ast4320: lecture 2 - uio

AST4320: LECTURE 2

M. DIJKSTRA

1. Growth of Small Perturbations in a Expanding & Static Media

The goal of this lecture is to derive an equation describing the time evolution of thedensity in a small perturbation in expanding & static media. This discussion closely followsChapter 11.2 in Longair, but in these notes I present intermediate steps not given in thisbook.

1.1. Basic Equations. We denote the density at a position x and time t with ⇢(x, t), thevelocity of the an ‘ideal’ fluid with v(x, t), the pressure of the fluid with p(x, t), and thegravitational potential with �(x, t). Throughout, I will drop writing the dependence onboth x and t. Pressure and density provide us with a complete description for an idealfluid.

The three equations which describe the dynamics of this ideal fluid in a gravitationalfield are:

@⇢

@t+r · (⇢v) = 0; Continuity equation(1)

@v

@t+ (v ·r)v = �1

⇢rp�r�; equation of motion/Euler equation(2)

r2� = 4⇡G⇢ Poisson equation.(3)

We first use the identity r · (⇢v) = ⇢r · v + v ·r⇢ to recast the continuity equation as

(4)@⇢

@t+ (v ·r)⇢+ ⇢r · v = 0.

The reason for doing this is that now both the continuity and the Euler equation containan operator ( @

@t + v ·r) ⌘ ddt . Here d

dt denotes the ‘total’ time derivative which gives thetime evolution of a quantity in a frame that is moving along with the fluid1. If we use total

1In the book ‘Physical Cosmology’ by J. Peacock (Chapter 2.1) an interesting example of the relevance

of the total derivative is given; ’The changes experienced by an observed moving with the fluid are inevitably

a mixture of temporal and spatial changes. If I start to feel rain as I cycle towards my destination, it might

be a good idea to cycle harder in the hope of arriving before the downpour really starts, but it could also be

that it is raining near my destination, and I should stop and wait for the local shower to finish. ...’

1

2 M. DIJKSTRA

derivates, Eq 1-3 change to

d⇢

dt+ ⇢r · v = 0; Continuity equation(5)

dv

dt= �1

⇢rp�r�; equation of motion/Euler equation(6)

r2� = 4⇡G⇢ Poisson equation.(7)

Especially the Euler equation illustrates why it is more natural to write the equation thisway: this equation simply says that acceleration equals the sum of all forces (per unitmass).

1.2. The Unperturbed Equations. We define the quantities of the ‘unperturbed’ fluidto be ⇢0, v0, p0, and �0. These quantities must obey the continuity, Euler & Poissonequations, i.e.

d⇢0dt

+ ⇢0r · v0 = 0; Continuity equation(8)

dv0

dt= � 1

⇢0rp0 �r�0; equation of motion/Euler equation(9)

r2�0 = 4⇡G⇢0 Poisson equation.(10)

Actually, it is interesting to point out that in a static medium, these solutions do not existbeyond the solution ⇢0 = p0 = v0 = 0 (Google ‘Jeans swindle’). We will ignore this fornow. We are most interested in the solution in an expanding medium.

1.3. The Perturbed Equations. We define the quantities of the ‘perturbed’ fluid to be⇢ ⌘ ⇢0 + �⇢, v ⌘ v0 + �v, p ⌘ p0 + �p, and � ⌘ �0 + ��. These quantities must alsoobey the continuity, Euler & Poisson equations. We are going to just plug everything in,recalling that ( @

@t + v ·r) ⌘ ddt . In the lecture, we focussed on the continuity equation for

the perturbed fluid, which reads

@⇢

@t+ (v ·r)⇢+ ⇢r · v = 0 )(11)

(@⇢0 + �⇢)

@t+ ([v0 + �v] ·r)(⇢0 + �⇢) + (⇢0 + �⇢)r · (v0 + �v) = 0(12)

We are going to explicitly write out all terms in this equation. I have color-coded someterms to simplify the analysis of this equation:

@⇢0@t

+@�⇢

@tI

+ (v0 ·r)⇢0 + (v0 ·r)�⇢II

+ (�v ·r)⇢0 + (�v ·r)�⇢+(13)

+⇢0r · v0 + ⇢0r · �v + �⇢r · v0 + �⇢r · �v = 0

AST4320: LECTURE 2 3

The red terms correspond to the unperturbed continuity equation (Eq 4), and so theirsum is 0. The blue terms contain products of two arbitrarily small quantities, and canbe ignored. Finally, the green term contains spatial derivatives of the unperturbed densityfield ⇢0. We assume this unperturbed density field to be homogeneous, and we can thereforeignore its spatial derivative. Note that sum of the terms with the ‘I‘ and ‘II‘ under themsimply correspond to the total derivative of �⇢. We are thus left with

d�⇢

dt+ ⇢0r · �v + �⇢r · v0 = 0.(14)

We can simplify this further if we define the over density � ⌘ �⇢/⇢0. Substituting this intothe equation above gives

d⇢0�

dt+ ⇢0r · �v + ⇢0�r · v0 =(15)

�d⇢0dt

+ ⇢0d�

dt+ ⇢0r · �v + ⇢0�r · v0 = 0.

Recall from the unperturbed continuity equation (Eq 8) that d⇢0dt = �⇢0r · v

0

. The firstand fourth term then cancel out, and we are left with

d�

dt= �r · �v .(16)

In Exercise 1 you are asked to go through a similar analysis for the Euler and Poissonequations. The three equations we are left with are:

d�

dt= �r · �v Continuity equation(17)

d�v

dt+ (�v ·r)v0 = � 1

⇢0r�p�r��; equation of motion/Euler equation(18)

r2�� = 4⇡G�⇢ Poisson equation.(19)

1.4. Switch to Comoving Coordinates. We next switch to ‘co-moving’ coordinates r

which expand along with the Universe. These coordinates relate to the physical coordinatesx via x ⌘ a(t)r. Spatial derivatives in comoving coordinates are denoted with rc ⌘ a(t)r.The total velocity of the perturbed fluid can be written as v = �x

�t .

v =�x

�t=

�a(t)r

�t= a(t) �r�t + r

�a�t .(20)

Recall v = �v + v0.

We can identify the second terms on the right hand side as the velocity arising from theexpansion of the Universe. The first term on the other hand, denotes additional ’peculiar’velocity. We have �v = a(t) �r�t ⌘ a(t)u. If we substitute this into the perturbed continuityequation (Eq 17), we get

4 M. DIJKSTRA

d�

dt= �r · au = �rc · u Continuity equation(21)

(22)

We now turn our attention to the perturbed Euler equation (Eq 18). On the first line,we only substitute �v = a(t)u. In the second line, we replace the proper/physical spatialderivatives with comoving derivates and we replace2 v0 = r

dadt ⌘ ra (Eq 20):

dau

dt+ (au ·r)v0 = � 1

⇢0r�p�r�� )(23)

dau

dt+ (au ·r)ar0 = � 1

a⇢0rc�p�

rc��

a)

adu

dt+ u

da

dt+ (au ·r)ar0 = � 1

a⇢0rc�p�

rc��

a.

The third line simple expands the first term. In Assignment 1 you are asked to showthat the third term on the right hand side equals

(au ·r)ar0 = au assignment 1(24)

With this simplification,

adu

dt+ 2au = � 1

a⇢0rc�p�

rc��

a.(25)

Now we take the comoving spatial derivative on both sides of this equation:

adrc · u

dt+ 2arc · u = � 1

a⇢0r2

c�p�r2

c��

a.(26)

Note that we used that⇣rc

ddt

⌘=

⇣ddtrc

⌘(convince yourself). Next use Eq 21 which

states that rc · u = �d�dt , and we are left with

�ad2�

dt2� 2a

d�

dt= � 1

a⇢0r2

c�p�r2

c��

a. )(27)

d2�

dt2+ 2

a

a

d�

dt=

1

a2⇢0r2

c�p+r2

c��

a2.

2Note that a ⌘ da

dt =

@a@t because the scale factor has no spatial dependence.


1.5. The Final Equation. We get to our final solution in 3 steps

1. Get rid of the �p term by assuming that the fluctuations are adiabatic, in whichpressure and density perturbations are related to the adiabatic sound speed, c2s ⌘ @p/@⇢.

2. Use the perturbed Poission equation to replace r2c��/a

2 = r2�� = 4⇡G�⇢.

These two steps give us

d2�

dt2+ 2

a

a

d�

dt=

c2sa2⇢0

r2c�⇢+ 4⇡G�⇢ )(28)

d2�

dt2+ 2

a

a

d�

dt=

c2sa2

r2c� + 4⇡G⇢0�.

3. We seek solutions of the form � / exp(ikc · r � !t), where kc ⌘ ak denotes the’comoving wavenumber’3. Note that kc · r = ak · r = k · x, and r2

c/a2 = r2. We therefore

finally get:

(29)d2�

dt2+ 2

a(t)

a(t)

d�

dt= �(4⇡G⇢0 � k2c2s ) ,

where k ⌘ 2⇡/� denotes the wavenumber of the perturbation.

The next lecture we will describe applications of this equation in more detail.

3The ‘comoving’ wavenumber of a photon that is emitted corresponds to the ‘rest-frame’ wavelength.

For example, an photon with E = 1 keV will always have a rest-frame energy (wavenumber corresponding

to) E = 1 keV. The proper energy/wavenumber will change as the background Universe expands

AST4320: LECTURE 3

M. DIJKSTRA

1. Jeans Length/Mass in Static & Expanding Media. Spherical Collapse.

The goals of this lecture are to (i) derive expressions for the Jeans length/mass, anddiscuss applications; (ii) to study the time-evolution of a density perturbation in the non-linear regime, as illustrated by the ‘spherical collapse’ model.

In the previous lectures we derived that for a small perturbation the time evolutionof the density contrast � ⌘ �⇢/⇢0 / exp(ik · x� !t) obeys the following second orderhomogeneous di↵erential equation:

(1)d2�

dt2+ 2

a(t)

a(t)

d�

dt= �(4⇡G⇢0 � k2c2s ) ,

where k ⌘ 2⇡/� denotes the wavenumber of the perturbation, cs denotes the adiabaticsound speed, and a(t) denotes the scale factor.

For a Universe with ⌦m = 1.0 and ⌦⇤ = 0.0 we obtained that

2a

a=

4

3t(2)

.

1.1. Jeans Length Static Medium. First we focus on the static case a = 0. We arethen left with

(3)d2�

dt2= A�, A = (4⇡G⇢0 � k2c2s )

We will try to find a solution of the form � = C exp(�t). Then, d�dt = �� and d2�

dt2= �2�.

We therefore have � = ±pA. Solutions with A < 0 correspond to imaginary values of �,

and thus correspond to oscillatory solutions (i.e. exp ipAt = cos

pAt+ i sin

pAt). On the

other hand, solutions with A > 0 correspond real values of �. In this case

(4) �(t) = C1 exp(pAt) + C2 exp(�

pAt), A � 0

where the first/second solution shows a solution that is exponentially growing/decaying.This second solution is not physical: gravity will not drive apart overdense regions. The

1

2 M. DIJKSTRA

condition A � 0 corresponds to 4⇡G⇢0 � k2c2s > 0, i.e. k2c2s < 4⇡G⇢0. We substitutek = 2⇡/� to get 4⇡2c2s/�

2 < 4⇡G⇢0, or

(5) � > cs

r⇡

G⇢0⌘ �J

, where �J denotes the ‘Jeans length’. Modes in the perturbation larger than �J will growas exp(

pAt) ⌘ exp(t/tcoll), where we defined the collapse time as tcoll =

1p4⇡G⇢

. Modes

shorter than the Jeans length do not collapse, instead the density oscillates around ⇢0.We can similarly define a Jeans-mass as

(6) MJ =4⇡

3

⇣�J

2

⌘3⇢0 =

⇡5/2

6G3/2⇢1/20

⇣ kT

µmp

⌘3/2.

