Cosmology Class - 2002/20031 Galaxy Formation and non Linear collapse By Guido Chincarini University...

Cosmology Class - 2002/2003 1

Galaxy Formation and non Linear collapse

By Guido Chincarini

University Milano - Bicocca

Cosmology Lectures

This part follows to a large extent Padmanabhan


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.


Visualization

b b

b b b b b b

Tot Background Perturbation

i b i i b i i

1 0 1 0

r,t t 1 r,t t r ,t


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


How would it move a particle on a shell?

2bTot

Total2

b3 3

Force

4 G tGMd rr

dt r r 3G M r,tGM

r rr r

The Universe Expands

33b b b

r r2 2b b0 0

r t a t x

4 4M t r t a t x

3 3

M r,t 4 q,t q dq 4 t q q,t q dq


And we can look at a series of shells and also assume that the shells contracting do not cross

each other

i

i

r3 2 3

Tot i b i i b i

0

r2

i

0 iti i 3

b b b

3 3i b i b i i

Mass within any shell at the initial value i

4 4( r ) 4 r dr ( r )3 3

4 r dr( pert.)1 1

14 Volumer3

4 4( r ) ( r )(1 )3 3

M M

M

M M


Remember

• At a well defined time tx 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(tx) x = a(t) x and in particular:

• ro= a(to) x = a(t) x =r => r/ro=a(t)/a(to) and for ro=x

• x=r a(to)/a(t).


Situation similar to the solution of Friedman equations

½(dr/dt)2 – GM/r =E

i

2i

2i

i

i i i i i i

2 2 2i i i

i

t

E 0 r always 0 the shell exp ands foreverr

E 0 r could be zeroor negative collapse

At t I assume a small overdensity and the shell exp ands with

thebackground

a ar a x a x r H t r H r

a a

r H rK

2 2

GMU

r

2

2 ib i i i b i i

3H4t r 1 & t3 8 G


22 2i

b i i i i i i

2 2i i

i i i i i

1i i i i i i i i

1 1i i i i

3H4 4U t r 1 r 13 38 G

H r1 K 1 &

2

E K K 1 K 1

That is collapse for

E 0 1 1

And remember that I was defined in relation to theBackground surrounding the pertubation at the time ti


For the case of interest E <0we have collapse when:

1[ 1]i i

1

1

1

1i

i

i crit i

1 1

1

0

i

i

i

1

1

( 1)m i

i i i

r

r


Derivation of rm/ri

• 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 rm at which point the collapse begins for an overdensity larger than the critical overdensity as stated in the previous slide.

i

i im i i i

m m i i m

ii i i i

m

1ii i i i i i i i

m

m i i11

i i ii i

At this point r 0, therefore

r rGM GM GME U K 1

r r r r r

rK 1 E Energy conservation

r

rK 1 E K 1

r

r 1 1

r ( 1)1


The Perturbation evolvesAnd shells do not cross and I conserve the mass

3 2

33 3

2

322 2

2 2 22

3 3 33

2

3

(1 cos )

( sin ) ;

3 3, ; 1

4 4 1 cos

1 1

6 6

, 3 6 3 6 91 ,

24 1 cos 4 1 cos 1 cos

9,

2 1 cos

i

b

b

r A

t B A G B

r t and forr A

a t so thatGt GB Sin

r t G B Sin Sin Sinr t

t A

Sinr t

M

M M

M

2

3

sin91

2 1 cosbt t


A & B

m im m 1

i i i

ii 1

i i

332 i

i 13i i

i i

3 2

ii 31 2 13i i i i i ii i

b

2i

i

r 1For r r r 2 A and because

r ( 1)

11A r and

2 (

3H

1)

1A 1 1B r

4G 2 ( 1) G r 13

111 1r

42 1 4 H 1G r 1

B

G 38

M

i

311 22

i i i i

1

2 H 1


Starting with small perturbations

33

2 4 26

3

2

31

3 3

1..... 1 .....

