the galaxy formation paradigm paradigm r. giovanelli astro620/spring ‘07 remember to mention

The Galaxy Formation The Galaxy Formation ParadigmParadigm

R. GiovanelliAstro620/Spring ‘07

Remember to mention .pdf file

When did galaxies form: a rough estimate -1

Consider a small ( <<1) spherical perturbation of radius r:

For H0=100h km/s/Mpc,

For h=0.7 and matter=0.25,

i.e. an ~L* galaxy coalesced from matter within a comoving volume of roughly ~ 1-1.5 Mpc radius

mattermatter GHrrrM )8/3()3/4()3/4()( 233

32121016.1/)( Mpcmattersun rhMrM

311104.1/)( Mpcsun rMrM

When did galaxies form: a rough estimate -2

The typical radius of the visible parts of an L* galaxy as z=0 is 10-20 kpc

However, we know that is embedded in a DM halo 5-10 times larger, say 75 kpc

The density enhancement represented by such a galaxy is thus

matter ~ (1500/75)3 ~ 8000

Since matter evolves like (1+z)3 , the density fluctuation was of

order ~ 1 (“epoch of formation”) when (1+z)3 ~ 8000 (*), i.e.

an L* galaxy could not have separated from Hubble flow before z~20

(*) If we apply what we’ll learn later in the analysis of the spherical infall model, such a perturbation would “turn around” at a z~10

Jeans Instability-1Jeans Instability-1

Consider a 3-D fluid of density, pressure and velocity field and potential field given by:

which obey

Continuity Eqn.:

Euler’s Eqn.:

Poisson’s Eqn.:

Eqn. of State:

),(),,(),,(),,( trtrvtrptr

)],([)/(

4

1)(

0)(

2

trptrp

G

pvvt

v

vt

Gas dynamics eqns. are written in Eulerian form.In Cosmology, it is preferred to write them in Lagrangian form, where derivatives refer to changes in a gas parcel as it moves with velocity v:

vtdt

d


),(),,(),,(),,( 0000 trtrvtrptr

),(),(),( trtrtr o

plus a small perturbation to (and similar ones for the other variables): , etc

Let the solution be a combination of an unperturbed or background component:

In an expanding Universe, it is convenient to use comoving coordinates r, so that

where a(t) is the cosmological scale factor

rtax)(

After some algebra, the equation for the evolution of the perturbation is:

oc

s Ga

c

dt

d

a

a

dt

d42 2

2

2

2

2

Where , and cs the speed of sound. ac

))(()( tkrkic

ccekC

The solution of the perturbation equation can be expressed as a superposition of plane waves:

which yields:

)4(2 222

2

sco ckGdt

d

a

a

dt

d

where “c” subscripts imply quanties expressed in the comoving reference frame.


First, we consider solutions in a static medium

0)4( 222

2

os Gkct

Time-evolving solutions are allowed; a dispersion relation is obtained:

where k=|k|.os Gkc 4222

Gravitational Instability in a Static Medium-1

In a static medium,

ckkaa ,0,1

The perturbation is a sound wave of when

i.e. when either the mean density is very low or when the wavelength of the perturbation is small.

222 kcs

os Gkc 422

k/2

Gravitational Instability in a Static Medium-2

2/12

os

JJ G

ck

The value of k for which =0 is the Jeans’ wavenumber and the corresponding wavelength is Jeans’ length

and Jeans’ Mass 3)2/(3

4J

oJM

•For J, is real and the solution is a sound wave

pressure provides support against gravitational collapse

•For J, is imaginary and the solution evolves exponentially,

with a timescale (~ the same for all scales J)

2/122/11 ])/(1[)4(|| JoG

Which for J is the free-fall collapse time 2/1)( off G

Interlude: Cosmic Evolution

Each cosmic component is characterized by an equation of state

and each evolves with the scale factor a(t)

In a flat Universe, the scale factor evolves as

2cwp

consta w )1(3

)]1(3/[2)( wtta

In a matter-dominated Universe (w ~ 0)

and

In a radiation-dominated Universe (w =1/3)

and

In a -dominated Universe (w = -1)

and

3/2)( tta

2/1)( tta

Hteta )(

33 )1()( ztamatter

44 )1()( ztarel

const

relmatter

024 /103.41 Khz mattereq

at

Ko =1.68 for 3 neutrinos

(that’s a zeq ~4000)

Gravitational Instability in an Expanding Universe - 1

The perturbation equation allows evolving solutions, a dispersion equation and the definition of a Jeans’ length , both in the case of a flat, matter dominated and a radiation dominated Universe.

