Topics in Galaxy Formation

(3) Non-linear Stages of Structure Formation

• The Non-linear Collapse of Spherical Density Perturbations

• The Zeldovich Approximation

• The Press-Schechter Formalism

• Cooling Processes

• Problems with the Standard Picture

1

The Non-linear Collapse of Density Perturbations

The collapse of a uniform spherical density perturbation in an otherwise uniformUniverse can be worked out exactly, a model sometimes referred to as spherical top-hatcollapse. The dynamics are the same as those of a closed Universe with Ω0 > 1. Thevariation of the scale factor of the perturbation ap is given by the parametric solution

ap = A(1 − cos θ) t = B(θ − sin θ) ,

A =Ω0

2(Ω0 − 1)and B =

Ω0

2H0(Ω0 − 1)3/2.

The perturbation reached maximum radius at θ = π and then collapsed to infinitedensity at θ = 2π. The perturbation stopped expanding, ap = 0, and separated out ofthe expanding background at θ = π. This occurred when the scale factor of theperturbation was a = amax, where

amax = 2A =Ω0

Ω0 − 1at time tmax = πB =

πΩ0

2H0(Ω0 − 1)3/2. (1)

2

The Non-linear Collapse of Density Perturbations

The density of the perturbation at maximum scale factor ρmax can now be related tothat of the background ρ0, which, for illustrative purposes, we take to be the criticalmodel, Ω0 = 1. Recalling that the density within the perturbation was Ω0 times that ofthe background model to begin with,

ρmax

ρ0= Ω0

(

a

amax

)3= 9π2/16 = 5.55 , (2)

where the scale factor of the background model has been evaluated at cosmic timetmax. Thus, by the time the perturbed sphere had stopped expanding, its density wasalready 5.55 times greater than that of the background density.

Interpreted literally, the spherical perturbed region collapsed to a black hole. It is muchmore likely to form some sort of bound object. The temperature of the gaseousbaryonic matter increased until internal pressure gradients became sufficient to balancethe attractive force of gravitation. For the cold dark matter, during collapse, the cloudfragmented into sub-units and then, through the process of violent relaxation, theseregions came to a dynamical equilibrium under the influence of large scale gravitationalpotential gradients.

3

The Non-linear Collapse of Density Perturbations

The end result is a system which satisfies the Virial Theorem. At amax, the sphere isstationary and all the energy of the system is in the form of gravitational potentialenergy. For a uniform sphere of radius rmax, the gravitational potential energy is−3GM2/5rmax. If the system does not lose mass and collapses to half this radius, itsgravitational potential energy becomes −3GM2/(5rmax/2) and, by conservation ofenergy, the kinetic energy, or internal thermal energy, acquired is

Kinetic Energy =3GM2

5(rmax/2)− 3GM2

5rmax=

3GM2

5rmax. (3)

By collapsing by a factor of two in radius from its maximum radius of expansion, thekinetic energy, or internal thermal energy, becomes half the negative gravitationalpotential energy, the condition for dynamical equilibrium according to the VirialTheorem. Therefore, the density of the perturbation increased by a further factor of 8,while the background density continues to decrease.

4

The Non-linear Collapse of Density Perturbations

The scale factor of the perturbation reached the value amax/2 at timet = (1.5 + π−1)tmax = 1.81tmax, when the background density was a further factorof (t/tmax)2 = 3.3 less than at maximum. The net result of these simple calculationsis that, when the collapsing cloud became a bound virialised object, its density was5.55 × 8 × 3.3 ≈ 150 times the background density at that time.

These simple calculations illustrate how structure forms according to the large scalesimulations. They show that galaxies and clusters must have formed rather later thanthe simple estimates we gave in the second lecture. According to Coles and Lucchin,the systems become virialised at a time t ≈ 3tmax when the density contrast wasabout 400.

Using these arguments, galaxies of mass M ≈ 1012 M⊙ could not have beenvirialised at redshifts greater than 10 and clusters of galaxies cannot have formed atredshifts much greater than one.

5

The Zeldovich Approximation

The next approximation is to assume that the perturbations were ellipsoidal with threeunequal principal axes. In the Zeldovich approximation, the development ofperturbations into the non-linear regime is followed in Lagrangian coordinates. If x andr are the proper and comoving position vectors of the particles of the fluid, theZeldovich approximation can be written

x = a(t)r + b(t)p(r) . (4)

The first term on the right-hand side describes the uniform expansion of thebackground model and the second term the perturbations of the particles’ positionsabout the Lagrangian (or comoving) coordinate r. Zeldovich showed that, in thecoordinate system of the principal axes of the ellipsoid, the motion of the particles incomoving coordinates is described by a ‘deformation tensor’ D

D =

a(t) − αb(t) 0 00 a(t) − βb(t) 00 0 a(t) − γb(t)

. (5)

Because of conservation of mass, the density in the vicinity of any particle is

[a(t) − αb(t)][a(t) − βb(t)][a(t) − γb(t)] = ¯a3(t) , (6)

where ¯ is the mean density of matter in the Universe.