In the last step I used that p = nkBT and ⇢ = nmpµ, in which n denotes the number

density of particles, µ their mean molecular weight. Under these assumptions, c2s = kBTµmp

.

Substituting some numbers gives

(7) MJ = 50 M�⇣ n

103 cm�3

⌘�1/2⇣ T

10 K

⌘3/2.

1.2. Jeans Length Expanding Medium. In this case a 6= 0, and we therefore have afunction of t in front of the term d�

dt . The time evolution of the scale factor can be obtainedfrom the Friedmann equation.

(8)⇣ aa

⌘2⌘ H2 =

8⇡G⇢tot3

� kcurvc2,

where kcurv denotes the curvature. We assume a flat Universe, which corresponds tokcurve = 0. We divide ⇢ = ⇢m + ⇢rad + ⇢⇤, where ⇢m = ⇢m,0a

�3 denotes the matterdensity (⇢m,0 denotes the present-day mass density), ⇢rad = ⇢rad,0a

�4 denotes the radiationdensity, and ⇢⇤ = ⇢⇤,0 denotes the vacuum energy density. Substituting these expressionsinto the Friedmann equation we get

⇣ aa

⌘2⌘ H2 =

8⇡G

3(⇢rad,0a

�4 + ⇢m,0a�3 + ⇢⇤)(9)

During matter domination we can ignore the radiation term, i.e.

⇣ aa

⌘2⌘ H2 =

8⇡G

3(⇢m,0a

�3 + ⇢⇤).(10)

It is common to analyse the solution for a Universe with ⌦m = 1 and ⌦⇤ = 0.0 because (i)a simple functional form exists for a(t) (as we see below), and (ii) at high redshift (z � 2)this provides an excellent description1 of standard model with ⌦m = 0.27 and ⌦⇤ = 0.73.In this case,

1The redshift evolution of ⌦m in a flat Universe with ⌦⇤ + ⌦m = 1 scales as ⌦m(z) = ⌦m,0


⇣ aa

⌘2=

8⇡G

3(⇢m,0a

�3) =⌦m,0

H20

a�3.(11)

, where we introduced the density parameter ⌦m,0 =8⇡G⇢m,0

3H20

. Writing this out we have

a2 =⌦m,0

H20

a�1 ) .(12)

da

dt=

s⌦m,0

H20

a�1/2 =1

H0a�1/2,) .

a1/2da =dt

H0) 2

3d(a3/2) =

t

H0+ C1 ) a(t) =

⇣3C1/2 +

3t

2H0

⌘2/3

where we used ⌦m = 1, and C1 denotes an integration constant. We have a boundarycondition that a(t = 0) = 0, and so

a(t) =⇣ 3t

2H0

⌘2/3) da

dt=

2

3

⇣ 3t

2H0

⌘�1/3 3

2H0=

1

H0

⇣ 3t

2H0

⌘�1/3.(13)

Note that this translates to t0 =2H03 for a(t) = 1. If we substitute expressions for a(t) and

a(t) into Eq 1, we are left with

d2�

dt2+

h21

H0

⇣ 3t

2H0

⌘�1/3⇣ 3t

2H0

⌘�2/3id�dt

= �(4⇡G⇢0 � k2c2s ) )(14)

d2�

dt2+h21

H0

⇣ 3t

2H0

⌘�1id�dt

= �(4⇡G⇢0 � k2c2s ) )

d2�

dt2+h21

H0

⇣2H0

3t

⌘id�dt

= �(4⇡G⇢0 � k2c2s ) )

d2�

dt2+

4

3t

d�

dt= �(4⇡G⇢0 � k2c2s ).

Important: the unperturbed density ⇢0 also evolves with time as ⇢m = ⇢m,0a�3 =

⇢m,0

⇣3t

2H0

⌘�2, where we used Eq 11. So we can write

4⇡G⇢0 = 4⇡G⇢m = 4⇡G⇢m,0

⇣ 3t

2H0

⌘�2=

4⇡G⇢m,04H20

9t2=

2⌦m

3t2=

2

3t2(15)

Find solutions of the form � = Ctn: d�dt = Cntn�1, d2�

dt2= Cn(n� 1)tn�2. To simplify the

analysis further, we will look for solutions in cases where the wavelength of the Perturbationgreatly exceeds the Jeans-length. In this case, we could not care less2 about the term k2c2s .Substituting everything into the di↵erential equation (and ignoring the term k2c2s ) we have

2No really.

4 M. DIJKSTRA

Cn(n� 1)tn�2 +4

3Cntn�2 = Ctn

⇣ 2

3t2

⌘. )(16)

n(n� 1) +4

3n =

2

3.

This equation has two solutions: n = 2/3 and n = �1. The decaying solution with n = �1

is not physical. So we are left with the solution � = Ct2/3 / a. This leads us to a veryimportant conclusion: in a Universe that is undergoing expansion the density contrast ofa plain wave perturbation much larger than the Jeans length increases as

(17) � =�⇢

⇢/ a / (1 + z)�1

This growth rate that occurs an an ‘algebraic’ (power-law) rate should be contrasted withthe exponential growth of the static medium. In the exercise of new week you will doa similar analysis in a flat Universe with a cosmological constant. Once the cosmologicalconstant starts accelerating the overall dynamics of the Universe, not surprisingly, theoverall the growth of the over density is suppressed.

(18) MJ ⇠ 1019M�

1.3. The Cosmological Jeans Mass. . We derived that

(19) MJ = 50 M�⇣ n

103 cm�3

⌘�1/2⇣ T

10 K

⌘3/2.

If we substitute the gas density and temperature right after decoupling: T = 2.7(1 + zrec),nHI ⇠ 2⇥ 10�7(1 + zrec)

3 (see assignment 2 for more details), then we

(20) MJ = 5.2⇥ 105M�

Following recombination the gas density decreases as a�3, and the gas temperature fornon-relativistic gas decreases as T / a�2 (see assignment 2), and we have

Appendix

A quick & dirty derivation of the Friedmann equation can be obtained was follows(Chapter 7.2.1 in Longair). We can compute the gravitational deceleration of a galaxy ata proper/physical distance x from Earth due to all matter between Earth and the distantgalaxy. Gauss theorem that it does not matter how the mass is distributed inside thesphere of radius x. The total gravitational forces is therefore simply Fg = �GmM

x2 , where

M = 4⇡3 x3⇢ and m denotes the mass of the galaxy. The equation of motion there is

(21) mx = �GmM

x2= �4⇡Gx⇢

3.


If we replace the physical coordinate x = a(t)r, in which r denotes a comoving coordinate,and write ⇢ = ⇢0a

�3, then

(22) ra = �4⇡Gar⇢0a�3

3= �4⇡G⇢0

3a2.

We used dardt = r da

dt + adrdt = r da

dt , because the comoving coordinate is simply moving alongwith the overall - in this case - expansion and contraction of the comoving coordinatesystem. We can remove r on both sides of the equation.

If we multiply both sides with a we have

aa = �4⇡G⇢0a

3a2⌘ Baa�2 )(23)

1

2

d

dta2 = B

d

dta�1 )

a2 =2B

a+ cint =

8⇡G⇢0a

+ cint,

where c is an integration constant. If we identify cint = kcurvc2a2 then we have the Friedman

equation.

AST4320: LECTURE 4

M. DIJKSTRA

1. Spherical Top-Hat Model

We follow the full time evolution of a spherical over density inside an Einstein-de-Sitteruniverse with ⌦m = 1.0 and ⌦⇤ = 0.0. We assume that over density is a sphere of radiusR(t) ⌘ b(t)R0 ⌘ b(t) (i.e. we define R0 = 1). Kirchho↵’s law states that this over densitye↵ectively behaves as a closed-universe with ⌦m > 1.0. Note that we call the ‘local’scale factor b(t) for clarity. We will compute the connection with the scale factor of thebackground universe [a(t)] later.

1.1. Preliminaries. Useful equations describing the background Einstein-de Sitter uni-verse with (⌦⇤,⌦m) = (1.0, 0.0):

(1) t0 =2

3H0

(2) a(t) = a0

⇣ t

t0

⌘2/3= a0

⇣3H0t

2

⌘2/3=

⇣3H0t

2

⌘2/3,

where we used a0 = 1. Moreover, the background density evolves as

(3) ⇢m(t) =1

6⇡Gt2.

1.2. Non-Linear Collapse. The radius of the sphere evolves as

(4) R = �GM

R2,

where M is the total mass inside the sphere, and is therefore M = 4⇡3 ⇢m,0R

30 = 4⇡

3 ⇢m,0.Solving this equation is not trivial. In assignment 2 we will show that the followingparameterised solutions satisfy Eq 4.

R = A(1� cos ✓)(5)

t = B(✓ � sin ✓)

A3 = GMB2.

Equation 5 shows:1

2 M. DIJKSTRA

• The sphere reaches its maximum at ✓ = B and time t = ⇡B.

• Sphere completely collapses at ✓ = 2⇡ and time t = 2⇡B.

• To first order in ✓ we have R = A✓2/2 (cos ✓ = 1 � ✓2

2! + ...), and t = B✓3/6

(sin ✓ = ✓ � ✓3

3! + ...). Thus, to first order we have

R(t) =A

2

⇣6tB

⌘2/3=

(GM)1/3

2(6t)2/3 =

⇣GM36t2

8

⌘1/3=(6)

=⇣G4⇡

3 ⇢m,036t2

8

⌘1/3=

⇣⌦m,036t2H2

0

16

⌘1/3=

⇣3H0t

2

⌘2/3.

where we used ⌦m,0 = 1.0. Note that R(t) = b(t) = a(t). That is, Eq 5 shows thatduring the early stages of the evolution the sphere simply evolves along with therest of the Universe as expected.

• The density contrast � ⌘ �⇢/⇢m equals:

�m =⇢sphere⇢m

� 1 = 6⇡Gt2M

43⇡R

3� 1 =(7)

9GMt2

2R3� 1 =

9GM

2

B2(✓ � sin ✓)2

A3(1� cos ✓)3� 1 =

9

2

(✓ � sin ✓)2

(1� cos ✓)3� 1.

Our goal is now to show that this non-linear evolution of the sphere connects perfectlyto the solution from linear theory. To do this we Taylor expand the expression for �m in ✓.

First, we Taylor expand sin ✓ = ✓ � ✓3

3! +✓5

5! + ..., cos ✓ = 1� ✓2

2! +✓4

4! + .... Substituting weget

�m =9

2

(✓ � sin ✓)2

(1� cos ✓)3� 1 =

9

2

(� ✓3

3! +✓5

5! + ...)2

( ✓2

2! �✓44! + ...)3

� 1 =(8)

9

2

�✓3

3!

�2(1� ✓2

20 + ...)2�✓22

�3(1� ✓2

12 + ...)3� 1 =

93628

1� 2✓2

20 + ...

1� 3✓212 + ...

� 1 =

⇣1� 2✓2

20+ ...

⌘⇣1 +

3✓2

12+ ...

⌘� 1 =

3

20✓2 + ... =

3

20

⇣6tB

⌘2/3+ ....

To first order, the density inside the sphere goes as t2/3 / a, exactly as in the linear regime.

We also know that collapse occurs at t = 2⇡B. In the linear regime this would correspondto �m = 1.69 (in reality, the sphere is infinitely dense of course). This number �m = 1.69will come in many applications.