The main difference between these solutions and those for a static medium is that the growth rate is not exponential.

In a flat Universe two evolving, power law solutions are allowed forJ:

matter dominated radiation dominated

- a growing mode

- a decaying mode

3/2t1

t

t1

t

In the evolution of perturbations in an expanding Universe, two agents counteract gravity:

- pressure (as in the static case) - universal expansion itself: fluctuation will not grow if collapse time is greater than Hubble time


* Consider first the evolution of perturbations of scale > the Hubble radius rH ;

Pressure doesn’t play a role on scales > rH , thus the perturbation grows as 3/2t

t

i.e. as in a matter dominated Universe

and as in a radiation dominated one.

* We can understand that as follows:

- consider the perturbation as a slightly overdense sphere within a flat (k=0) Universe of scale factor ao; matter outside the sphere does not affect evolution within the sphere, so the sphere evolves like a Friedman Universe with density higher than critical, i.e. k=1

- Let the density of the perturbation be 1 and its scale factor a1

oo GH

GaH

)3/8(

)3/8(/1 1211

* The evolution of the scale factor in

the perturbed region and in the flat background are then respectively


oo GH

GaH

)3/8(

)3/8(/1 1211

* Let’s compare the densities

when the expansion rates are equal (H1 = H2 )

21

1

8

3

aG ooo

o

* If the perturbation is v small, a1 ~ ao 12 )( ao

•Remembering the evolution of the scale factor in radiation dominated and matter dominated Universes

i.e. we recover the behavior of the

growing mode

3/2

2

ta

ta

The amplitude of a perturbation with > rH always grows


In a Universe with multiple components, the evolution of a perturbation depends on the type of component that is perturbed and on the type of component that is gravitationally dominant. - In the expression for Jeans’ length the speed of sound (or the velocity dispersion) is that of the perturbed component; however, the density is that of the gravitationally dominant component.

2/12

os

JJ G

ck

Consider, for example, a DM perturbation of scale which has entered the horizon, and is such that

Pressure is unable to prevent collapse

However, if the Universe is radiation dominated the expansion time is determined by rad ; then the Universe expands too fast and prevents the DM perturbation from collapsing.

2/1)( DMpressure Gv

t

/122/1exp )()( DMrad GGt

In the radiation dominated era, perturbations with scale < rH cannot grow


Evolution of Dark Matter Perturbations

A. Evolution of the density perturbation

1. > rH , i.e. a < aenter perturbation scale > than Hubble radius

2. Perturbation enters horizon, radiation dominates: aenter < a < aeq

3. Matter dominates: aeq < a

2a

const

a

Log a



B. Evolution of Jeans’ Length2/1

2

os

JJ G

ck

For DM, cs is replaced by the velocity dispersion of the DM particles. We distinguish three epochs:

1. Before anr , DM particles are relativistic and, while radiation is the dominant gravitational form

2. Between anr and aeq , the velocity dispersion of DM particles decays as a-1 , so

3. After aeq , matter becomes the gravitationally dominant form

22/142/1 )()( aaG radJ

aaaGv radJ 2/1412/1 )()(

2/12/1312/1 )()( aaaGv DMJ



B. Evolution of Jeans’ Mass

For DM, cs is replaced by the velocity dispersion of the DM particles. We distinguish three epochs:

1. Before anr , DM particles are relativistic and, while radiation is the dominant gravitational form

2. Between anr and aeq , the velocity dispersion of DM particles decays as a-1 , so

3. After aeq , matter becomes the gravitationally dominant form

3)2/(3

4J

oJM

3633)( aaaM JDMJ

constaaM JDMJ 333)(

2/32/333)( aaaM JDMJ



anr aeq

MJ

J


Evolution of Baryonic Matter Perturbations

2/32/1

2/32/3

32

,

,

,

aMaaa

aMaaaa

aMaaa

JJrec

JJreceq

JJeq

Proceeding through analogous scaling relations

With one major difference wrt DM:-Before the epoch of recombination, radiation and baryonic matter are tightly coupled, radiation providing the pressure of the coupled fluid.-After recombination, the pressure drops by many orders of magnitude to practically zero (that of a few 1000K baryon fluid).- As a result, both the Jeans’ Length and Mass drop precipitously. Numerically:

