. Lecture 6 Hierarchical Clustering Press Schechter Formalism

31. Press Schechter Formalism halo mass function n(M,a) Gaussian random field random spheres of mass M linear-extrapolated δrmsat a:     2 / 1  2 2 2 ( ) 2 ( ) exp( / 2 ) P   nonlinear σ   ( , ) ( ) ( ) M a M D a  0 linear fraction of spheres with δ>δc =1.68: a(t) a0=1      2 / 1  2 2 2 ( , ) 2 [ ( , )] exp[ / 2 ( , )] F M a d M a M a  c  δ   δc   / 1  2 2 2 ( ) exp( ) 2 / dx x  c     / ( , ) M a    c x c  ( ) ( ) D a 0M PS ansaz: F is the mass fraction in halos >M (at a) derivative of F with respect to M: / 1   2   ln 2  d dM   2   0 ( , ) exp( ) 2 / n M a dM     c c ln M d M M Mo & White 2002

32. Press Schechter Formalism cont. / 1   2   ln 2  d dM   2   0 ( , ) exp( ) 2 / n M a dM     c c ln M d M M ~ M 2 log n(M) Example:     n ( ) ( / )  P k M M  M ) 2 / M M      0 * k c ln d  3 ( / ) n 6    0  ln d ~2 M / 2 e ~ ~ ~ M*(a)    2 2 ( )  exp( / n M M M M M M    * self-similar evolution, scaled with M* log M  approximate     c ( ) defined by ( , ) M a M a  c  ( ) ( ) D a 0M * * c  / 1 ) 13 5 ( ) ( ~ 10 M a M D a M a  time Pk * * 0  ~    2  1 2 2 ( ) 2 ( ) ( ) ( ) R dk k P k W kR  0  in a flat universe Top Hat ( ) ( / ) a ) 1 ( g D a a g  ~ 3) ( ) ( / ) ( ) [sin( 3  ) cos( )] /( W x x R W k kR kR kR kR    1  1 ( 2 / ) a 5 R R     7 / 4 ) ( ) ( ) ( ( ) m g a a a a             m m  2 1 ( / ) a 70    kR=π 3  a  ( ) m a   m 3  a    m  x k R

33. Press Schechter cont. Better fit using ellipsoidal collapse (Sheth & Tormen 2002)   4 . 0 1/2 ( , ) 1 ( 4 . 0  4 . 0  / ) erfc(0.85 /2 ) F M a     1 2 , 3 , 22 %, 7 . 4 %, . 0 54 % Comparison of PS to N-body simulations log M2n(M) factor 2 log M

34. Press-Schechter in ΛCDM 14 log 13 13 . 3 . 1  ( ) 2 M z z   * 13 2σ log M/Mʘ 12 1σ 11 10 9 Mo & White 2002

35. Press-Schechter Mo & White 2002

36. Merger Tree t

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