AST4320: LECTURE 5

M. DIJKSTRA

1. Virialization

As usual, please let me know if there are any typos or if anything is unclear.

1.1. Preliminaries. In previous lectures we found that in the linear regime a densityperturbation � ⌘ �⇢/⇢ grows with time as

(1) �lin

/ t2/3 / a(t)

in an Einstein-de-Sitter Universe for which ⌦ = 1.0 and ⌦⇤

= 0.0. Here a(t) denotesthe usual scale factor. This cosmological model describes our Universe very well at highredshift where the dark energy has no dynamical impact yet. In this same Einstein-de-SitterUniverse the background density evolves as

(2) ⇢m

=1

6⇡Gt2.

The full non-linear evolution of a spherical perturbation of radius R was given by a pa-rameterised solution:

R = A(1� cos ✓)(3)

t = B(✓ � sin ✓),

where A3 = GMB2, in which M denotes the total mass of the perturbation. We showedthat the non-linear over density is then given by

(4) �non�lin

=⇢sphere

⇢m

� 1 =9

2

(✓ � sin ✓)2

(1� cos ✓)3� 1.

The perturbation reaches its maximum radius at ✓ = ⇡, at which R = 2A and t = ⇡B.The sphere collapses into a singularity for ✓ = 2⇡ and t = 2⇡B.

1.2. Derivation of Virial Theorem. Consider a system of N particles. The accelerationof particle ‘i’ due to the gravitational pull of all other particles is given by

(5) r

i

= �X

j 6=i

Gmjeij

|ri

� r

j

|2 = �X

j 6=i

Gmj(ri � r

j

)

|ri

� r

j

|3

1

2 M. DIJKSTRA

where we have used that eij ⌘ (ri

�rj

)

|ri

�rj

| denotes a vector of unit length in the direction of

the separation between particles ’i’ and ’j’. Next, we take the scalar (‘dot’) product withmiri on both sides of the equality sign:

(6) miri · ri = �X

j 6=i

Gmimjri · (ri � r

j

)

|ri

� r

j

|3

In order to rewrite Eq 2 we first compute

d2

dt2(r

i

· ri

) =d

dt

⇣ d

dtr

i

· ri

⌘=

d

dt

⇣2r

i

· ri

⌘= 2r

i

· ri

+ 2ri

· ri

.(7)

We therefore can recast Eq 2 as:

(8)1

2

d2

dt2(r

i

· ri

)� r

i

· ri

= �X

j 6=i

Gmjri · (ri � r

j

)

|ri

� r

j

|3

Now sum over all particles ‘i’

(9)1

2

X

i

mid2

dt2(r

i

· ri

)�X

i

miri · ri = �X

i

X

j 6=i

Gmimjri · (ri � r

j

)

|ri

� r

j

|3 .

The quantity under the right hand side that is being summed over ‘i’ and ‘j’ is an N⇥Nmatrix M whose diagonal elements ’Mii’ (or ’Mjj ’) are zero (because r

i

� r

i

= 0). Thesum of element ’ij’ and ’ji’ of this matrix is

Fij ⌘ Mij +Mji =Gmimjri · (ri � r

j

)

|ri

� r

j

|3 +Gmjmirj · (rj � r

i

)

|rj

� r

i

|3 =(10)

=Gmimj

|rj

� r

i

|3⇣r

i

· (ri

� r

j

) + r

j

· (rj

� r

i

)⌘=

Gmimj

|rj

� r

i

|3 (ri � r

j

)2 =Gmimj

|rj

� r

i

|

We have (note that because matrix elements with i = j are zero anyway, we can re-move/insert the restriction j 6= i on the sum over j as we please. I have chosen to leave itin)

(11)X

i

X

j 6=i

Fij =X

i

X

j 6=i

(Mij +Mji) = 2X

i

X

j 6=i

Mij ,

where in the last step we used that in the second step, we are summing over all elementsover the matrix twice (we are only doing it in a di↵erent order). We can therefore replacethe right hand side of Eq 5 with 1

2

Pi

Pj 6=i Fij . Therefore, Eq 5 becomes

(12)1

2

X

i

mid2

dt2(r

i

· ri

)�X

i

miri · ri = �1

2

X

i

X

j 6=i

Gmimj

|rj

� r

i

| .

The second term on the left hand side is twice the kinetic energy of the system: T =1

2

Pimiv2i . The term on the left hand side is the total potential energy U : the factor


1

2

accounts for the fact that the double summation actually double counts the potentialenergy1. We therefore have

(13)1

2

X

i

mid2

dt2(r

i

· ri

)� 2T = U.

We have now proven the virial theorem which states that if 1

2

Pimi

d2

dt2 (ri · ri) = 0 then

(14) 2T = |U |.

While it is not possible to proof that There are good reasons why 1

2

Pimi

d2

dt2 (ri · ri) = 0, itis possible to argue that in a time averaged sense, this term will go to 0 provided we averageover a long enough time. We show this next. We define the quantity G ⌘ d

dt

Pimiri · ri =

2P

imiri · ri. This quantity is known is the ’virial’. The time average of the time-derivativeof the virial is given by

DdGdt

E= lim

T!1

1

T

Z T

0

dt0dG

dt0= lim

T!1

1

T[G(T )�G(0)].(15)

The virial G(t) can continuously change with time. However, for a system in equilibriumwe do not expect G(T ) to blow up to infinity at some point in time (this would not reallybe an equilibrium). If this is the case then [G(T )�G(0)] should remain finite, whereas Tcan be increased arbitrarily, in which case dG

dt ! 0 and the virial theorem is satisfied2.

1.3. Virialization. In the spherical collapse model virialization occurs at Rfir

= 0.5Rmax

.This is because the total energy in particle inside the perturbation is equal to the potentialenergy of the perturbation at turn-around (because then the kinetic energy is 0). That is,

(16) Etot

= �3GM2

5Rmax

.

When the sphere contracts to R = 0.5Rmax

then the total binding energy is Unew

= � 6GM2

5Rmax

.

The additional �U = � 3GM2

5Rmax

that the sphere lost has been transformed into kinetic energy

T . We therefore have T = 3GM2

5Rmax

and Unew

= � 6GM2

5Rmax

which satisfies the virial theorem.The radius at which virialization occurs is the ‘virial radius’, denoted with R

vir

.

We now want to compute the real overdensity of the perturbation at collapse (i.e. at✓ = 2⇡ in Eq 3). Because virialization keeps the density of the sphere fixed at the valueit had at ✓ = 3⇡/2, we need to evaluate �

non�kin

by evaluating ⇢sphere

at ✓ = 3⇡/2 [at

1

To clarify further: in the double sum we add the potential energy of particle ‘i’ due to particle ’j’ to the

same potential energy of particle ’j’ due to particle ’i’. We are then double counting the potential energy: if

these two particles were the only two particles of he system, the total potential energy of the system would

be only U = �Gm1m2|r1�r2|

because this is how much energy we have to insert into the system to increase the

distance between the two particles to 1 at which point the total energy of the system would be zero.

2

Even this argument is not entirely convincing, one can imagine that it is not reasonable to have T �the age of the Universe so in practise even T will be finite.

4 M. DIJKSTRA

which point R = A = 0.5Rmax

and we have virialization] and ⇢m

at 2⇡. We can get to thisanswer in a simple way: we know that the sphere virialized R = 0.5R

max

. The density ofthe sphere ⇢

m

is therefore 23 = 8⇥ larger than it was at turnaround. Similarly, we needto evaluate ⇢

m

at tcollapse

= 2⇡B while turn-around happened at tturn�around

= ⇡B. Thatis t

collapse

= 2tturn�around

and ⇢m

decreased by a factor of 22 = 4 (because ⇢m

/ t�2, seepreliminaries). We therefore have

�collapsenon�lin

=⇢virializationsphere

⇢collapsem

� 1 =8⇢turn�around

sphere

1

4

⇢turnaroundm

� 1 =(17)

32⇢turn�around

sphere

⇢turnaroundm

� 1 = 32(�turn�around

non�lin

+ 1)� 1 ⇠ 32(5.55)� 1 ⇠ 178,

where we used that the non-linear overdensity at turn-around was � = 4.55 (see previouslecture). An object virializes with ⇠ 178 times the mean density of the Universe at thatmoment.

1.4. Useful Quantities Related to Virialization. The virial mass isMvir

= 178⇢m

4⇡3

R3.Some useful numbers to have in mind: The virial radius of an object of mass M that col-lapsed at redshift z:

(18) Rvir

= 1 kpc⇣ M

108 M�

⌘⇣1 + z

10

⌘�1

.

The circular velocity

(19) vcirc

⌘r

GM

Rtot

⇡ 20 km s�1

⇣ M

108 M�

⌘1/3⇣1 + z

10

⌘1/2

.

The virial temperature

(20) Tvir

⌘ mp

v2circ

2kB

⇡ 2⇥ 104 K⇣ M

108 M�

⌘2/3⇣1 + z

10

⌘.

1.5. Relevance for Galaxy Formation. Gas that reaches virial radius is shock-heatedto the virial temperature. The fate of the gas then depends on the ‘cooling time’ definedas (also see slides):

(21) tcool

⌘3

2

nkT

n2C(T ),

where C(T ) is the cooling curve shown on the slides. We compared the cooling time to thecollapse time

(22) tcoll

⌘ 1p4⇡G⇢

vir

.

The condition tcool

< tcoll

corresponds to e�cient cooling: gas can cool, looses pressuresupport and can collapse to the centre of the dark matter halo. On the other hand, when


tcool

> tcoll

gas cooling is slow, and collapse is slowed down by pressure. We showed thatthis physical mechanism

1.6. Supplemental Reading. The following texts/books can be helpful:

• Expressions for virial radii, temperatures, and circular velocities for cosmologiesother than Einstein-de-Sitter, see http://adsabs.harvard.edu/abs/2001PhR...349..125B,or http://arxiv.org/pdf/astro-ph/0010468v3.pdf (page 16).

• For the discussion on the relevance of the cooling times & collapse times for galaxyformation: The ‘classical’ papers by Rees & Ostriker 1977(http://adsabs.harvard.edu/abs/1977MNRAS.179..541R), and White & Rees 1978(http://adsabs.harvard.edu/abs/1978MNRAS.183..341W). Most textbooks will gooddiscussion on this (I liked using Peacock’s cosmological physics: Chapter 17.3).Peacock gives the expression for galaxy mass in fundamental constants.

AST4320: LECTURE 6

M. DIJKSTRA

1. Gaussian Random Fields & The Press Schechter Formalism

1.1. Support for Previous Lecture. The plot shown in the slides where the coolingtime was compared to the collapse time contains lines of constant mass. These masses arevirial masses:

(1) M =4

3⇡⇢R

3

vir

.

We defined the circular velocity v

2

circ

= GM

Rfir, and kT

fir

= 1

2

m

p

v

2

circ

(where m

p

denotes

proton mass). We can combine these last two expressions to get R

vir

=GMmp

2kTvir. In other

words, the virial mass M / ⇢R

3

vir

/ ⇢

M

3

T

3vir. This proportionality can only be obeyed if

⇢

M

2

T

3vir

=constant. This explains the why lines of constant mass lie in this diagram the way

they do: for example, for a constant Tvir

we have ⇢M

2 =constant, i.e. M / ⇢

�1/2 as is thecase in the Figure.

