Introduction to Cosmology

Introduction to 
COSMOLOGY
CERN Lectures
July 2022
Daniel Baumann
University of Amsterdam &
National Taiwan University

Disclaimer:
Teaching all of cosmology in 3 hours is an impossible task.
I will focus on the big picture and suppress technical details.
More can be found in:

OUTLINE:
The Expanding
Universe
Structure
Formation
Quantum Origin
• Hubble-Lemaitre law
• Friedmann equation
• Matter and radiation
• Dark energy
• Nucleosynthesis
• Cosmic microwave background
• Density fluctuations
• Gravitational clustering
• Galaxy formation
• Power spectrum
• Cosmic sound waves
• CMB anisotropies
• Horizon problem
• Cosmological inflation
• Quantum fluctuations
• Primordial perturbations

Goal of this Lecture
We wish to derive, and then solve, the equation governing the evolution of
the entire Universe.
This is possible because, on large scales, the Universe is homogeneous and
isotropic, and therefore allows for a simple mathematical description.
100 Mpc
Sloan Digital Sky Survey

0 0.5 1 1.5 2
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
×103
km/s
0 5 10 15 20 25 30 35
0
5
10
15
20
×103
Hubble’s Law
In 1929, Hubble discovered the velocity-distance relation of galaxies:
V = H0R
Inferred from redshifts Inferred from luminosities

Hubble’s Law
In General Relativity, this can be interpreted as the expansion of space:
time
R(t) = a(t)R0
Scale factor
R0
V ⌘ Ṙ =
ȧ
a
R ⌘ H R
Hubble parameter

Hubble’s Law
The observed value of the Hubble constant is
H0 = (68 ± 2) km s 1
Mpc 1
where Mpc = 3.2 ⇥ 106
light-years.
⇠ 1026
m
The corresponding Hubble time and Hubble distance are
tH ⌘ H 1
0 = (14.38 ± 0.42) Gyrs
dH ⌘ cH 1
0 = (4380 ± 130) Mpc
These are the characteristic age and size of the universe.

Friedmann Equation
Let us now determine the evolution of the scale factor.
We will do this using Newtonian gravity (but comment on GR corrections).
The force on a test particle on the surface of the sphere is
F =
GMm
R2
Consider a spherical region in a homogeneous universe:
⇢(t)
R(t)
M
m

Friedmann Equation
The acceleration of the test particle is
d2
R
dt2
=
GM
R2
Integrating this gives the conservation of energy:
1
2
✓
dR
dt
◆2
GM
R
= U = const.
Substituting M =
4⇡
3
R3
⇢ and R(t) = a(t)R0 , we find
✓
ȧ
a
◆2
=
8⇡G
3
⇢ +
2U
R2
0
1
a2
Friedmann
Equation
Hubble
parameter

U > 0 U < 0 U = 0
Friedmann Equation
✓
ȧ
a
◆2
=
8⇡G
3
⇢ +
2U
R2
0
1
a2
There are three diﬀerent cases:
expands forever stops and recollapses limiting case
In GR, these correspond to:
negatively curved positively curved flat
Observations favor a ﬂat universe.

Friedmann Equation
For a flat universe, the Friedmann equation reduces to
H2
=
8⇡G
3
⇢
From the measured value of the Hubble constant, we infer the average
density of the universe today:
⇢0 =
3H2
0
8⇡G
= 0.8 ⇥ 10 29
grams cm 3
= 1.3 ⇥ 1011
M Mpc 3
= 5.1 protons m 3

Rµ⌫
1
2
Rgµ⌫ = 8⇡G Tµ⌫
H2
=
8⇡G
3
⇢
Friedmann Equation
In GR, the Friedmann equation is derived from the Einstein equation:
Curvature tensor Energy-momentum tensor
where the mass density is replaced by the energy density .
" ⌘ ⇢c2
(In natural units (c ⌘ 1), we use ⇢ interchangeably.)
"
and
To solve the Friedmann equation, we need to know how the energy density
evolves as the universe expands.

Cosmic Inventory
Radiation
(photons, neutrinos)
Dark matter
Atoms
Dark energy
The universe is filled with four diﬀerent types of energy:
We need to determine how each of these components evolves and sources
the expansion of the universe.

Fluid Equation
Consider a fluid in a box:
E, P
V
dE = dQ PdV
The First Law of thermodynamics implies
Homogeneity doesn’t allow any heat flow ( ), so that
dQ = 0
dE
dt
= P
dV
dt

Fluid Equation
d⇢
dt
= 3
ȧ
a
(⇢ + P)
Cosmological fluids are described by a constant equation of state:
w ⌘ P/⇢
The fluid equation then becomes
˙
⇢
⇢
= 3(1 + w)
ȧ
a
⇢ / a 3(1+w)
Fluid
Equation
Using E = ⇢V and V / a3
, we find
dE
dt
= P
dV
dt

Matter
For a pressureless fluid (= matter), we get
P = 0 ⇢ ⌘
E
V
/ a 3
E = const
Feeding ⇢ / a 3
into the Friedmann equation, we find
✓
ȧ
a
◆2
/ ⇢ / a 3
a / t2/3
Most of the matter in the universe is non-luminous dark matter.
Ordinary matter is less than 5%.