32214 )/())(/(102.3/ eqmassmassbaryonsunJ aahMM

2/32214 )/())(/(102.3/ eqmassmassbaryonsunJ aahMM 2/32/124 )/())(/(10/ eqmassmassbaryonsunJ aahMM

eqaa

receq aaa

eqaa

For:


anr aeq

MJ

J

Evolution of Baryonic Matter Perturbations

Solid=baryonsDotted=DM


1. Before entering the horizon, all perturbations grow like a2 , as they exceed the Jeans’ length – which is ~ the Hubble radius.

2. After entering the horizon and for as long as radiation is the dominant gravitational form (a < aeq ), the perturbation is prevented from growing by the cosmic expansion.

3. At a=aeq , DM becomes the gravitationally dominant form and the DM component of the perturbation resumes growth, its amplitude increasing like a.

4. The baryonic component of the perturbation is prevented from growing by the tight coupling with radiation, via Thomson scattering, until a=arec .

5. After recombination, the baryonic component can resume growth; between aeq and arec , however, the DM component of the perturbation has grown by a factor arec /aeq ~ 21mass h2 .

6. When baryons decouple from photons, they fall into the deep potential wells created by DM.

7. When baryon = DM, both components of the perturbation grow like a.

Nonlinear Regime: the Spherical Infall Model-1

The description of the evolution of perturbations given so far only applies to small amplitudes, i.e. to the linear regime.

The next step is the analysis of the Spherical Infall Model .

To summarize (see notes for details and derivations):- A slightly overdense region initially expands, but at a slightly lower rate than the universal rate-The expansion of each shell of the overdense region eventually stops and the motion “turns around” initiating collapse- The collapse is followed by a process of violent relaxation that reshuffles, randomizing ,the orbits of the infalling particles, leading to virialization.-Through dissipative processes, baryons lose energy and fall deeper in the potential well of DM.-If the cooling time of the baryon gas is smaller than the collapse time, fragmentation will take place and smaller units can collapse


iiturn zz )1(57.01

6.516/91 2 turn

A couple of important results of the spherical infall model analysis:

1. A perturbation of amplitude i at redshift zi will turn around at

E.g. an overdensity of 1% at z=1000 will turn around at z~4.7 at which time will be overdense by a factor of 4.6 wrt the background

2. The turn around overdensity is always 4.6 times the background density, independent on initial conditions


We define a virialized system as one where E=U+K=-K, i.e. |U|=2K

At turn around time, all the energy is in U, so that E=U~3GM2 /5Rmax

We define virial velocity , virial radius:

and the redshift of virialization

virRGMU 5/3|| 2 2/turnvir RR 2/1)5/6( turnvir RGMv

)1(63.0)1(36.01 turniivir zzz

3

3

3

)1(180

)(1

1568)(6.582

viro

virvir

turnturnbgturnvir

z

zz

zz

… as for the density at the epoch of virialization:


The spherical infall model also yields scaling relations between size, mass and velocity that precur the observed relations we all use and love:

13/1123/12 )1()10/())(164( virsunmattervir zMMhkpcR

2/13/1126/12 )1()10/())(/125( virsunmattervir zMMhskmV

We can also define a Virial Temperature via

… yielding:

2)2/1()2/3( virvirial VT

)1()10/())(1068.3( 3/2123/125virsunmattervir zMMhKT

Cooling-1

The cooling function of a primordial baryonic fluid, i.e.

(T) energy loss by radiation p.u. time p.u. volume

is a function of T and contributed to by 3 main processes:

1. Compton cooling with the CMB photons (important only at z>10)2. Thermal bremsstrahlung (free-free) at T > 106 K3. Bound-bound and recombination transitions of H and He at 104 and 105 K

[If elements > He are present, cooling rates below 10^6 K increase dramatically.We define cooling time

which is useful to compare to the Gravitational collapse time

By setting tcool = tdyn , we obtain a relationship between density and temperature.

)(

3

|/| 2 Tn

kTn

dtdE

Et

b

bcool

bdyn nGt 2/1)(

the galaxy formation paradigm paradigm r. giovanelli astro620/spring ‘07 remember to mention

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