1.2. Introduction. Temperature fluctuations in the cosmic microwave background areGaussian. In the standard cosmological model these T-fluctuations are sourced by densityfluctuations, which must also be Gaussian. Gaussian random fields play a key role intheory of structure formation, and we summarise some properties here. First, we considera single-variate Gaussian distribution. This has many of the features which return inthe more general (and cosmologically relevant) multivariate Gaussian distribution. Wethen summarise properties of a Multi-variate Gaussian. This is followed by discussingproperties of Gaussian random fields in Fourier space. Here, we introduce the powerspectrum, which plays a central role in structure formation theories. This discussion followsthat in the lecture of Eiichiro Komatsu: http://www.mpa-garching.mpg.de/

~

komatsu/

cmb/lecture_NG_iucaa_2011.pdf.

1.3. Single-Variate Gaussian Random Fields. We have a single variable x drawn fromthe probability distribution function (PDF) p(x), which is normalised to

R1�1 dx p(x) = 1.

If x obeys Gaussian statistics then

(2) p(x) =1p2⇡�

exp⇣� x

2

2�2

⌘

1

2 M. DIJKSTRA

, where �2 denotes the variance. The PDF contains the full information of the field. Varioususeful statistical quantities of the field are ’moments’ of the field. The first four are:

hxi =Z 1

�1dx xp(x) = 0(3)

hx2i =Z 1

�1dx x

2

p(x) = �

2

hx3i =Z 1

�1dx x

3

p(x) = 0

hx4i =Z 1

�1dx x

4

p(x) = 3�4

.

These 4 moments indicate that the field has zero mean (first moment), variance �2 (secondmoment), zero skewness (third moment), zero kurtosis (fourth moment: the kurtosis

4

⌘hxi � 3hx2i2 = 0). For a single-variate Gaussian random field, all odd moments vanish, alleven moments are given in terms of �2n.

1.4. Multi-Variate Gaussian Random Fields. The general expression for a multi-variate (here there are N variables) Gaussian-PDF is given by

(4) p(x1

, x

2

, ..., x

N

) =1

(2⇡)N/2|⇠|exp

⇣� 0.5

X

ij

x

i

(⇠�1

ij

)xj

⌘,

where ⇠ij

denotes the covariance matrix or the two-point correlation function. Moments ofthis PDF are

hxi

i =Z 1

�1dx

1

Z 1

�1dx

2

...

Z 1

�1dx

N

| {z }Rd

Nx

x

i

p(x1

, x

2

, ..., x

N

) = 0(5)

hxi

x

j

i =Z

d

N

x x

i

x

j

p(x1

, x

2

, ..., x

N

) = ⇠

ij

hxi

x

j

x

l

i =Z

d

N

x x

i

x

j

x

l

p(x1

, x

2

, ..., x

N

) = 0

hxi

x

j

x

l

x

m

i =Z

d

N

x x

i

x

j

x

l

x

m

p(x1

, x

2

, ..., x

N

) = ⇠

ij

⇠

lm

+ ⇠

il

⇠

jm

+ ⇠

im

⇠

jl

.

This is very similar to the single variate case: odd moments vanish. Even moments aregiven in terms of ⇠. Note that hx

i

x

i

i = ⇠

ii

= �

2. Moreover, hxi

x

i

x

i

x

i

i = 3�4 exactly as inthe single-variate case.


Next, we will use that the Universe is isotropic (invariant under rotation) and homoge-neous (invariant under translation). Let x now be a variable that depends on position q.The two-point correlation function ⇠

ij

is

⇠

ij

= hx(qi

)x(qj

)i = hx(qi

)x(qi

+ r

ij

)i,(6)

where r

ij

is a vector that connects the two points. Homogeneity (translational invariance)requires that ⇠

ij

is a function of rij

) alone, and not of qi

. Homogeneity and isotropy thenrequires that ⇠

ij

is a function of |rij

| alone.

1.5. Multi-Variate Gaussian Random Fields in Fourier Space. Working in Fourierspace has its advantages. The Fourier transform of x(q) is given by

x(k) =

Zd

3

q exp(�ik · q)x(q).(7)

The PDF of the Fourier components is given by

p(x(k1

), x(k2

), ..., x(kN

)) =1

(2⇡)N/2|C|1/2exp

⇣� 0.5

X

ij

x(ki

)(C�1

ij

)x⇤(kj

)⌘,(8)

where x

⇤ denotes the complex conjugate of x. Now, similar to the previous cases

hx(ki

)x⇤(kj

)i = C

ij

.(9)

Assuming spatial homogeneity, we can show that (Assigment 3)

C

ij

= (2⇡)3�D

(ki

� k

j

)P (kj

)(10)

P (kj

) =

Zd

3

r exp(�ik · r)⇠(r).

Here, P (kj

) is known as the ’power spectrum’. Assuming additional isotropy, we havethatP (k

j

) = P (|kj

|). Using these expressions can ‘simplify’1 the PDF of the Fouriercomponents:

p(x(k1

), x(k2

), ..., x(kN

)) =1

(2⇡)N/2|⇧i

P (|ki

)|)|1/2exp

⇣� 0.5

X

i

|x(ki

)|2

P (|ki

|)

⌘.(11)

The reason that we call this simplified is that there is no N ⇥N matrix anymore. Instead,all information is now encoded in the N -dimensional vector P (k

i

), the power spectrum.This power spectrum is a popular way to describe the information of a (Gaussian) densityfield.

1

It still looks a bit ugly.

4 M. DIJKSTRA

1.6. Press-Schechter Formalism. We are interested in the probability that an atom atposition q is part of a collapsed object with mass > M . This probability is denoted withP (> M). The Press-Schechter ansatz is that

(12) P (> M) = P (� > �

crit

|M)

where P (�|M) denotes the PDF of the density field smoothed on some scale R - whichcorresponds to a mass scale M . The idea is that if � > �

crit

on some scale M , then � = �

crit

on some larger mass-scale M 0 and the atom would be part of the larger collapsed mass M 0.

The temperature fluctuations of the Cosmic Microwave Background tell us that themass-density field was Gaussian at early times. The mass-density field smoothed oversome mass-scale M is also a Gaussian random field. We can the compute the probability

(13) P (� > �

crit

|M) =1

�(M)p2⇡

Z 1

�crit

d�

0 exp⇣� �

02

2�2(M)

⌘,

where �

2(M) denotes the variance of the mass-density field smoothed over scale M . Thisintegral can be re-expressed as an error-function2:

(14) P (> M) = P (� > �

crit

|M) =1

2

h1� erf

⇣⌫p2

⌘i,

where ⌫ ⌘ �crit�(M)

. This idea has problem with it, if we take M ! 0, then P (> M) = 0.5.

This is because for a Gaussian random field, half the density fluctuations are under dense.In non-linear evolution, matter from the under dense regions will fall onto the massiveobjects (i.e. empty regions get emptier, dense regions become denser). This was a wellknown problem as soon as the theory was proposed. It was fixed my multiplying by afactor3 of 2. We will do that as well.

Because P (> M) denotes the fraction of atoms locked up in an object of mass > M ,@P

@M

dM denotes the mass fraction of the Universe locked up into collapsed dark matterhalos of masses in the range M ± dM/2. This fraction can be related to the dark matterhalo mass function, n(M)dM , which denotes the number density of dark matter halos withmasses in the range M ± dM/2. To see this connection, we

• Note that Mn(M)dM denotes the mass density in dark matter halos with massesin the range M ± dM/2.

• Note that Mn(M)dM/⇢

m

denotes the fraction of the mass density (i.e. mass frac-tion) in dark matter halos with masses in the range M ± dM/2.

2

The error function is define as erf(x) ⌘ 2p⇡

R 2

0exp(�t

2)dt.

3

This factor of 2 follows naturally from extension of the theory, and relates to the fact that the standard

theory does not take into account at all that collapsed objects of some smaller mass M

00can end up in a

more massive object of mass M . This problem is known as the ‘cloud-in-cloud’ problem.


We therefore have (where we have include Press & Schechters infamous factor of 2)

Mn(M)dM/⇢

m

=@P

@M

dM )(15)

n(M) =⇢

m

M

@P

@M

=⇢

m

M

@

@M

h1� erf

⇣⌫p2

⌘i=

=⇢

m

M

p2p2

@

@⌫

h1� erf

⇣⌫p2

⌘i@⌫

@M

=⇢

m

M

1p2

@

@x

h1� erf(x)

i@⌫

@M

=

= �⇢

m

M

1p2

2p⇡

exp(�x

2)�

crit

�

2(M)

@�

@M

What remains to be done is to characterise our Gaussian random field, which we do withthe power spectrum P (k) which enters our expression through its influence on �(M). Wewill sketch this relation next.

1.7. Relation Between P (k) and �(M). We know that ⇠(r) and P (k) are Fourier trans-forms of each other:

(16) ⇠(r) =

Z 1

0

d

3

k

(2⇡)3P (k) exp(ik · r)

We also know from our analysis on Gaussian random fields that �2 = ⇠(r = 0). Therefore,

(17) �

2 =

Z 1

0

d

3

k

(2⇡)3P (k) =

isotropy

1

2⇡2

Zdk k

2

P (k) /Z

dk k

n+2

where we assumed that P (k) = Ak

n. The value �2 is variance in mass at one location. Weare interested in the variance �2(M), smoothed over some mass scale M , which correspondsto a spatial scale R. Smoothing the density field over some scale R corresponds to removingall fluctuations on scales smaller than R, or in other words, suppressing the power spectrumon wave numbers larger than k = 2⇡/R. We therefore have

(18) �

2(M) = �

2(R) =

Z2⇡/R

0

d

3

k

(2⇡)3P (k) /

ZR

�1

dk k

n+2 / R

(�n�3) / M

(�n�3)/3

.

The main point is that we can compute �(M) (and hence predict the dark matter halomass function) once we specified P (k). The Press-Schechter Theory does a remarkable jobat reproducing simulated mass functions that properly account for all gravitational e↵ects.

1.8. Useful Reading. Useful reading material includes:

• Lecture on Gaussian random fields was taken from lecture given by Eiichiro Ko-matsu http://www.mpa-garching.mpg.de/

~

komatsu/cmb/lecture_NG_iucaa_2011.

pdf. This discussion also contains extensive discussion on non-Gaussianity.• Press-Schechter Theory: Textbooks by J. Peacock (Chapter 17), and M. Longair(see index) have descriptions on which this lecture was based.

AST4320: LECTURE 7

M. DIJKSTRA

1. Fixing the Fudge Factor 2 in the PS Formalism

1.1. Background Information. The Press-Schechter formalism is based on the followingprinciples

• the mass-density field is Gaussian (this is highly accurate at early times).

• time evolution governed by linear theory: i.e. �(t) / t2/3.• an object has collapsed when its linear overdensity exceeds �

crit

= 1.69.

In our analysis we will often encounter �2(M) which denotes the variance of the massdensity field smoothed on a scale M (i.e. R, M and R can be interchanged via M =4⇡3

⇢m

R3). The mass-dependence of �2(M) can be obtained from the mass-power spectrumP (k) as

(1) �2(M) = �2(R) =

Z2⇡/R

0

d3k

(2⇡)3P (k) /

Z R�1

dk kn+2 / R(�n�3) / M (�n�3)/3,

where we assumed that P (k) / kn.

1.2. The Fudge Factor. In the Press-Schechter formalism the mass-fraction in boundobjects with mass > M is given by

(2) P (> M) = 2P (� > �crit

|M) = 1� erf(⌫/p2),

where ⌫ ⌘ �crit

/�(M). This factor of 2 is a fudge because it was introduced entirely toguarantee that P (> 0) = 1.