6

The Zeldovich Approximation

The clever aspect of the Zeldovich solution is that, although the constants α, β and γ

vary from point to point in space depending upon the spectrum of the perturbations, thefunctions a(t) and b(t) are the same for all particles. In the case of the critical model,Ω0 = 1,

a(t) =1

1 + z=

(

t

t0

)2/3

and b(t) =2

5

1

(1 + z)2=

2

5

(

t

t0

)4/3

, (7)

where t0 = 2/3H0. The function b(t) has exactly the same dependence upon scalefactor (or cosmic time) as was derived from perturbing the Friedman solutions.

If we consider the case in which α > β > γ, collapse occurs most rapidly along thex-axis and the density becomes infinite when a(t) − αb(t) = 0. At this point, theellipsoid has collapsed to a ‘pancake’ and the solution breaks down for later times.Although the density becomes formally infinite in the pancake, the surface densityremains finite, and so the solution still gives the correct result for the gravitationalpotential at points away from the caustic surface. The Zeldovich approximation cannotdeal with the more realisitic situation in which collapse of the gas cloud into the pancakegives rise to strong shock waves, which heat the matter falling into the pancake.

7

The Zeldovich Approximation

The results of numerical N-body simulations have shown that the Zeldovichapproximation is quite remarkably effective in describing the evolution of the non-linearstages of the collapse of large scale structures up to the point at which caustics areformed.

N-body Simulation Zeldovich Approximation

8

Non-Linear Development of theDensity Perturbations

It is evident from the power-law form of the two-point correlation function for galaxiesξ(r) = (r/r0)

−1.8 that on scales much larger than the characteristic length scaler0 ≈ 7 Mpc, the perturbations are still in the linear stage of development and soprovide directly information about the form of the processed initial power spectrum.

On scales r ≤ r0, the perturbations become non-linear and it might seem more difficultto recover information about the processed power-spectrum on these scales. Animportant insight was provided by Hamilton and his colleagues who showed how it ispossible to relate the observed spectrum of perturbations in the non-linear regime,ξ(r) ≥ 1, to the processed initial spectrum in the linear regime.

9

Non-Linear Development of theDensity Perturbations

The variation of the spatial two-pointcorrelation function with the square ofthe scale factor as the perturbationsevolve from linear to non-linearamplitudes to bound systems.

The corresponding evolution of thespatial two-point correlation function asa function of redshift, normalised toresult in a two-point correlation functionfor galaxies which has slope −1.8.

10

The Modified Initial Power Spectrum

Max Tegmark and hiscolleagues have shownhow many other pieces ofdata are consistent withthis picture. Note:• Overlap of WMAP

and SDSS powerspectra.

• Statistics ofgravitational lensing.

• Power spectrum ofneutral hydrogenclouds.

11

The Press-Schechter Formalism

According to the Cold Dark Matter scenario for galaxy formation, galaxies and largerscale structures are built up by the process of hierarchical clustering. Press andSchechter (1974) provided an analytic formalism for the process of structure formationonce the density perturbations had reached such an amplitude that they could beconsidered to have formed bound objects.

The analysis begins with the assumption that the primordial density perturbations areGaussian fluctuations, that is, the phases of the waves which make up the densitydistribution are random and the distribution of the amplitudes of the perturbations of agiven mass M can be described by a Gaussian function

p(δ) =1√

2πσ(M)exp

[

− δ2

2σ2(M)

]

, (8)

where δ = δ/ is the density contrast associated with perturbations of mass M .Being a Gaussian distribution, the mean value of the distribution is zero and its varianceis σ2(M), that is, the mean-squared fluctuation is

〈δ2〉 =

⟨(

δρ

ρ

)2⟩

= σ2(M). (9)

12

The Press-Schechter Formalism

Press and Schechter assumed that, when the perturbations grow in amplitude to avalue greater than some critical value δc, they develop rapidly into bound objects withmass M .

We assume that the perturbations have a power-law power-spectrum P(k) = kn andwe know the rules which describe the growth of the perturbations with cosmic epoch.For illustrative purposes, let us assume that the background world model is the criticalEinstein-de Sitter model, Ω0 = 1,ΩΛ = 0, so that the perturbations develop asδ ∝ a ∝ t2/3.

For fluctuations of a given mass M , the fraction F(M) of those which become boundat a particular epoch are those with amplitudes greater than δc

F(M) = 1√

2πσ(M)

∫ ∞

δcexp

[

− δ2

2σ2(M)

]

dδ = 12 [1 − Φ(tc)] , (10)

where tc = δc/√

2σ and Φ(x) is the probability integral defined by

Φ(x) =2√π

∫ x

0e−t2 dt . (11)

13

The Press-Schechter Formalism

We can relate the mean square density perturbation on a particular scale to the powerspectrum of the perturbations.

σ2(M) =

⟨(

δ

)2⟩

=⟨

δ2⟩

= AM−(3+n)/3 , (12)

where A is a constant. We can now express tc in terms of the mass distribution

tc =δc√

2σ(M)=

δc√2A1/2

M(3+n)/6 =

(

M

M∗

)(3+n)/6

, (13)

where we have introduced a reference mass M∗ = (2A/δ2c)3/(3+n).

