1. Cosmology Class - 2002/20031 Galaxy Formation and non Linear collapse By Guido Chincarini University Milano - Bicocca Cosmology Lectures This part follows to a large extent Padmanabhan

2. Cosmology Class - 2002/20032 Density perturbations We have seen that under particular condition the perturbation densities grow and after collapse my generate, assuming they reach some equilibrium, an astronomical object the way we know them. Density perurbations may be positive, excess of density, or negative, deficiency of density compared to the background mean density. Now we must investigate two directions: –The spectrum of perturbations, how it is filtered though the cosmic time and how it evolves and match the observations. –How a single perturbation grow or dissipate and which are the characteristic parameters as a function of time. Here we will be dealing with the second problem and develop next the formation and evolution of the Large Scale structure after taking in consideration the observations and the methods of statistical analysis.

3. Cosmology Class - 2002/20033 Visualization

4. Cosmology Class - 2002/20034 Definition of the problem We use proper radial coordinates r = a(t) x in the Newtonian limit developed in class and where x is the co_moving Friedmann coordinate. Here we will have:  b = Equivalent potential of the Friedmann metric   (r,t) = The potential generated due to the excess density:  It is then possible to demonstrate, see Padmanabahn Chapter 4, that the first integral of motion is: ½(dr/dt) 2 – GM/r =E

5. Cosmology Class - 2002/20035 How would it move a particle on a shell? The Universe Expands

6. Cosmology Class - 2002/20036 And we can look at a series of shells and also assume that the shells contracting do not cross each other

7. Cosmology Class - 2002/20037 Remember At a well defined time t x I have a well set system of coordinates and each object has a space coordinate at that time. I indicate by x the separation between two points. If at some point I make the Universe run again, either expand or contract, all space quantities will change accordingly to the relation we found for the proper distance etc. That is r (the proper separation) will change as a(t) x. Or a(t x ) x = a(t) x and in particular: r o = a(t o ) x = a(t) x =r => r/r o =a(t)/a(t o ) and for r o =x x=r a(t o )/a(t).

8. Cosmology Class - 2002/20038 Situation similar to the solution of Friedman equations ½(dr/dt)2 – GM/r =E

9. Cosmology Class - 2002/20039 And remember that  I was defined in relation to the Background surrounding the pertubation at the time t i

10. Cosmology Class - 2002/200310 For the case of interest E <0 we have collapse when:

11. Cosmology Class - 2002/200311 Derivation of r m /r i The overdensity expands together with the background and however at a slower rate since each shell feels the overdensity inside its radius and its expansion is retarded. Perturbation in the Hubble flow caused by the perturbation. The background decreaseds faster and the overdensity grow to a maximum radius r m at which point the collapse begins for an overdensity larger than the critical overdensity as stated in the previous slide.

12. Cosmology Class - 2002/200312 The Perturbation evolves And shells do not cross and I conserve the mass

13. Cosmology Class - 2002/200313 A & B

14. Cosmology Class - 2002/200314 Starting with small perturbations

15. Cosmology Class - 2002/200315 3/5 of the perturbation is in the growing mode and this is the growth in the linear regime which could be compared to the non linear growth. We did that as an approximation for small perturbation but we could develop the equation in linear regime for small perturbations.

16. Cosmology Class - 2002/200316 If at the redshift z i I had a density contrast  i the present value would be  0. riri

17. Cosmology Class - 2002/200317 Using the approximations for A & B I use the value I derived for A and B in the case of small perturbations. Note the definition of  0 which is the contrast at the present time. The equation show how the perturbation are developing as a funcion of the cosmic time. We would like to know, however, an estimate of when the growth of the perturbations make it necessary to pass from the linear regime to the non linear regime.