6 2

9 3 37 3, 1

2 20 2800 201 cos

63

6( sin ) ,

6 20

Sin Cos

Sinr t O

tt B

t B B r tB


2

3

213

1311 22

23 23

2

3

2

3

22 33

3

2

2

3

63 13 2

, 6 1;20 20 3

2 1

1 63 3 4 38

20 20 3 202 16 2

3

3 1,

5 1

i

i i

i i i i

i i

ii i

ii

ii

tB

r t t for t H

H

t t

t tt

t

tr t a t

t

z

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.


2

3

11 1 1 3 311

2 2 1 5 100 02 331 213 3 3 2 30 3

3 3 3 454 4 52 02 2 21

13 1 3, ,

5 1 5 1

1

ii i

i

i i

zx xii A riA ri zi i

i

zi t tiiB B

H zi i i i

ztr t a t for any t r t or

t z z

in a different way for and small

0320

00

31

1 5

t

ii i i

i

a t xr x and I define from above z

a z

If at the redshift zi I had a density contrast i the present value would be 0.

2i

b i i

3Ht

8 G

ri


Using the approximations for A & B

23

0

3

2 03

20

2

3

2

3203

20

3

23

0

1( )

1

3(1 cos ) (1 cos )

10

33( sin ) ( sin )45

9 ( sin )( , ) ( )

2 (1 cos )

5 4(1 ) ( )

3 3( sin )

(1 ) ( )

i

i

b

ztt z

xr A

tt B

r t t

tz t

tz t

• 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.


3 2

2

3

(1 cos )

( sin ) ;

( , ) 9 ( sin )( , ) 1 1

( ) 2 (1 cos )b

r A

t B A G B

r tr t

t

M

2

3

00

0

3

203

20

2

323

20

3

sin9

2 1 cos

3 3(1 )

( ) 5 5

3( ) (1 cos )

10

3 3( sin )

5 4

5 4(1 ) ( )

3 3( sin )

b

i ii

i

t t

az

a t

xr t

tt

tz t

The easy case

See next slides for details:

See next slides for details:

And therefore I can also write

r

=

Time of turn arounddr/dt = 0, r=rm

Again a summary


A detail – see Notes Page 28


2233 36 13 ( )20 5 1

ilinear

i

t ta t

B t z

That is 3/5 of the perturbation grows as t2/3 and for =1 I canalso write:

2

33 3 (1 ) ;55

1( ) 3 35 5( ) 1

ii i o

o

i i

i

i

i

i

i

is the present value of the density contrast as

predicted by the linear theory if the density contrast at

tz a

z was

nd in generalt

za t

a t z

And in units of a(to) I can write ri = ai/ao x = x/(1+zi). That is ro = x


When does the Non Linear Regime begins?

2 23 3

02

3

2 22

0

2 223 33 3 33

13 3 5 4, 1 ; (1 ) 17

5 5 1 3 3( sin )

3 3 3 ( sin ) 3 3 3 ( sin ) 3 3, 1 1 ( sin )

5 5 4 5 5 4 5 431

1 15

iLinearLinear i i

b i

Linear i i i i

i i

ztr t z Slide

t z

zr t z z

z B

22

33

2

3 3( 1) 0.341

5 4 22 9

( 1) 1 0.4662 2

.56821.013

Non Linear Linear

Linear

Non Linear

Linear

Non Linear

eginning of growth z z

z

z

z

z

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.


0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

~2/3

~/2


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?


Toward VirializationThe student could also read the excellent paper by Lynden Bell on Violent Relaxation

rm

rvir

2

m Spherem

2 2

m

2

m

2 2

vir m

vir m

E U KAt virialization E K

2K U 0

3at t t K 0 E U G

5 r

v 3K E G

2 5 r

U when virialized3 6

G G5

2 K v

6v

r 5 r

1r r

2G

5 r

M

M M

M

M

M M


How long does it take to collapse?