1.3. Deriving the Factor of 2. The PS-theory without the fudge factor underestimatesP (> M) (because otherwise there would be no need to increase P (> M) by the factor2). This is because it actually is possible for � to be less than �

crit

on a mass-scale M-centered on some mass element at q - while it is part of a collapsed object with a mass> M . Visually, this is shown in Figure 1.

The assumption that P (> M) = P (� > �crit

|M) is thus not correct. We can fix this,and interestingly the fix provides the infamous factor of 2. We show this next.

Again consider a mass-element at q (as in Fig 1). We are going to evaluate the densityat q but smoothed over the largest possible scale, and then at increasingly smaller scales.We are going to do this in a number of steps

1

2 M. DIJKSTRA

qR

R’

q

Figure 1. The density at q smoothed on a scale R is � < �crit

. The PSformalism would say that this point cannot be part of a collapsed object ofmass > M . Note however, that neighbouring regions may all have � > �

crit

.If we smooth the density field on some larger scale R0 > R, then it is possiblethat the smoothed density at q � > �

crit

and the point would be part of acollapsed object of mass M 0 > M . The original PS formalism (without thefactor of 2) failed to account for these objects.

• First, we smooth the density field in a sphere of radius R ! 1 centered on q. Eq 1then shows that �(R ! 1) = 0. On this scale the density field is Gaussian withzero variance, and � on this scale is � = 0. This point is labeled as ‘1’ in Figure 2.

• Next, we reduce R. This corresponds to adding independent Fourier modes to theintegral over k. We now have a finite �2(M

2

). The subscript ‘2’ refers to the stepnumber in this analysis. We now draw a random � from the Gaussian PDF withvariance �2(M

2

). This �2

is labeled in Figure 2 with ‘2’.• Next, we reduce R further. This again corresponds to adding independent Fouriermodes. The new variance �2(M

3

) ⌘ �2(M2

) + d�2

23

, where d�2

23

is the variance weadded by adding the new Fourier modes. We generate the new �

3

= �2

+d�23

whered�

23

is generated from a Gaussian distribution with variance d�2

23

• Repeat....

This analyse shows that � makes a random change for each reduction in R (the r.m.ssize of this change in step n is given by the newly added variance d�n,n�1

). Each point in


M (i.e. R)

M’M

12

3

Figure 2.

the Universe has its own random trajectory - each random trajectory is called a ‘random

walk’.

Figure 2 shows the random walk associated with point q shown in Figure 1. The randomwalk crosses the barrier at mass scale M 0, and q is thus part of a collapsed object of massM 0. We continue the random walk to smaller R (and thus M). In the case of Figure 1, weknow that the random walk brings � to � < �

crit

at mass scale M . Note that because eachstep is drawn from a Gaussian distribution, which is symmetric around zero, we could havegenerated the random walk- mirrored around the line �

crit

, and which ends at = 2�crit

��- with equal probability. We therefore know that the probability of ending at � on massscale M , after crossing �

crit

on mass scale M 0 is equal to ending at on mass scale M .

We know that P (�|M) is a Gaussian with variance �2(M). We would like to di↵erentiatebetween points at �|M (this is short-hand notation for density � smoothed on mass scaleM) that did, and that did not cross the barrier at some larger mass-scale M 0. We can

4 M. DIJKSTRA

obtain the probability that a point has �|M without having crossed any barrier on largerscales, P

nc

(‘nc’ stands for ‘no crossing’) from

(3) Pnc

(�|M) = P (�|M)� Pcross

(�|M),

where Pcross

(�|M) denotes the probability that a point reached �|M after having crossedthe barrier. We know from our random-walk analysis (Fig 2) that P

cross

(�|M) = P (|M)!We therefore have(4)

Pnc

(�|M) = P (�|M)�P ([2�crit

��]|M) =1p

2⇡�(M)

⇣exp

h� �2

2�2(M)

i�exp

h� [2�

crit

� �]2

2�2(M)

i⌘.

The probability that a mass element at q is therefore embedded within a collapsed objectof mass > M is then

(5) P (> M) = 1�Z �crit

�1d� P

nc

(�|M)

I left it as en exercise to show that now we are left with

(6) P (> M) = 2P (� > �crit

|M) = 1� erf(⌫/p2).

That is, the factor of 2 now comes in naturally.

1.4. Useful Reading. Useful reading material includes

• On Press-Schechter Theory, random walks shown in Figure 2 and how to get thefudge factor of 2 I consulted books by J. Peacock (Cosmological Physics), and M.Longair (Galaxy Formation). More extended (& formal) descriptions can be foundin the book by Mo, Van den Bosch & White. Di↵erent books give di↵erent de-scriptions & derivations. I anything is unclear, I would recommend trying di↵erentbooks until you find one that works best for you (for this subject).

AST4320: LECTURE 8

M. DIJKSTRA

1. The Two-Point Correlation Function ⇠(r)

1.1. Introduction. In the previous lecture we used the Press-Schechter Formalism toconvert a Gaussian random density field, combined with the linear theory of growth ofstructure, to predict number density of collapsed objects as a function of their mass M . Wedenoted this number density with n(M), which is also referred to as the halo mass function.The goal of this lecture is to go study one other key characteristic of the fluctuations inthe density field, namely the two-point correlation function ⇠(r) (see our Lecture notes onGaussian random fields).

1.2. The Meaning of ⇠(r). Pick two volume elements dV labelled ‘1’ and ‘2’, separatedby a distance r12, and correlate the number of galaxies in both:

dV 2hngal,1ngal,2i = dV 2

hngal(1 + �gal,1)ngal(1 + �gal,2)i =(1)

= dV 2n2galh1 + �gal,1 + �gal,2 + �gal,1�gal,2i =

= dV 2n2gal

⇣h1i+ h�gal,1i+ h�gal,2i+ h�gal,1�gal,2i

⌘.

If the galaxy distribution is Gaussian like the mass density field (which is a reasonableassumption as we will see at the end of § 1.3), then from our lectures on Gaussian randomfields we know that h�gali = 0, and h�gal,1�gal,2i = ⇠gal(r12). We therefore have

dV 2hngal,1ngal,2i = dV 2n2

gal

⇣1 + h�gal,1�gal,2i

⌘⌘ dV 2n2

gal(1 + ⇠gal(r12)).(2)

If we pick the volume element to be su�ciently small that the number density of galaxiesngaldV ⌧ 1, then ngaldV denotes the probability that a volume element contains a galaxy.We can see that ⇠gal(r12) denotes excess probability over random that the two volumeelements have galaxies inside them.

1.3. The Two-Point Function of Collapsed Objects and Bias. Our goal is now tocompute ⇠(r) of mass inside collapsed objects. These collapsed objects is where galaxiesare thought to form. Understanding the clustering properties of mass in collapsed objectstherefore can help us understand clustering properties of galaxies.

First recall that in the Press-Schechter formalism a mass element is part of a collapsedobject when its linear over density �lin � �crit = 1.69. How does this requirement a↵ect⇠(r)? The number density of collapsed objects (in other words, dark matter halos) within

1

2 M. DIJKSTRA

the mass range M ± dM/2 is denoted by n(M)dM . Press-Schechter theory provided uswith an expression for n(M)

n(M) = �

r2

⇡

⇢mM

dln �

dM⌫ exp(�⌫2/2) ⌘ k⌫ exp(�⌫2/2), ⌫ ⌘ �crit/�(M).(3)

We first address how the number density of collapsed objects changes if we move into apart of the Universe where the average overdensity with some (arbitrary) volume V equals� = ✏ > 0. In this region we expect a boost in the number density of objects because oftwo e↵ects:

Figure 1. This Figure shows the mass overdensity � as a function of po-sition R along a random line through the universe. The mass density fieldfluctuates, and the red solid line denotes a long-wavelength fluctuation.Because of this long-wavelength fluctuations, there are large regions in Uni-verse for which the average overdensity is not 0 (as is the case for the iniverseas a whole, recall that h�i = 0 for a Gaussian random density field). Forexample, in the blue box the mean overdensity is � = ✏. The number densityof collapsed objects in a volume where the mean overdensity is elevated byan amount known as ‘bias’.


(1) If the local overdensity (averaged over the volume V) is � = ✏, then the local densityof dark matter (and baryonic) particles - and consequently dark matter particlesin collapsed objects - is higher by a factor of (1 + ✏).

(2) If the local mean overdensity is � = ✏, then the collapse barrier for most dark matterparticles lies closer on average, at �crit � ✏ (see Figure 1).

We consider two e↵ects separately: the first e↵ect boosts n ! n(1 + ✏). The seconde↵ect requires more work: the barrier for collapse �crit lies lower at �crit � ✏. The resultingchange in n(M) at a fixed mass can be obtained by applying the chain rule:

n ! n+dn

d⌫

d⌫

d✏✏.(4)

We know that the ‘local’ collapse barrier is lowered to �crit � ✏, and therefore

⌫ = [�crit � ✏]/� )

d⌫

d✏= �

1

�.(5)

Substituting this into Eq 3 we get

n ! n�

✏

�

dn

d⌫= n�

✏

�

⇣k exp(�⌫2/2)� k⌫2 exp(�⌫2/2)

⌘=

= n�

✏

�

n

⌫

⇣1� ⌫2

⌘= n

h1�

✏

�

⇣1� ⌫2

⌫

⌘i= n

h1 +

✏

�

⇣⌫2 � 1

⌫

⌘i⌘ n+ �n.(6)

So we have

�n

n⌘

⇣⌫2 � 1

�⌫

⌘✏ ⌘ bL✏(7)

The quantity bL denotes the ‘Lagrangian’ bias. This bias accounts for the boost in the localnumber density of collapsed objects as a result of the lowering of the collapse threshold.

The total boost in the number density of collapsed object now changes as

n ! n(1 + ✏)(1 + bL✏) ⇡ n(1 + ✏[1 + bL]) ⌘ n(1 + bE✏)(8)

, where we have defined the Eulerian bias bE

bE = 1 + bL = 1 +⌫2 � 1

�⌫.(9)

The Eulerian bias parameter includes the boost in the number density of collapsed objects- in response to a an enhancement of the local density by a factor of 1 + ✏ - due to booththe increase in the local density, and due to the local lowering of the collapse barrier. Sosimply speaking, this accounts for the total boost.

4 M. DIJKSTRA

Now we return to the galaxy correlation function

⇠gal(r12) = h�gal,1�gal,2i.(10)

Note that here we defined ngal = n(1 + �gal). From the analysis above we know that�gal = bE✏ (here we used that galaxies reside in collapsed objects). We therefore have

⇠gal(r12) = b2Eh✏1✏2i = b2E⇠(r12) ,(11)

where ⇠(r12) denotes the 2-point correlation function of the overall mass density field1. Thisis an important result: the correlation function of galaxies - and more generally of massthat resides inside collapsed objects - is the same as that for the overall mass distributionmultiplied by the bias parameter squared2.

There are four regimes that we look at

(1) ⌫ = 1 ) bE = 1. Collapsed objects with ⌫ = 1 have the same 2-pt function ⇠(r)as all matter (irrespective of whether it is part of a bound object or not). Notethat ⌫ = 1 corresponds to objects with masses where the PS mass function starsto turn-over from power-law to exponential decrease: i.e. ⌫ = 1 corresponds to thecharacteristic mass of the PS mass-function.

(2) ⌫ < 1 ) bE < 1. Collapsed objects with ⌫ < 1 have a 2-pt function ⇠(r) that issuppressed to that of all matter. These objects are said to ’cluster less’, or said tobe ‘anti-biased’ tracers of the total matter distribution. Objects with ⌫ < 1 lie inthe power-law low-mass end of the PS mass function.