Radiation
For a relativistic fluid (= radiation), we get
P =
1
3
⇢ ⇢ ⌘
E
V
/ a 4
E / a 1
Most of the radiation in the universe are photons from the early universe
(= cosmic microwave background). Starlight is less than 0.1%.
✓
ȧ
a
◆2
/ ⇢ / a 4
a / t1/2
Feeding into the Friedmann equation, we find
⇢ / a 4

Matter and Radiation
The universe was first dominated by radiation, then by matter:
10−10
10−8
10−6
10−4
10−2
1
10−5
1
105
1010
1015
1020
1025
1030
ρr ∝ a−4
ρm ∝ a−3
a(t)
ρ(a)

Acceleration Equation
Combining the Friedmann equation with the fluid equation
d
dt
"✓
ȧ
a
◆2
=
8⇡G
3
⇢
#
+
d⇢
dt
= 3
ȧ
a
(⇢ + P)
ä
a
=
4⇡G
3
(⇢ + 3P)
Acceleration
Equation
we get
There are two diﬀerent cases:
P >
1
3
⇢ P <
1
3
⇢
Expansion slows down Expansion speeds up
ä < 0 ä > 0

Dark Energy
In 1998, it was discovered that the universe is accelerating.
The source of the acceleration is unknown. We call it dark energy.
34
36
38
40
42
44
46
Distance
modulus
CFA1
CFA2
CFA3S
CFA3K
CFA4p1
CFA4p2
CSP
SDSS
SNLS
SCP
CANDELS
+CLASH
GOODS
PS1
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Redshift
°0.5
0.0
0.5
Residuals
With DE
Without DE

Dark Energy
In 1998, it was discovered that the universe is accelerating.
The source of the acceleration is unknown. We call it dark energy.
The simplest form of dark energy is a cosmological constant:
Rµ⌫
1
2
Rgµ⌫ + ⇤gµ⌫ = 8⇡G Tµ⌫
Moving this term to the RHS, it becomes a vacuum energy:
E / V
P = ⇢ ⇢ ⌘
E
V
= const
✓
ȧ
a
◆2
/ ⇢ = const a / eH0t
Feeding this into the Friedmann equation gives

Dark Energy
10−10
10−8
10−6
10−4
10−2
1 102
104
10−5
1
105
1010
1015
1020
1025
1030
ρr
ρm
ρΛ
a(t)
ρ(a)
Today, the universe is dominated by dark energy (Why now?):
today

Cosmological Constant Problem
The observed value of the dark energy density is much smaller than our
(naive) theoretical expectation:
⇢QFT ⇠
⇤UV
Z
0
dk3
(2⇡)3
1
2
~! ⇠ ⇤4
UV o ⇢obs = (2 ⇥ 10 3
eV)
Quantum zero-point fluctuations
Sum over wavelengths
Explaining the observed dark energy density remains one of the most
important open questions in fundamental physics.

The Hot Big Bang
The universe started in a hot and dense state:
As the universe expands, it cools.
Many interesting things happened.
temperature
earlier later
time
hotter colder
T(t) / a(t) 1

The universe is filled with almost equal
amounts of matter and antimatter
For some mysterious reason, there was initially a fraction more matter
than antimatter. This matter survived the annihilation.
10 000 000 001 10 000 000 000
matter
Without this asymmetry we wouldn’t exist.
As the universe cools, matter
and antimatter annihilate.
+ =
light
t = 10 19
s

t = 10 5
s Quarks and gluons condense into nuclei:
temperature
time
u
d
u
u
d
d
proton
neutron
10 μs

t = 1 s Neutrinos decouple and neutrons freeze out:
temperature
time
1 s
Free-streaming neutrinos
• 40% of the energy density
• Significant eﬀect on the expansion

t = 3 min Light elements (H, He, and Li) form:
temperature
time
H He
3 min
• Heavier nuclei were fused inside stars.
• Big Bang nucleosynthesis (BBN) predicts the
correct abundances of the light elements.
25%
75%

Atoms form and the first light is released:
370 000 yrs
e-
e-
e-
e-
e-
e-
H He
e
temperature
time
Free-streaming
photons
t = 370 000 yrs

• 410 photons per cubic centimeter
• cooled by the expansion: 2.7 K
• faint microwave radiation: CMB
This afterglow of the Big Bang is still seen today:
0
100
200
300
400
Intensity
[MJy/sr]
100 200 300 400 500 600
−0.1
0
0.1
Frequency [GHz]
Cosmic Microwave Background

temperature
time
1 billion yrs
t > 1 billion yrs Matter collapses into stars and galaxies:

This history of the universe is an observational fact:
10 μs 370 000 yrs
1 s 3 min
QCD phase
transition
Neutrino
decoupling
BBN
e
-
Photon
decoupling
Structure
formation
1 billion yrs
• The basic picture has been confirmed by many independent observations.
• Many precise details are probed by measurements of the CMB.