Since the amplitude of the perturbation δ(M) grows as δ(M) ∝ a ∝ t2/3, it followsthat σ2(M) = δ2(M) ∝ t4/3, that is, A ∝ t4/3. Therefore,

M∗ ∝ A3/(3+n) ∝ t4/(3+n) , (14)

which can be rewritten

M∗ = M∗0

(

t

t0

)4/(3+n)

, (15)

where M∗0 is the value of M∗ at the present epoch t0.

14

The Press-Schechter Formalism

The fraction of perturbations with masses in the range M to M + dM isdF = (∂F/∂M) dM . In the linear regime, the mass of the perturbation is M = ¯V

where ¯ is the mean density of the background model. Once the perturbation becomesnon-linear, collapse ensues and ultimately a bound object of mass M is formed. Thespace density N(M) dM of these masses is V −1, that is,

N(M) dM =1

V= − ¯

M

∂F

∂MdM , (16)

the minus sign appearing because F is a decreasing function of increasing M .

We now have everything we need to determine the mass distribution and how it evolveswith time. Noting that

dΦ

dx=

2√π

e−x2, (17)

we find

N(M) =1

2√

π

(

1 +n

3

)

¯

M2

(

M

M∗

)(3+n)/6

exp

−(

M

M∗

)(3+n)/3

, (18)

in which all the time dependence of N(M) has been absorbed into the variation of M∗

with cosmic epoch.

15

The Press-Schechter Formalism

The evolution of thePress-Schechter massfunction with cosmicepoch for the Ω0 = 1

model. The usefulness ofthis function is that itgives a good descriptionof the results of thenumerical analyses.

16

The Press-Schechter Formalism

Comparison of thePress-Schechterformalism with the resultsof the supercomputersimulations by Springel,White and their colleages.The dotted line shows thePress-Schechter resultsand the red boxes theresult of the simulations.

17

The Press-Schechter Formalism

For example, the functioncan be used to predictthe development of themass spectrum of objectsof different mass withcosmic epoch. Thisprovides an importanttest of the picture ofstructure formation.

18

The Importance of Dissipation Processes

So far we have mostly considered the development of perturbations under the influenceof gravity alone. In addition, we need to consider the role of dissipation, by which wemean energy loss by radiation, resulting in the loss of thermal energy from the system.

In a number of circumstances, once the gas within the system is stabilised by thermalpressure, loss of energy by radiation can be an effective way of decreasing the internalpressure, allowing the region to contract in order to preserve pressure equilibrium. Ifthe radiation process is effective in removing pressure support from the system, thiscan result in a runaway situation, known as a thermal instability. This is the processwhich may be responsible for the cooling flows which are present in the hot gas in thecentral regions of rich clusters of galaxies

19

The Cooling Rate per Unit Volume

The cooling rate per unitvolume Λ(T) of anastrophysical plasma withnumber density 1 nucleuscm−3 by radiation fordifferent cosmicabundances of the heavyelements as a function oftemperature.

20

The Stability of Perturbations against Cooling

This cooling curve can beconverted into a numberdensity-temperaturediagram on which variousloci can be plotted, aswell as the baryonicmasses of the systems. Ifthe systems falls abovethe cooling locus, coolingwill be more importantthan gravitationalcollapse.

21

Millennium Simulations including Baryonic Matter

The panels show:

• The dark matter

• The gas density

• The shock waves

• The temperature of the gas

• The Sunyaev-Zeldovich temperature decrement

• The kinematic Sunyaev-Zeldovich effect

Animations by Volker Springel. Run simulations

22

Evolution of Population of Rich Clusters of Galaxies

The abundances ofclusters changedramatically with thecosmological model(Borgani and Guzzo,2001). The rich clustersare identified for differentmodels of structureformation. Thedifferences are importantat relatively smallredshifts.

23

Problems with the Standard Picture

• The predicted excess of dwarf galaxies see simulations. Possibly not observedbecause the potential wells are too shallow to prevent the sweeping out of all thediffuse baryonic gas.

• The structure of galaxies and clusters see simulations. The simulations predict thatthe distribution of matter tends to be more ‘cuspy’ than observed.

• The early formation of elliptical galaxies

• The mass-metallicity relation for galaxies

• The global star-formation rate

24

Elliptical Galaxies at Large Redshifts

There seems to belittle change in thebaryonic massesof massive ellipticalgalaxies out toredshift z = 2.

25

Elliptical Galaxies at Large Redshifts

The same result isfound if we add inour 3CR radiogalaxies.

26

Elliptical Galaxies at Large Redshifts

There seems to be littlechange in the spectra ofelliptical galaxies out toredshift z = 2.

27

The Metallicity Issue

The most massive galaxieshave greater metallicitiesthan less massive galaxies.Yet, in the hierarchicalclustering picture, themassive galaxies are builtout of lower mass galaxies.

28

The Global Star Formation Rate

The maximum rate of starformation seems to haveoccurred at z ∼ 1, by whichredshift the giant ellipticalgalaxies were quiescent.This is often referred to asdown-sizing. The starformation is associated withgalaxies of mass significantlyless than the most luminouselliptical galaxies.

29