(3) ⌫ > 1 ) bE > 1. Collapsed objects with ⌫ > 1 have a 2-pt function ⇠(r) thatis enhanced to that of all matter. These objects are said to ’cluster stronger’, orsaid to be ‘biased’ tracers of the total matter distribution. These objects lie in theexponential tail of the halo mass function and are therefore more rare.

(4) ⌫ � 1 ) bE ⇠

⌫� . Note that ⌫ ⌘ �crit/�(M) � 1. i.e. �(M) ⌧ 1. Therefore

bE � ⌫ � 1. The rarest objects are very strongly clustered!

The slides show some examples of the observed bias of galaxies, and some other appli-cations.

1.4. Acoustic Scale and Baryon Acoustic Oscillations. The acoustic peak is a fea-ture at R 150 cMpc (cMpc stands for comoving Mpc) in the two-point correlation functionof matter, and therefore of galaxies. Basic insight into why this feature exists can beobtained by considering a point perturbation is the very Early Universe. The baryons,photons and dark matter all respond in di↵erent ways to this perturbation. The descrip-tion I gave in the lecture was based on that by D. Eisenstein (who is an expert in this area).The website can be found at https://www.cfa.harvard.edu/

~

deisenst/acousticpeak/

1Note that ✏ was just another symbol that denoted local mass overdensity, we used � for this in many

other lectures.

2This also justifies that we assumed in § 1.2 that the distribution of galaxies was described by a Gaussian

random field.


acoustic_physics.html. I have uploaded a PDF version of this site as well (EisensteinA-cousticpeak.pdf).


• The non-technical description if the BAO (EisensteinAcousticpeak.pdf).• Helpful additional discussion and (working) movies that support the notes by D.Eisenstein can be found on http://scienceblogs.com/startswithabang/2008/

04/25/cosmic-sound-waves-rule/.• The discussion/derivation of Eulerian & Lagrangian bias was taken from J. Peacock.‘Cosmological Physics’. Chapter 17. Feel free to consult your preferred books onalternative views/discussions (I am sure that Mo, Van den Bosch & White will beextremely useful as well).

AST4320: LECTURE 9

M. DIJKSTRA

1. More on ⇠(r) and the Powerspectrum P (k)

1.1. Introduction. In the previous lecture we looked in some detail at the two-pointfunction of matter, and of matter that is enclosed in bound objects. We found that

• the two-point function of mass inside bound objects of mass M was related to theoverall matter two-point function via

(1) ⇠bound

(r|M) = b2E

(M)⇠(r),

where bE

(M) denotes the (Eulerian) ’bias’ parameter.• the bias parameter is mostly a function of mass: i.e. ⇠

bound

(r|M) mostly has thesame dependence on radius as ⇠(r), i.e. there is little scale-dependence to thebias (which can be important if you want to do precision cosmology with galaxy-clustering).

• the two-point functions have a characteristic feature imprinted on them at r ⇠ 150cMpc, which corresponds to the total distance a pressure wave could travel from apoint perturbation in the cosmological density-field from t = 0 to t = t

rec

, wheretrec

denotes the age of the Universe at recombination.

Today we look at some more details of ⇠(r), namely its properties in 2-dimensions. Wealso look at its Fourier counterpart the 2D power spectrum. Finally, we start discussingwhat sets the shape of the matter power spectrum of our Universe.

1.2. Two 2D-Two-Point Function I: The Alcock-Paczynski Test. The two-pointcorrelation function ⇠(r) gives us the excess probability over random of finding two objects(e.g. galaxies) at some separation r. However, when observing astrophysical objects, wedo not know the 3D separation r perfectly. In practise we write ⇠(r) ⌘ ⇠(r?, rII). Here, r?denotes the ‘transverse’ separation and r|| denotes the line-of-sight separation. These areindicated schematically in Figure 1. The transverse separation corresponds to the angularseparation (in radians) times the angular diameter distance D

A

(z) to redshift z.

(2) r? ⌘ ↵DA

(z).

The expression for DA

(z) is given below. The line-of-sight separation is inferred from thevelocity separation as:

(3) r|| ⌘ �v/H(z),1

2 M. DIJKSTRA

Figure 1. This Figure shows how a 3D separation r is usually separatedinto a transverse (r?) and line-of-sight (rII) separation (on the left). Theobserved is located in the bottom of the figure. Isotropy requires thatcontours of constant ⇠ are circular in the r?�r|| plane (shown on the right).

where H(z) denotes the Hubble parameters at redshift z. The Hubble parameter andangular diameter distance are given by

H(z) = H0

p(1 + z)3⌦

m

+ ⌦⇤

(4)

DA

(z) =

Z z

0

dz

H(z).

Isotropy requires that ⇠(r?) = ⇠(r||) if r? = r||. In other words, isotropy requires thatcontours of constant ⇠ should be circles in the r? � r|| plane. Recall that the conversionfrom the observable quantities (↵,�v) into (r?, r||) requires knowledge of H(z) and D

A

(z).Adopting the wrong set of cosmological parameters will give the wrong conversion, and willgenerally not yield circular contours. If we enforce the contours of ⇠ in the r? � r|| plane

to be circular, then this translates to constraints1 on DA(z) and H(z), and therefore oncosmological parameters.

1

We get circular contours not only for the correct cosmological parameters, but also if we simultaneously

adopt the incorrect DA

(z) by a factor of x and H(z) by a factor of x�1

(this results in the same boost x in

both r? and r||, which preserves the circularity of the contours). Enforcing circular constraints therefore

constrains the product H(z)DA

(z).


1.3. Two 2D-Two-Point Function II & 2D Power Spectrum I: Redshift Space

Distortions. The Alcock-Paczynski test is only accurate if peculiar velocities are negligi-ble. As in previous lectures, we decompose the velocity v of an object into an ‘unperturbed’velocity v

0

and a peculiar velocity u. That is, v = v

0

+ u. Peculiar velocities cause socalled redshift-space distortions’: consider an overdensity � > 0. The gravitational poten-tial generated by this overdensity attracts nearby matter. Recall from linear theory that� = �r · u. From linear theory we know that if � > 0, then � > 0 and therefore r · u < 0.In other words, the overdensity is surrounded by a peculiar velocity vectors pointing at thecentral over density �. This is illustrated visually in the left panel of Figure 2.

Real space Redshift space

Figure 2. .

Redshift space corresponds to the space in which the radial coordinate of an objectcomes entirely from its measured velocity. Peculiar velocities thus induce distortions in thedistribution of galaxies in redshift space in the line-of-sight direction. This is illustratedvisually in the right panel of Figure 2: the peculiar velocity of a galaxy on the far side ofthe perturbation is pointing toward is: hence, the total recessional velocity of this galaxyappears smaller. It would therefore appear closer to the overdensity in redshift space thanit is in reality. Peculiar velocities induced by an overdensity thus ‘squash’ (compress) thetwo-point correlation along the line-of-sight direction. We denote this two point functionin redshift space with ⇠

s

(xs

). More generally, a subscript ‘s’ on any quantity means it ismeasured in redshift space.

The squashing of ⇠s,g(xs

) along the line-of-sight direction a↵ects the redshift space powerspectrum P

s,g(ks

). Note that the extra label ’g’ is two emphasis that we now look at thegalaxy two-point function and galaxy power spectrum. Note that large r|| correspond to

4 M. DIJKSTRA

small k|| (and vice versa), and the galaxy power spectrum in redshift space is squashedalong the transverse direction. This is illustrated visually by Figure 3.

Figure 3. .

The deformation of the 2D galaxy power spectrum in redshift space is quantitativelygiven by the famous Kaiser formula:

Ps,g(ks

) = Pg

(k)[1 + �µ2

ks

]2(5)

� ⇡⌦0.6m,0

bE

, µks

⌘ [ks

· e||]/ks, ks

= |ks

|,

where e|| denotes a unit-vector in the line-of-sight direction. So we therefore have

Ps,g(ks

) = Pg

(k)[1 + �]2 µks

= 1(6)

Ps,g(ks

) = Pg

(k) µks

= 0.

Importantly, � depends on the cosmological mass density parameter ⌦m,0 and the bias

of the galaxies we are studying. The deformation of the 2D power spectrum in redshifttherefore encodes cosmological information as well as additional constraints on the bias ofthe galaxies.

The Kaiser formula is su�ciently important to derive it - partly in these notes, andpartly in assignment 5. In assignment 5 we will first show that the Kaiser formulaimplies that

�s,g(ks

) = �g

(k)[1 + �µ2

ks

].(7)


We will also study in assigment 5 that this equation is the Fourier counterpart of thefollowing equation (this more di�cult to show as it involves using a few other results weobtained from linear theory. The assignment will walk you through them):

�s,g(xs

) = �g

(x)� d

dx

⇣u · exH

0

⌘,(8)

where ex denotes a unit vector in the direction of x. Before we derive this expression next,we look at what this expression means: if � > 0, then all peculiar velocities u point in thedirection of the point at x. Thus, taking a positive step �x in the direction of ex we seeu change from 0 (at the overdensity itself) to a negative value which is pointed parallel toex. We therefore have du·ex

dx < 0, and hence �s,g(xs

) > �g

(x), i.e. redshift space distortionsincrease the number density of galaxies at x

s

relative to the real number density at x.

Our starting point in the derivation of the equation above is the realisation that redshiftspace distortions cannot change the total number of galaxies: they only change their appar-ent locations along the line-of-sight. We can therefore relate number densities in redshiftspace and real space via

ns

(xs)d3x

s

= n(x)d3x(9)

d3xs

= dxs

x2s

d⌦, d3x = dx x2d⌦, d⌦ = sin ✓d✓d�.

The angular part of the volume element is not a↵ected by redshift space distortions, andd⌦ is the same in both real and redshift space. We therefore have

ns

(xs) = n(x)x2

x2s

dx

dxs

⌘ n(x)J,(10)

where J denotes the Jacobian of the coordinate transformation from real to redshift space2.We next derive the relation between x and x

s

. For simplicity we focus on the low-redshiftUniverse where the recessional velocity of a galaxy is just cz, in which z is the redshift andc the speed of light. We then have

cz = H0

x+ u · ex

⌘ H0

xs

, x = |x|,(11)

where the last equality in the equation on the left is how the line-of-sight coordinate inredshift space is defined. We therefore have:

xs

= x+u · e

x

H0

= xh1 +

u · ex

xH0

i.(12)

2

Perhaps a simpler way of seeing this relation is that number density of galaxies in redshift space changes

entirely due to the change in volume of the volume element d3x when it is transformed into redshift space.

The new volume of this volume element in redshift space is J�1

times larger.

6 M. DIJKSTRA

Then from the first equality it is apparent that

dx

dxs

=⇣dx

s

dx

⌘�1

=⇣1 +

d

dx

u · ex

H0

⌘�1

(13)

Using the last equality in Eq 12 we get an expression for the Jacobian J

J =dx

dxs

x2

x2s

=⇣1 +

d

dx

u · ex

H0

⌘�1

⇣1 +

u · ex

H0

x

⌘�2

.(14)

Now look at these terms closely:

• u ·ex

is the peculiar velocity along e

x

, which is just |u| = u. For comparison, H0

x isthe recessional velocity of point x due to the Hubble expansion. At a cosmologicaldistance, say z = 0.1, this recessional velocity is cz = 3⇥104 km/s. For comparison,typical peculiar velocities are at most u ⇠ 10�100 km/s. This term is clearly small.