18. Cosmology Class - 2002/200318  The easy case See next slides for details: And therefore I can also write r   =  Time of turn around dr/dt = 0, r=r m Again a summary

19. Cosmology Class - 2002/200319 A detail – see Notes Page 28

20. Cosmology Class - 2002/200320 That is 3/5 of the perturbation grows as t 2/3 and for  =1 I can also write: And in units of a(t o ) I can write r i = a i /a o x = x/(1+z i ). That is r o = x

21. Cosmology Class - 2002/200321 When does the Non Linear Regime begins? We define the transition between the linear and the non linear regime when we reach a contrast density of about  = 1. The above computation shows that at this time the two solurion differ considerably from each other.

22. Cosmology Class - 2002/200322   ~2  /3 ~  /2

23. Cosmology Class - 2002/200323 What I would like to know: At what z do I have the transition between linear and non linear? What is the ratio of the densities between perturbation and Background? At which z do we have the Maximum expansion? How large a radius do we reach? And how dense? At which redshift do I have the maximum expansion? At which redshift does the perturbation collapse? And what about Equilibrium (Virialization) and Virial parameters? What is the role of the barionic matters in all this?

24. Cosmology Class - 2002/200324 Toward Virialization The student could also read the excellent paper by Lynden Bell on Violent Relaxation rmrm r vir

25. Cosmology Class - 2002/200325 How long does it take to collapse?

26. Cosmology Class - 2002/200326 What is the density of the collapsed object?

27. Cosmology Class - 2002/200327

28. Cosmology Class - 2002/200328 Density Time bb  b  a -3 8 5.55  =2  /3=2.09  =  =3.14  =2  1.87 a nl a max 1.59 2.0

29. Cosmology Class - 2002/200329 What happens to the baryons? During the collapse the gas involved develops shocks and heating. This generates pressure and at some point the collapse will stop. The agglomerate works toward equilibrium and the thermal energy must equal the gravitational energy. And for a mixture of Hydrogen and Helium we have:

30. Cosmology Class - 2002/200330 Derivation

31. Cosmology Class - 2002/200331 And from Cosmology we have:

32. Cosmology Class - 2002/200332  h=1 eventually

33. Cosmology Class - 2002/200333 Virial velocity and Temperature

34. Cosmology Class - 2002/200334 Example Assume a typical mass of the order of the mass of the galaxy: M=10 12 M  and h=0.5 Assume also that the mass collapse at about z=5 then we have the values of the parameters as specified below. Once the object is virialized, the value of the parameters does not change except for the evolution of the object itself. For collapse at higher z the virial radius is smaller with higher probability of shocks. Temperature needs viscosity and heating and ? Do we have any process making galaxies to loose angular momentum?

35. Cosmology Class - 2002/200335 Temperature and density The temperature is very high and should emit, assuming the model is somewhat realistic, in the X ray. This however should be compared with hydro dynamical simulations to better understand what is going on. The density at collapse [equilibrium] is fairly high. Assuming a galaxy with a mass of about 10 12 solar masses, a diameter of about 30 kpc and a background of 1.88 10 -29  h 2 =.9 10 -29 the mean density would be about 5 10 4. Very close! Coincidence? Obviously we should compute a density profile.

36. Cosmology Class - 2002/200336  h=0.5 M = 10 12 M  M = 10 10 M 

37. Cosmology Class - 2002/200337  h=0.5 M = 10 12 M  M = 10 10 M 

38. Cosmology Class - 2002/200338  h=0.5 M = 10 12 M  M = 10 10 M 

39. Cosmology Class - 2002/200339  h=0.5

40. Cosmology Class - 2002/200340 Cooling and Mass limits - Is any mass allowed? Assume I have the baryonic part in thermal equilibrium, the hot gas will radiate and the balance must be rearranged as a function of time. The following relations exist between the Temperature, cooling time and dynamical (or free-fall) time: Here n is the particle density per cm 3 (n in units cm -3 ), L( T) the cooling rate of the gas at temperature T.