2 2

3 3

2 2

3 3

22 33

22

00

33

5 4 5 4(1 ) 2

3 3 3 3( sin ) ( sin )

(1 ) 0.363

214

0.631

(1 ) 0.5653

31

5

11

(

11

4

1

collapse

c

i i

i

ollapse

m

m

ii i

i ii i

z Colz

z

lapse at

z

z

zz

or z

z

zz

0) 0.63 (1 )1.683collapse mz


What is the density of the collapsed object?

3 3m

collapsed object vir m

3maximum exp ansion irvir

2 2

coll

m m b m b m b m3

3

mb m b collapse

collapse

3

mm b m b collapse

collapse

4r r3 8

4 rr3

sin9 9t t t 5.55 t

2 161 cos

1 zt t

1 z

1 z8 44.4 t 44.4 t

1 z

4

MM

3

b collap

3

0 colls apsee 11

4.4 t0.

77.6 t 1 z63


2

3

3

3

3

0

92 ; 1 5.6

16

(1 )8 ; 0.63

(1 )

(1 )8 44.4 ( ) 44.4 ( )

(1 )

177.6 ( ) 177.6 1

m mm

vir b

vir m coll

m vir m

mcoll m b m b coll

coll

b coll coll

r

r

r z

r z

zt t

z

t z


Density

Time

b

b a-3

2

3

sin9

2 1 cosbt t

8

5.55

=2/3=2.09==3.14

=2

1.87anl amax

223

2 3maxnl 3

max nl

sin1 za 1.231.87

a 1 z sin 3.14

1.59

2

3max

coll

a 6.281.59

a 3.14

2.0


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:

2 23 1 3

2 2 5

( 0.25) 0.572 3

pvir

H H He HeH

H He

GkT m v v

r

m n m nY m

n n

M


Derivation

1( 0.25)

32 3 2 12

0.25

1( 0.25) 0.57

3 2 1 0.3753 3 1

0.3752 2 2 4

H H He He H

HeH He

H

He He

H H

HHH H

He He

H H

m n m n m YY

nn nn

m nY

m nm Y

Y mm nY Y

n mY Y

n n


And from Cosmology we have:

29 2 30

10 10

0

00 0

1

3 112 2 3

0

0

1.8810

0.6510

1.686 (1 )

( )( )

( 10 ) 0.95( )4

3

coll

i ii

h gcm

t h yr

z

ax r x t t r

a t

x r h Mpc

1312

MM M


h=1 eventually

21 3

112 33 3

1 12213

1

329 2

21 3

1 3169 (1 )

2 10 1.686 (1 )

10 1.9910

3.0910

443 177.61.8810 13

169 (1 )

vir m collcoll

vir

coll

coll

coll

xr r z h kpc

z

or

r

h z

z h kpc

1312

1312

M

M

M

M


Virial velocity and Temperature

1 1

2 2

1

28 12 33

121 2

1 213 312

11

3212

1

30 12

224 2 52

3 65 5

3 6.67 10 10 1.99 10

5 169 1 3.09 10

123.5 (1 ) /

95 /

0.57 1.67 10 123.5 10

3 1.38

vir m

coll

coll

vir H

G Gv r r

z h

z h km s

v h km s

vT m

k

M M

M

M

M

M

16

2 26 3 3

12

10

1.010 (1 )collz h K

M


Example

Assume a typical mass of the order of the mass of the galaxy:M=1012 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?

2

3vir

3 310 1 82 2

coll 0 coll

1 11 12 3

vir

169r 0.5 17 kpc

1 5

t t 1 z 0.65 10 h / 6 h 0.5 8.8 10

not very much time for stellar evolution

v 123.5 6 0.5 km s 240 km s


Temperature and density

26 63

vir

3coll

0

T 10 6 0.5 3.8 10 K

177.6 1 z 38362

• 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 1012 solar masses, a diameter of about 30 kpc and a background of 1.88 10-29 h2 = .9 10-29 the mean density would be about 5 104 . Very close! Coincidence?

• Obviously we should compute a density profile.