• The term ddx

u·ex

H0

is a bit more tricky: again the term du·ex

dx = dudx denotes the velocity

gradient of the peculiar velocity. It can be shown that this term must be smallerthan H

0

from the following: consider a mass element a distance �x away from thepoint at x in the direction of e

x

. The total velocity di↵erence between these twopoints equals �v = H

0

�x + �u. Only if the points are stationary with respectto each other do we have H

0

= |�u/�x| ⇡ du/dx. So only when the peculiarvelocity is large enough to balance the Hubble expansion are the two terms equal:this cancellation corresponds to turn-around in our analysis of non-linear evolution,which is well into the non-linear regime. In other words, we have du/dx ⌧ H

0

inthe linear regime3.

The above analysis suggests that we can simplify

J ⇡⇣1� d

dx

u · ex

H0

⌘.(15)

Substituting this into Eq 10 we get

ns

(xs) = n(x)J = n(x)⇣1� d

dx

u · ex

H0

⌘.(16)

The final step in our derivation is to recast ns

(xs) ⌘ ns

(1+�s,g(xs

)) and n(x) ⌘ n(1+�g

(x)).Now, the average number density in redshift space is just the same as in real space (redshiftspace distortions have simply shu✏ed objects around along the line of sight direction).Substituting this into the above equation, and we find:

�s,g(xs

) = �g

(x)� d

dx

⇣u · exH

0

⌘.(17)

3

We can use a similar argument to convince ourselves that

ddx

u·ex

H0

� u·ex

H0

x: the di↵erence is basically x

vs dx.


1.4. The Power Spectrum & Transfer Function. The notes on the power spectrumwill appear in the next set of notes, when we have finished discussing this. For this discus-sion I followed Extragalactic Astronomy and Cosmology: An Introduction by P. Schneider(Chapter 7 discusses the transfer function).

AST4320: LECTURE 10

M. DIJKSTRA

1. The Mass Power Spectrum P (k)

1.1. Introduction: the Power Spectrum & Transfer Function. The power spectrumP (k) emerged in several of our previous lectures:

• It fully characterised the properties of a Gaussian random (density) field.• It determined �(M) in Press-Schechter theory.• We constrain(ed) the slope d logP (k)/d log k from the observed galaxy two-pointcorrelation function in assignment 5.

Now we will discuss some key properties of P (k). Linear theory allowed us to describethe time evolution of a density perturbation � with wavenumber k. This time evolutionwas

(1) �(k, t) = �(k, t = 0)D+(t),

where we derived that D+(t) / t2/3 / a in the Einstein-de Sitter Universe (⌦m = 1.0,⌦⇤ = 0.0). Therefore,

(2) P (k, t) = P (k, t = 0)D2+(t) ⌘ P0(k)D

2+(t).

One of the goals of modern cosmology is to calculate P0(k).

There are no preferred length scales in the very early Universe, and the only functionalform for P0(k) with no scale length is a power law:

(3) P0(k) = Akn.

We may think of this P0(k) as the ‘primordial’ power spectrum as it described density fluc-tuations at t = 0. As we will see next, perturbations with di↵erent wave numbers evolveddi↵erently in the very early Universe. This modifies the matter power spectrum from thepower law form given above. These modifications are encoded in the so-called ‘transferfunction’ T (k). This transfer function T (k) encodes the information on the evolution ofsome density perturbation �(k), and therefore a↵ects the power spectrum as

(4) P0(k) = AknT 2(k) .

Our goal is now to obtain some intuition for T (k), and expressions for some limitingbehaviour of T (k). Before proceeding it is useful to discuss the concept of a ‘horizon’, asit plays a key role in shaping the transfer function.

1

2 M. DIJKSTRA

1.2. The Horizon. There are many horizons in cosmology. We focus on the so called’particle horizon’, which corresponds to the ‘maximum proper distance over which there can

be causal communication at time t’ (Quote lifted from M. Longair’s book). The particlehorizon thus corresponds to the maximum proper distance a photon could have travelledbetween t = 0 and t = t. This distance corresponds to

(5) rH(t) = a(t)

Z t

0

cdt0

a(t0).

The integral gives the total comoving distance traversed by the photon. The term a(t)converts that into a proper distance at time t. During radiation domination the scale fac-tor a(t) / t1/2 ⌘ Ct1/2, where C is a constant. We can solve the integral to give rH = 2ct.This is a factor of two larger than the maximum distance a photon could travel in a staticmedium. The factor of 2 accounts for the fact that space itself is expanding during thephoton’s flight.

Now consider the evolution of a perturbation of some proper length/wavenumber L.The time evolution of the wavelength of this perturbation is given by L = L0(a/a0) =L0(t/t0)1/2. Consider a perturbation for which L > rH: because L > rH there is no way toknow what lies outside of L. It is therefore impossible to determine what the mean densityof the background should be, and therefore whether the perturbation L is overdense orunder dense. We need general relativity to describe the time evolution of perturbationslarger than the horizon scale. I will comment on this later.

While we currently cannot say much about the expected time evolution of the perturba-tion L, we can predict that L will become smaller than the horizon in the future: If L > rHat some time t1, then L2 = L(t2/t1)1/2 at some later time t2 while the horizon at this timerH,2 = rH(t2/t1). The perturbation size equals the horizon scale when

rH(t2/t1) = L(t2/t1)1/2 ) rH(t2/t1)

1/2 = L ) t2 = t1(L/rH)2 > t1.(6)

After the perturbation enters the horizon we can apply our classical (non-relativistic) per-turbation theory.

1.3. The Transfer Function T (k). Figure 1 shows the time evolution of a perturbation�(k) (with corresponding wavelength or ‘size’ of the perturbation L = 2⇡/k) that entersthe horizon during the radiation dominated era at scale factor aenter). There are three keyevents during the evolution of this perturbation:

• When the matter density starts to dominate the Universal energy density - thishappens at the redshift of matter-radiation equality zeq ⇠ 24000 - the dark matterperturbation grows as � / a. This is what we derived in previous lectures.

• At redshifts z > zeq - i.e. a < aeq - the Universal energy density is dominatedby radiation. During radiation domination the scale factor grows as a / a1/2 (asopposed to a / t2/3 that we found during matter dominance). This di↵erent time


Figure 1. This Figure shows (schematically) the time evolution of an over-density � on some scale L that enters the horizon at aenter ⌧ aeq. Thetime evolution goes through three di↵erent phases: (i) � / a2 before hori-zon entry which follows from general relativistic perturbation theory; (ii)� =constant after horizon entry, and up until aeq. This stalling of the growthof the perturbation is known as the Meszaros e↵ect; (iii) when matter startsto dominate the Universal energy density � / a as we derived in previouslecture. This Figure illustrates that the Meszaros e↵ect suppress the growthof this perturbation by a factor of (aenter/aeq)2 compared to uninhibitedgrowth.

dependence of the scale factor - combined with the fact that radiation dominatesthe Universal (mass-)energy density - gives rise to a drastically di↵erent predictedtime evolution for �. As we will see next, � barely grows at all during radiationdominance. This ‘stalling’ of the growth of density perturbation in the radiation-dominated era is known as the ‘Meszaros’ e↵ect. The fluctuations are said to befrozen in the background. Mathematically, the Meszaros e↵ect is easy to under-stand. Recall that the density evolution of a perturbation � was given by thefollowing di↵erential equation:

(7) �m + 2a

a�m = 4⇡G⇢m�m.

Divide both sides by H2 = 8⇡G⇢tot

3 , where ⇢tot = ⇢rad + ⇢m. Using H = aa we find

4 M. DIJKSTRA

(8)�mH2

+2

H�m =

3⇢m�m2[⇢m + ⇢rad]

.

If we now use that deep in the radiation dominated era ⇢rad � ⇢m, then the termon the RHS can be ignored. This is because we are multiplying a small number� with another small number, and can see this term e↵ectively as a second orderterm. The di↵erential equation then simplifies to

(9)�mH

+ 2�m = 0.

If we further use that H = a/a = 1/[2t] then we are left with

(10) �m +�mt

= 0 ) �m = A+B log t = A+ C log a.

The perturbation thus only grows logarithmically with the scale factor. This growthis represented by the horizontal line in Figure 1.

• Before the perturbation enters the horizon, at a < aenter, it grows � / a2. As Imentioned in the lecture, this follows from general relativistic perturbation theory.This is beyond the scope of this lecture. Because the result from GR is not intu-itive, we might as well have replaced the words ‘general relativistic perturbationtheory’ with ‘magic’ 1.

Figure 1 shows that for the perturbation that entered the horizon during radiation

dominance at aenter the growth was suppressed by a factor of T (k) =⇣aenter

aeq

⌘2. This

suggests that T (k) ! 1 if aenter ! aeq. Indeed, for perturbations that enter the horizonduring matter domination we do not have any inhibition of growth: � / a2 during radiationdominance, and � / a during matter dominance (see footnote). This discussion clearlysuggest that there is a particular scale of interest, namely that for which aenter = aeq.This corresponds to the smallest scale for which there is no suppression of growth by theMeszaros e↵ect. The size of this perturbation is therefore equal to the horizon scale atmatter-radiation equality:

L0 = rH(aeq) = aeq

Z teq

0

cdt

a(t)= ... =

assignment 5

c

H0

1p2⌦m,0zeq

⇡ 80 cMpc,(11)

1I found a prescription later today that might provide some more insight than requiring magic: density

perturbation �k generate perturbations in the gravitational potential �k which correspond to metric per-

turbation in general relativity (matter warps space time). The equation that described the time evolution

of this metric perturbation corresponds to the perturbed Poisson equation from lecture 2, which in Fourier

space reads �k2

�k = 4⇡Ga2⇢�k. If we require that the metric perturbation cannot evolve for perturbations

outside the horizon, then we must have that a2⇢�k is independent of time. We therefore must have �k / a2

during radiation domination (⇢rad

/ a�4

), and �k / a during matter domination (⇢rad

/ a�3

).


where ⌦m,0 denotes the present-day mass density parameter, H0 is the present-day Hub-ble constant, and ‘cMpc’ denotes comoving Mpc (just to emphasis that L0 is a comovingquantity). The corresponding wavenumber is k0 = 0.1 cMpc�1.

Finally, we would like to express T (k) as a function of k. We found that the suppression

T (k) for perturbations entering at a < aeq was given by T (k) =⇣aenter

aeq

⌘2. A perturba-

tion of length L ⌧ L0 enters the horizon at aenter(L) ⇡ aeq(L/L0) = aeq(k0/k), where inthe first approximation we assumed that the horizon scale evolves as rH / a out to aeq.

Under this approximation we therefore have T (k) =⇣k0

k

⌘2for k � k0. One of the slides

shows the T (k) obtained from a more precise calculation (Bardeen et al. 1986). I havealso explicitly plotted the slope d log T/d log k. The approximate calculation given abovecaptures the limiting behaviour of T (k) and identifies the physical reason for the turn-overin the transfer function.

1.4. The Power Spectrum P (k). The power spectrum is given by P0(k) = AknT 2(k).We have specified T (k). The slope of the ‘primordial’ power spectrum has been inferred(from the Cosmic Microwave Background) to be close to 1. This value is predicted naturallyby inflation theories. The power spectrum therefore scales as

(12) P (k) /⇢

k k ⌧ k0k�3 k � k0,

with a turn-over at k = k0.Several additional comments on P (k)

• The normalisation constant A is obtained by matching to observations of the Cos-mic Microwave Background.

• Baryons a↵ect precise shape of P (k) (see paper by Eisenstein & Hu 1998). Oneexample of how baryons a↵ect the mass power spectrum was given in our discussionof the acoustic peak in the two-point correlation function (the single acoustic peakin ⇠(r) corresponds to a series of oscillations in the power spectrum).