41. Cosmology Class - 2002/200341 Mechanisms No cooling:t cool > H -1 Slow cooling via ~ static collapse:H -1 > t cool > t free-fall Efficient cooling: t cool < t free-fall (In the last case the cloud goes toward collapse and could also fragment – instability - and form smaller objects, stars etc.). Cooling via: Brehmsstrahlung Recombination, lines and continuum cooling Inverse Compton [the latter important only for z > 7 as we will see]

42. Cosmology Class - 2002/200342 t cool =8 10 6 (n cm-3 ) -1 [T 6 -1/2 +1.5 f m T 6 -3/2 ] -1 In the following I use f m =1 (no Metal) for solar f m =30 T~10 6 Brehmsstrahlung Line Cooling

43. Cosmology Class - 2002/200343 T<10 6

44. Cosmology Class - 2002/200344 T<10 6 - Continue If M =2.8 10 11 n -1/2 T 3/2 >2.8 10 11 9.37 = =2.8 10 12 Then  Cooling not very Efficient. Vice-versa if  Temperature Density T  n 1/3 M=2.8 10 12 t dyn >H o -1 T = 10 6

45. Cosmology Class - 2002/200345 T > 10 6

46. Cosmology Class - 2002/200346 T > 10 6 - Continue If the radius is too large the cooling is not very efficient and chances are I am not forming galaxies. In other words in order to form galaxies and have an efficient cooling the radius of the cloud must shrink below an effective radius which is of the order of 105 kpc. For fun compare with the estimated halos of the galaxies along the line of sight of a distant quasar. More or less we estimate the same size. Or we could also follow the reasoning that very large clouds would almost be consistent with a diffuse medium. Try to follow these reasoning to derive ideas on the distribution of matter in the Universe.

47. Cosmology Class - 2002/200347 T > 10 6 - Continue Or an other way to look at it is (see notes Page 46): For  > 1 R > 105 kpc t cool > t dyn Vice versa for  < 1 ; in this case cooling is efficient The cloud must shrink for efficient cooling

48. Cosmology Class - 2002/200348 Summary For a given Mass of the primordial cloud and T < 10 6 we have the following relation: The Mass of the forming object is smaller than a critical mass. M < 2.6 10 12 solar masses. For a given Radius of the primordial cloud and T > 10 6 we have the following relation: The Radius of the primordial cloud must be smaller than a critical Radius in order to have efficient cooling and form galaxies. R < 105 kpc.

49. Cosmology Class - 2002/200349 Continue The dashed light blue line next slide

50. Cosmology Class - 2002/200350 T>10 6 Temperature Density T  n 1/3 M=2.6 10 12 t dyn >H o -1 T = 10 6 T  n R ~ 105 kpc t cool = H o -1 t cool =t dyn

51. Cosmology Class - 2002/200351 DM & Baryons together Cooling: Only the baryonic matter is at work – The gas initially is not at virial Temperature. Dynamics: Dark Matter dominates. Shocks and Heating. t cool > t dyn gas may be heated to Virial Temp equilibrium t cool < t dyn May never reach Equilibrium may sink in the potential well, sink Fragment etc..

52. Cosmology Class - 2002/200352 Assumption & Definition We assume a gas fraction F of the total Mass. Assume Line cooling dominates. The gas is distributed over a radius r m /2 = r vir. We also assume t dyn ~ ½ t coll so that we have:

53. Cosmology Class - 2002/200353 Continue So that assuming spherical collapse and again using the reasoning with  we have [for f m metal abundance see slide 42]: Again masses of the order 10 11 – 10 12 are picked up preferentially

54. Cosmology Class - 2002/200354 And for the Compton Cooling N e Electron density. TGas temperature.  r Density of the radiation, MWB. T r Radiation Temperature.   comp Cooling Rate – and assume T >> T r.

55. Cosmology Class - 2002/200355 & The relevant equations are See notes for details: And t dyn at t collapse

56. Cosmology Class - 2002/200356 Combining That is  7.6 independent of mass. That is Compton Cooling is important for those objects collapsing at z > 7.6 An uinteresting game could be to consider what happened of these clouds just before re-ionozation and indeed find out how efficient these hot clouds could be in reionizing the intergalactic medium. Develop a chapter on the ionization of the Intergalactic Medium.