Virial Radius versus zcollapse

zcoll

0 2 4 6 8 10 12 14 16

r vir(k

pc)

0

20

40

60

80

100

h=0.5

M = 1012 M

M = 1010 M


Virial Velocity versus zcollapse

zcollapse

0 2 4 6 8 10 12 14 16

V (

km/s

)

10

100

1000

h=0.5

M = 1012 M

M = 1010 M


Tvir versus zcollapse

zcollapse

0 2 4 6 8 10 12 14 16

Tvi

r

1e+4

1e+5

1e+6

1e+7

1e+8

h=0.5

M = 1012 M

M = 1010 M


collapse /versus zcollapse

zcoll

0 2 4 6 8 10 12 14 16

(co

ll /

0

2e+5

4e+5

6e+5

8e+5

h=0.5


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 cm3 (n in units cm-3), T) the cooling rate of the gas at temperature T.

2

1

27

3

3 3 5 5

1 1 3 1510

162 2

3

2 ( )

p p p

vir

free fall dyn

coolp

m v m mG GT

k k r k R

t t yrG n

E kTt

E m T

M M


Mechanisms

• No cooling: tcool > H-1

• Slow cooling via ~ static collapse: H-1 > tcool > tfree-fall

• Efficient cooling: tcool < tfree-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]


tcool=8 106 (n cm-3)-1 [T6-1/2+1.5 fm T6

-3/2]-1

In the following I use fm=1 (no Metal) for solar fm=30

Cooling Time

Temperature (T6)

0.0001 0.0010 0.0100 0.1000 1.0000 10.0000 100.0000 1000.0000

t cool(y

r)

1e+0

1e+1

1e+2

1e+3

1e+4

1e+5

1e+6

1e+7

1e+8

1e+9

1e+10

T~106

Brehmsstrahlung

Line Cooling


T<106

3 31 12 2 2 2

3 322 2

3 12 2

12

3

1 2 12 3 3

/ ; / ; ;

; ;

cool dyn cool dynt t t T t T

v RT

R

const T T

M=M

for

M

M

32 33

22

3 31 144 112 2 2 26 61 1

2 2 123

5

5.710 2.8104

3

p

p

k TG mR T n T nm

nR

MM M

M


T<106 - Continue

3 16 1 1 2 3

6 2 2617 2

13 11 112 2

6

810 (1.5)1 40 9.37

510

2.810 9.37 2.8 10

n TSlide n T

n

n T

M M

If M =2.8 1011 n-1/2 T3/2 >2.8 1011 9.37 = =2.8 1012

Then Cooling not very Efficient. Vice-versa if

Tem

pera

ture

Density

T n1/3M=2.8 1012

t dyn>

Ho-1

T = 106


T > 106

1 1 1 12 2 2 2

12

1 12 2

12

3

/ ; / ; ;

;

cool dyn cool dynt t t T t T

RT R

RR consf t Tor

M=

M


T > 106 - Continue

16 1 2 1 16 2 2

617 2

1

212 1

2 11 1 6 22 261

2

3

1 1 1 122 2 2 2 26 6

1 12 2

6

8101 6.25

510

5

5 3 110

4 43

5.2210 16.9

16.9 16.9 6.25 105.6 1

p

pp

n Tn T

n

kT

G m kRR n TG mn m

R

n T cm n T kpc

If R n T

M

M

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.


T > 106 - Continue

Or an other way to look at it is (see notes Page 46):

1 12 216.9 16.9

0.16R kpc T n kpc

For > 1 R > 105 kpc tcool > tdyn

Vice versa for < 1 ; in this case cooling is efficientThe cloud must shrink for efficient cooling


Summary

For a given Mass of the primordial cloud and T < 106 we have the following relation:

The Mass of the forming object is smaller than a critical mass. M < 2.6 1012 solar masses.

For a given Radius of the primordial cloud and T > 106 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.