• The case n = 1 corresponds to a special case which yields scale-invariant fluctua-tions. This is discussed next.

1.5. Why n = 1 corresponds to Scale Invariance. We derived the relation betweenP (k) and the variance in the mass density field averaged over some mass-scaleM in previouslectures. The RMS (root mean square) amplitude of fluctuations smoothed over mass-scaleM is given by

(13)p�2(M) ⌘ �(M) / M�(n+3/6) /

n=1M�2/3.

For a < aenter � / a2, so

(14) �(M) / a2M�2/3.

6 M. DIJKSTRA

This equation shows that �M is smaller for larger M . However, higher masses correspondto larger scales. These higher mass fluctuations therefore enter the horizon at a later time,and the Meszaros e↵ect limits their growth by a smaller factor. We can relate the mass Mof a perturbation to the scale factor at which it entered the horizon as

(15) M = Mh(t) / ⇢mr3H / a�3t3 / t�3/2t3 / t3/2,

where we used that the mass density ⇢m / a�3 and that the proper horizon scale scales asrH / t. We have derived the time-dependence of horizon entry of fluctuations of mass M .Substituting this into Eq 14 we have

(16) �(M) / a2M�2/3 / a2t�1 = constant.

Fluctuations that enter the horizon2 at a < aenter therefore have the same RMS amplitudeat horizon entry. The subsequent growth of these perturbations is stalled until aeq, afterwhich they all grow as � / a. This identical growth ensures that the RMS amplitude ofthese fluctuations remains independent of scale at all times. Although I have not shownit in the lecture, you can do the same analysis for perturbations that enter the horizonat a > aenter and get the same result: namely that �(M)=constant! This shows thatall fluctuations enter the horizon with the same RMS amplitude. This remarkable scaleinvariance is a special property of the power spectrum with n = 1.

1.6. Some Concluding Remarks. A very brief & broad summary of the processes thatare relevant for the formation of structure in our Universe.

• At time t ⇠ 0 some process (inflation) generates the primordial power spectrumP (k) = Akn with n ⇠ 1.

• Processes like horizon entry of a perturbation combined with the Meszaros e↵ectthen modify the shape of the power spectrum into P (k) = AknT 2(k) at a aeqwhere T (k) = 1 for k ⌧ k0 ⇠ 0.1 cMpc�1, and T (k) = (k0/k)2 for k � k0. Baryonsprovide further smaller modifications of this power spectrum.

• At aeq < a < arec dark matter perturbations grow as � / a, while radiation pressureprevent baryons from collapsing on all scales smaller than the Jeans length (which isvery large during this epoch). Acoustic waves generated by density perturbationspermeate the primordial photon-baryon plasma, which introduce further smallercorrections to the power spectrum P (k).

• At a > arec the sound speed drops by five orders of magnitude, and the consequently,the Jeans mass drops by 10 orders of magnitude. Baryons are now free to collapseinto the potential wells generated by the dark matter (i.e �b ! �DM). This isimportant: the RMS amplitude of the density fluctuations in baryons at arec isonly �b ⇠ 10�5, which would not be enough to form non-linear objects (recall that� / a and that a increases only by a factor of 103).

• Press-Schechter theory allows us to take a Gaussian random density field - whichdescribes the density field post recombination still extremely well - and transform

2Convince yourself that this corresponds to fluctuations on mass-scales that are relevant for astrophysical

objects (galaxies, groups of galaxies, clusters of galaxies).


this into predictions for the abundance (number density) and clustering of (non-linear) collapsed objects. While this theory has many flaws, it predicts the numberdensity of collapsed objects as a function of mass M and redshift z remarkably well.It also highlights the important hierarchical aspect of structure formation: namelythat small (i.e. low mass) objects collapse first, and that larger (i.e. more massive)objects collapse later.

1.7. Shortcomings/Caveats. Our discussion of structure formation does not explain themost apparent visual appearance of structures in the Universe, namely the walls, filaments,and nodes that are apparent in observed galaxy distributions and numerical simulations(see slides). The main reason is that in our discussion of the non-linear growth of structurewe focussed on spherical top-hat model. In this model the evolution of the perturbationis determined entirely by its radius R(t) and the overdensity inside of it (�(t)). Most

R1R2

Figure 2. Geometry for a simple ellipsoidal perturbation.

perturbations in Gaussian random fields are not spherical. In fact, Gaussian random fieldtheory can be used to study shapes of peaks in the density field. Consider the simple case inwhich an overdensity is ellipsoidal instead of spherical, and that there are two characteristicaxes denoted with R1 and R2. Outside of this ellipsoid the overdensity is � = 0. Clearly,the mean overdensity inside the sphere of radius R1 is larger than that inside the sphereof radius R2. This implies that less growth in � is required for the perturbation reach thecritical linear overdensity for collapse �crit = 1.69. The structure therefore collapses alongits R1 axes while it still has a finite size in its R2 direction. Gravity therefore amplifiesthese deviations from spherical symmetry into flattened objects like walls (collapse along1 axis), filaments (collapse along two axes), and halos (collapse along all three axes). So

8 M. DIJKSTRA

while our discussion of structure formation in previous lectures does not produce theseobserved & simulated features in the mass distribution, the theory is easily adjusted to beable to explain these structures.


• For the discussion of Transfer functions I used the book by Peter Schneider ’Extra-galactic Astronomy and Cosmology: An Introduction’.

• Useful discussions on horizon scales in cosmology, and the di�culty of modellingsuper horizon scales are given in Longair ’Galaxy Formation’. Specifically Chapter12.2 + 12.3.

• The lectures by Frank van den Bosch have some very nice & clear slides. Seehttp://www.astro.yale.edu/vdbosch/astro610_lecture4.pdf for a very briefdiscussion why � / a2 before horizon entry. His site also has references to his book(Mo, Van den Bosch & White) which provides much more details to these lectures.Can be very useful.

AST4320: LECTURE 10

M. DIJKSTRA

1. Structure of Dark Matter Halos

1.1. The Isothermal Density Profile. We would like to understand the structure of

dark matter halos. In the standard cosmological model dark matter only interacts gravita-

tionally, is collisionless and (hence) has no pressure. Because dark matter particles do not

collide, they do not behave as an ideal fluid. Recall that ideal fluids were described com-

pletely by their density and pressure. In ideal fluids the velocity distribution of particles

is completely isotropic, and the pressure provides a measure of the mean speed of the gas

particles. In theory a dark matter halo can contain particles with more complex velocity

distributions (for example, an in falling low mass halo can survive as a separate sub halo

within a larger halo. This cannot happen for an ideal fluid).

If we want to derive the density profile of a dark matter halo then we cannot use the

fluid equations we used before. Instead, we need to introduce the so-called ‘distribution

function’, f(x,v), which corresponds to the (mass) density of particles in phase-space.

Phase-space is the space spanned by 3D physical space and 3D velocity space, i.e. a point

in 6D phase space has 6 coordinates (x, y, z, v

x

, v

y

, v

z

). The distribution function relates to

more commonly encountered quantities density [⇢(x)] and the velocity distribution [f(v)]

as

⇢(x) =

Zd

3v f(x,v)(1)

f(v) =

Zd

3xf(x,v).

Distribution functions help us accurately describe collision less systems such as stars and

dark matter particles. At first glance, going to 6D to describe a system does not look very

appealing at all. Fortunately, Boltzmann’s brilliance comes to the rescue here. Boltzmann

proposed that for systems in ‘thermodynamic equilibrium’ the phase function has the

following simple form:

f(x,v) / exp

⇣�E

kbT

⌘,(2)

where E denotes the total energy of a particle, kb is the Boltzmann constant, and T de-

notes the temperature of the system (note that ’temperature’ for a system of stars clearly

1

2 M. DIJKSTRA

does not mean ordinary temperature. Instead it provides a measure of the velocity dis-

persion of the stars.). The energy per dark matter particle is given by E = E(v,x) =

12mDM|v|2 + mDM�(x), where mDM denotes the mass of a dark matter particle, and �

denotes the gravitational potential (note that this is the ordinary gravitational potential,

and not the perturbed one that we worked with in perturbation theory).

The dark matter density at x is given by

⇢(x) /Z

d

3v f(x,v) = N

Z 1

04⇡v

2dv exp

⇣� mDMv

2

2kbT

⌘exp

⇣�mDM�(x)

kbT

⌘(3)

, where we have adopted the notation |v| = v. The potential energy of a dark matter

particle only depends on its position, and the term exp

⇣�mDM�(x)

kbT

⌘can be taken outside

of the integral. The integral over v converges, and can be replaced with a number C2(x)

(here we allow for the possibility that the velocity distribution changes with position). We

define C1(x) = C2(x)N , and we are left with

⇢(x) = C1(x) exp

⇣�mDM�(x)

kbT

⌘.(4)

Next we assume that the velocity distribution does not depend on x, i.e. particles have the

same velocity-distribution at all x: this is analogous to assuming the same temperature

everywhere for an ideal gas. Under this assumption C1(x) = C1 is a number. We can

invert this equation as

�(x) = � kbT

mDMln

⇣⇢

C1

⌘.(5)

This is useful because there is a second equation that relates the gravitational potential

and the density, namely the Poisson equation. The Poisson equation reads

r2� = 4⇡G⇢ ! 1

r

2

d

dr

r

2 d

dr

�(r) = 4⇡G⇢(r),(6)

where expressed the r2operator in spherical coordinates and assume spherical symmetry

in the density and gravitational potential. If we now substitute Eq 5 for �x, then we obtain

a second order di↵erential equation for ⇢(r):

kbT

mDMr

2

d

dr

r

2 d

dr

ln

⇣⇢

C1

⌘=

kbT

mDMr

2

d

dr

r

2 d

dr

ln ⇢ = 4⇡G⇢(r).(7)

We now have a second order di↵erential equation. It is straightforward to show (by sub-

stituting it into the di↵erential equation) that the following provides a solution to this


equation:

⇢(r) =

A

r

2A =

kbT

2⇡GmDM.(8)

This density profile that falls o↵ as r

�2is known as an isothermal density profile. If

we perturb a smooth distribution of dark matter particles, further assume that these dark

matter particles have the same velocity distribution (i.e. ‘temperature’) everywhere in

space

1, then this is the profile that gravitational collapse will lead to.

It may help to provide some intuition by comparing the equilibrium solution of an ideal

gas in pressure equilibrium with the gravitational potential generated by the same gas. For

gas in hydrostatic equilibrium with gravity we have

(10)

dP

dr

= �GM(< r)⇢

r

2.

For gas with the same temperature - isothermal gas - we have P = nkbT0 =

⇢

mpkbT0.

Note that M(< r) = 4⇡

Rr

0 x

2dx ⇢(x). I will leave it as an exercise to show that this is

the same di↵erential equation, and that isothermal gas hydrostatic equilibrium therefore

would settle into the same density-profile.


• The derivation of the isothermal density profile for the dark matter follows Binney

& Tremaine ’Galactic Dynamics’ (page 226-228).

• For a review on the cusp-core problem, see the paper by E. de Blok mentioned on

the slides.

1We have practically shown already what this distribution is. Recall that f(v)d3v = d3v

Rd3xf(x,v).

In this case

(9) f(v)dv

Zd3xf(x,v) = 4⇡v2dv exp

⇣� mDMv2

2kbT

⌘Zd3x exp

⇣�mDM�(x)

kbT

⌘.

The velocity distribution of dark-matter particles thus obeys a Maxwell-Boltzmann distribution with tem-

perature T .

ast4320: lecture 2 - uio

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