57. Cosmology Class - 2002/200357 An other example Assume that after maximum expansion we have a contraction of a factor f c and assume that after virialization the density does not change, then: For a galaxy with a mass of about 10 11 M  within a radius of 10 kpc we have  obs /  c,0 ~ 10 5 so that: So that (note however that since the time of collapse is rather long we should account for the variation of  b as a function of time) the redshift of formation is too close:

58. Cosmology Class - 2002/200358 Angular Momentum Mass: M Energy (At max. expansion)E ~ - GM 2 /R Angular Momentum:L=Mv  R=M  R 2 Angular velocity:  =L/MR 2 Equilibrium condition:  ( 2 support) R=GM/R 2 =  (the rotational Energy available) /  sup(needed to counterbalance the gravitational field. Angular Velocity  Support

59. Cosmology Class - 2002/200359 The facts The Observations show that we have ~ 0.05 for Elliptical galaxies and ~ 0.4 – 0.5 for disk galaxies. The gas is about 10 % of the Halo mass and during collapse the gas will dissipate and during his evolution to a disk could cool rapidly, fragment and form stars. N body simulations show that due to the irregular distribution of matter an object will acquire via tidal torques a value in the range of 0.1 – 0.01 with a mean value of about 0.05. That is of the same order for Ellipticals but much to low for disk galaxies.

60. Cosmology Class - 2002/200360 Comments to the facts That is the gas in forming a disk galaxy should collapse (during collapse we conserve the Mass) of a factor fc = R initial /R disk = ( disk / in ) 2 = (0.5/0.05) 2 = 100 in order to satisfy the observations. (See slide 33) That is in order to form a galaxy of 10 11 M , R = 10 kpc I have to beginn with a cloud of about 1 Mpc. However to such a cloud it will take to collapse: t coll = (  /2) (R 3 /2GM) 1/2 ~ 5.3 10 10 yr Much too long and the same would be for the 3 kpc core which should start from a 300 kpc radius region.

61. Cosmology Class - 2002/200361 Let’s look into some details Note that both the gas and the DM are virialized, and however at the beginning the disk did not collapse yet. R c,r c (disk after collapse), k 1, k 2, are characteristic radii and parameters accounting for the geometry and mass distribution.

62. Cosmology Class - 2002/200362 Continue Note that the Angular momentum per Unit mass acquired by the gas is the same as that gained by the DM since at the beginning all the matter of the perturbation is subject to the same tidal torques. The gas, during collapse from R c to r c, conserve angular momentum. L d /M d = L/M.

63. Cosmology Class - 2002/200363 Conclusion The Collapse factor has been reduced by about a factor of 10 due to the fact that the mass of the disk is about 10% of the mass of the halo DM.

64. Cosmology Class - 2002/200364 More about when – Primeval Galaxies The MWB tell us that at z ~ 1000 the perturbations were in the linear regime since otherwise we would have detected them. It is therefore very clear that galaxy formation occur after decoupling. We also determined that at the turn around time – top hat model -the spherical over dense region has a density which is 9  2 /16 times higher than the background density,  b. If the material contracts by a factor f c then the overdensity increases by a factor f c 3. We consider a galaxy with M=10 11 M , r ~ 10 kpc so that  obs /  c ~ 10 5.

65. Cosmology Class - 2002/200365 Z collapse So that: Note also that we should integrate to account for the change of  b during the collapse. The factor 2 2/3 since: t collapse ~ 2 t m ;  b  a –3  t –2 ; so that the density contrast increases by a factor 2 2 and z collapse factor 2 2/3. For dissipation_less collapse f c ~ 2. For a disk f c ~ 10 or more.