T

13T


ContinueThe dashed light blue line next slide

6

3 21 1 32

6

11 1 22

10

10

cool o

cool o

T

t n T H const T n

T

t n T H const T n


T>106

Tem

pera

ture

Density

T n1/3

M=2.6 1012

t dyn>

Ho-1

T = 106

T

n R ~ 105 kpct co

ol =

Ho

-1

19

140 ( ) 4000 12000

3 1184002 ; ( ) 7.310 ( ) ( )( )cool e

Spitzer page for T for T

nkTt T n n HI Exp

T T

tcool =tdyn


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.tcool > tdyn

gas may be heated to Virial Temp

equilibrium

tcool < tdyn

May never reach Equilibrium may sink in the potential well, sink

Fragment etc.

.


Assumption & Definition

• We assume a gas fraction F of the total Mass.

• Assume Line cooling dominates.

• The gas is distributed over a radius rm/2 = rvir.

• We also assume tdyn ~ ½ tcoll so that we have:

12

9123

2 2

323

2 312 2

3

21 ~ ~ 1.5102 2 200

6~ ; ~3 5

mdyn coll

vir pm

cool

G rt t yrkpcR

GvT m vk r

T Rt R

R

MM

M

MM

M


Continue

• So that assuming spherical collapse and again using the reasoning with we have [for fm metal abundance see slide 42]:

31 219 1 2

12

1 11 212

11

2.410 0.1 200

~ 1.6 0.1

~ 6.410 0 1

1

.

mcool m

coolm

dyn

crit m

rFt f yrkpc

t Ff

f

t

f

i

F

M

M

M M M

Again masses of the order 1011 – 1012 are picked up preferentially


And for the Compton Cooling

• Ne Electron density.

• T Gas temperature. r Density of the radiation, MWB.

• Tr Radiation Temperature.

comp Cooling Rate – and assume T >> Tr.

4

3 33

2 2 4 8

T e r rcompton

e

p e e p ecomp

comp T e r r T r

N T T

m

m N kTm m m kTkTt

N T T T


& The relevant equations areSee notes for details:

4

4124

32

0 0

10 10

(1 )

3~ 2.110 1

8 (1 )

1 1 (1 ) ; 12 2

0.6510

r c r

p ecomp

T c r

dyn coll

z

m m kt z yr

z

t t t z

t h yr

And tdyn at tcollapse

310 121 0.6510 (1 )2dyn collt z h


Combining

412

52 2310 2

2.110 11.610 1

1 0.6510 (1 )2

compton collcoll

collapse coll

t z hz

t z

That is < 1 only for zcoll > 7.6 independent of mass. That is ComptonCooling 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.


An other example

• Assume that after maximum expansion we have a contraction of a factor fc and assume that after virialization the density does not change, then:

• For a galaxy with a mass of about 1011 M within a radius of 10 kpc we have obs/ c,0 ~ 105 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:

2 33 3 3 30 0 0

9 ( ) 1 5.6 (1 ) 5.616obs c b m b m c c m cf t z f z f

1 13 3

,1

0 0 3

1 1 153

.0

6 mobs

mc c

c

zf fz


Angular Momentum

• Mass: M• Energy (At max. expansion) E ~ - GM2/R

• Angular Momentum: L=Mv R=MR2

• Angular velocity: =L/MR2

• Equilibrium condition: (2

support)R=GM/R2

= (the rotational Energy available) / sup(needed to counterbalance the gravitational

field.

1322

1 1 3 51 122 2 2 2 2 2sup

L EL R L

R G G R G

M M M M

Angular Velocity

Support


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.


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 = Rinitial/Rdisk= (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 1011 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:

tcoll= (/2) (R3/2GM)1/2~ 5.3 1010 yr

• Much too long and the same would be for the 3 kpc core which should start from a 300 kpc radius region.


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.

• Rc ,rc (disk after collapse), k1, k2, are characteristic radii and parameters accounting for the geometry and mass distribution.

5 52 2

1 52 2

; disk diskin disk

disk

i dd d

d

L E L E

GG

EL

L E

MM

MM

22

1 2; dd

c c

GGE k E kR r

MM


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 Rc to rc, conserve angular momentum. Ld/Md = L/M.

1 115 52 222 2

2

2

1

11 122 22

1

1

2

:

di d d d d c d

d c

i c d

d

d c

c d

c

EL Rk

L E k r

R

R k

r

that isk

k r

k

M

M M MM

M

M

MM M M

M


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.