66. Cosmology Class - 2002/200366 Program - Go to L_S_S Before doing the Gaussian Fluctuations and the evolution of the Power Spectrum it is wise to discuss the Large Scale Structure as done in the Power Point Lecture (to be improved). Develop Further this part since it seems to be very interesting and useful to the students,

67. Cosmology Class - 2002/200367 Typical mass in hierarchical models See eventually details on the Power Spectrum, what it is. Lecture LSS to be completed. Fluctuations of M within a sphere of Radius R described by the variance  (M), Gaussian distribution of density inhomogeneities. Contrast  o =  (M), M  3  k –3 &  (M) 2 = = C M –(3+n)/3  (M) = (M/M o ) –(3+n)/6 With the constant M o to be determined.

68. Cosmology Class - 2002/200368 Normalization Counting galaxies we see that  N/N ~ 0.9 at 10 h -1 Mpc and we measure M (R= 10 h -1 Mpc ) ~ 1.15 10^15 (h -1  ) M  We finally assume:

69. Cosmology Class - 2002/200369 Normalization and derivation of the redshift at which a given mass collapse

70. Cosmology Class - 2002/200370 Finally and However The reasoning was not completely correct since for  N/N ~ 1 at some mass scale M o we are already in the non linear regime. We did more or less: By definition  M/M =1/b  N/N and we would like to set, in agreement with the above,  (M o ) = 1/b. If the theory says that  (M) = 1/b (M/M c ) -n then we will take M c = M o. Suppose we now observe  o =1 as evolved considering non linear effects. It would be  o =0.57 using linear theory and coming from a fluctuation  i at t i.

71. Cosmology Class - 2002/200371 Or saying differently In our spherical model we had  =(  b -1  is unity at  =2  / 3. Using this value of  the density excess extrapolated at the present epoch is 0.57. That is the normalization of the spectrum should be  (M o ) = 0.57/b rather than  (M o ) = 1/b. Applying this correction we find:

72. Cosmology Class - 2002/200372 Typical Mass which collapse and becomes non linear at various redshifts.  =1,h=0.5 and density contrast  o (M)=  (M) with =2. Log (M/M  ) z b=1, n=-1 b=2, n=-1 b=1, n=-2 b=2, n=-2

73. Cosmology Class - 2002/200373 Timetable for Formation Gravitational potential fluctuations Z  10 3 Spheroids of GalaxiesZ ~ 20 The first Engines for active galactic nuclei Z  10 The intergalactic mediumZ ~ 10 Dark Matter Z  5 Dark halos of galaxiesZ ~ 5 Angular momentum of rotation of galaxiesZ ~ 5 The first 10% of the heavy elements Z  3 Cosmic magnetic fields Z  3 Rich clusters of galaxiesZ ~ 2 Thin disks of spiral galaxiesZ ~ 1 Superclusters, walls and voidsZ ~ 1

74. Cosmology Class - 2002/200374 Intergalactic (& ISM) Medium Observational evidence exists that the column density of Hydrogen is related to the color excess (HI & Dust). The empirical relation: Near the Sun we have: n H =10 6 m -3 so that for a distance d through the disk we have: N H = 3.09 10 25 (d/kpc) m -2 ; E (B-V) = 0.53 (d/kpc) and A V = 1.6 (d/kpc) 1 m 2 1 kpc nHnH

75. Cosmology Class - 2002/200375 Toward the GC A M ~ 0.6 ; A B = 34.5 The Probability for a B photon to reach us is: 10 –0.4 (34.5) = 10 –13.8 = 1.6 10 -14

76. Cosmology Class - 2002/200376 Standard ISM Extinction Band X eff /nm MM (E X-V /E B-V )(A X /A V ) U3655.611.641.531 B4455.481.1.324 V5514.8301. R6584.42-0.78.748 I8064.08-1.60.482 J12203.64-2.22.282 H16303.32-2.55.175 K21903.28-2.74.112 L34503.25-2.91.058 M4750-3.02.023 N-2.93.052