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 fc then the overdensity increases by a factor fc

3.

• We consider a galaxy with M=1011 M , r ~ 10 kpc so that obs/ c ~ 105.


Zcollapse

• So that:

• Note also that we should integrate to account for the change of b

during the collapse.

• The factor 22/3 since: t collapse ~ 2 tm ; b a –3 t –2 ; so that the density contrast increases by a factor 22 and zcollapse factor 22/3.

• For dissipation_less collapse fc ~ 2.

• For a disk fc ~ 10 or more.

2

33 3

13

13

95.6 1

16

1 30

5.6

obs c b m c turn around c

obsturn around

c c c

f t z f

zf f


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,


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 = <(2> = C M –(3+n)/3

(M) = (M/Mo) –(3+n)/6

With the constant Mo to be determined.


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:

3

6

;

n

o

N M Mb b b Bias factM

M Morb

N


Normalization and derivation of the redshift at which a given mass collapse

(3 )15 1 6

6

315 1

3

6

15 1

1.15100.9 ;

0.91.1510

0.9;

1.1510

; & 1; 11.686

n

o

n

o

n

oo coll

hN bN

hb

b h

for z

MM

M M

MM

M

MM M


Finally and However

• The reasoning was not completely correct since for N/N ~ 1 at some mass scale Mo 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, (Mo) = 1/b. If the theory says that (M) = 1/b (M/Mc)-n then we will take Mc = Mo.

• 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 ti.

6

315 11.1510 1 1.686 0.9n

collbz h z

M M


Or saying differently

• In our spherical model we had =( b -1is 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 (Mo) = 0.57/b rather than (Mo) = 1/b. Applying this correction we find:

6

315 11.1510 1 1.686 0.57n

collbz h z

M M


Typical Mass which collapse and becomes non linear at various redshifts. =1,h=0.5 and density contrast o(M)= (M) with =2.

0 2 4 6 8 10

8

10

12

14

Log

(M

/M)

z

b=1, n=-1

b=2, n=-1

b=1, n=-2

b=2, n=-2


Timetable for Formation

Gravitational potential fluctuations Z 103

Spheroids of Galaxies Z ~ 20

The first Engines for active galactic nuclei Z 10

The intergalactic medium Z ~ 10

Dark Matter Z 5

Dark halos of galaxies Z ~ 5

Angular momentum of rotation of galaxies Z ~ 5

The first 10% of the heavy elements Z 3

Cosmic magnetic fields Z 3

Rich clusters of galaxies Z ~ 2

Thin disks of spiral galaxies Z ~ 1

Superclusters, walls and voids Z ~ 1


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: nH =106 m-3 so that for a distance d through the disk we have:

• NH= 3.09 1025 (d/kpc) m-2; E (B-V) = 0.53 (d/kpc) and AV = 1.6 (d/kpc)

25 2 1

225

5.810 ( )

( )5.810

Htot

Htot

N E B V m mag

NE B V m

1 m2

1 kpc

nH


Toward the GC

• AM ~ 0.6 ; AB = 34.5

• The Probability for a B photon to reach us is:

• 10 –0.4 (34.5) = 10 –13.8 = 1.6 10 -14


Standard ISM Extinction

Band X eff/nm M (EX-V/EB-V) (AX/AV)

U 365 5.61 1.64 1.531

B 445 5.48 1. 1.324

V 551 4.83 0 1.

R 658 4.42 -0.78 .748

I 806 4.08 -1.60 .482

J 1220 3.64 -2.22 .282

H 1630 3.32 -2.55 .175

K 2190 3.28 -2.74 .112

L 3450 3.25 -2.91 .058

M 4750 -3.02 .023

N -2.93 .052

Cosmology Class - 2002/20031 Galaxy Formation and non Linear collapse By Guido Chincarini University...

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Transcript of Cosmology Class - 2002/20031 Galaxy Formation and non Linear collapse By Guido Chincarini University...