How did the structure in the universe form?
Light in the Universe
Light on Earth
The large-scale structure of the universe isn’t randomly distributed, but has
spatial correlations. What created these correlations?

Our best answer to these questions involves a fascination connection between
the physics of the very small and the very large:
Gravitational
clustering
Quantum
fluctuations
CMB
fluctuations
Galaxies
Cosmic
sound waves
Part II.1
Part II.2
Part III
10 32
s 370 000 yrs 1 billion yrs

Our best answer to these questions involves a fascination connection between
the physics of the very small and the very large:
This allows us to use cosmological observations to learn about short-distance/
high-energy particle physics.
Quantum
fluctuations
CMB
fluctuations
Galaxies
10 32

Gravitational Clustering
Consider a spherical overdensity in a homogeneous universe:
We are interested in the evolution of the density contrast:
¯
⇢(t)
⇢(t)
⌘
⇢ ¯
⇢
¯
⇢
Galaxies form when the density contrast reaches a critical value.

• In a static universe, the density contrast grows exponentially:
• In an expanding universe, the growth is slower:
during the matter era
during the radiation era
(t) / ln t
(t) / t2/3
(t) / et/⌧
The clustering of matter only begins after matter-radiation equality.
The evolution of the density contrast can be derived using Newtonian
gravity (see slides online). Here, I just quote the results:

Fourier Modes
In reality, density perturbations are not spherically symmetric.
A general density fluctuation can be decomposed into its Fourier modes:
Each Fourier mode satisfies the same equation of motion as a spherically
symmetric overdensity.
X
k
k sin(kx + k)
x
=
= k = 2⇡/

Power Spectrum
The power spectrum is the square of the Fourier amplitude:
The Fourier transform of the power spectrum is the two-point correlation
function. This is the main statistic of cosmological correlations.
P(k) = | k|2
⇠(r) =
Z
d3
k
(2⇡)3
P(k) eikr

10−4
10−3
0.01 0.1 1 10
k [Mpc−1
]
1
10
102
103
104
P
(k)
galaxy
clustering
Lyα
CMB
Power Spectrum
The observed matter power spectrum is Horizon scale at
matter-radiation equality
Large scales Small scales

µK +300
300
The CMB has tiny variations in its intensity, corresponding to small density
fluctuations in the primordial plasma:
These fluctuations aren’t random, but are highly correlated.

2 10 30
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
500 1000 1500 2000 2500
Multipole
90◦
18◦ 1◦
0.2◦
0.1◦
0.07◦
Angular separation
Large scales Small scales
The observed CMB power spectrum is

10 100 1000
Multipole Figure courtesy of Mathew Madhavacheril

2 10 30
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
500 1000 1500 2000 2500
Multipole
90◦
18◦ 1◦
0.2◦
0.1◦
0.07◦
Angular separation
What created the features in the power spectrum?
CMB Power Spectrum

Photon-Baryon Fluid
At early times, photons and baryons (mostly protons and electrons) are
strongly coupled and act as a single fluid:
Dark matter
Baryons
Photons
The photon pressure prevents the collapse of density fluctuations.
This allows for sound waves (like for density fluctuations in air).

The pattern of the CMB fluctuations is a consequence of these sound waves:
Superposition of many waves CMB correlations
Cosmic Sound Waves

Cosmic Sound Waves
Consider the evolution of a single localized density fluctuation:

Cosmic Sound Waves
This creates a radial sound wave in the photon-baryon fluid:
The wave travels a distance of 50 000 light-years (called the sound horizon)
before the universe becomes transparent to light.
Dark matter
Photons + baryons
rs
Sound
horizon

CMB Anisotropies
This sound horizon is imprinted in the pattern of CMB fluctuations:
Temperature Polarization
2 +2
1 +1
0 2 +2
1 +1
0
20
40
60
80
0
0.4
0.2
0.4
intensity of 11396 cold spots
✓s ✓s
1
2 ✓s
Planck (2015)

CMB Anisotropies
This sound horizon is imprinted in the pattern of CMB fluctuations:
90◦
18◦
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
2◦
0.2◦
0.1◦
0.07◦
Angular separation

CMB Anisotropies
The precise pattern of the CMB fluctuations depends on the composition of the
universe (and its initial conditions):
Atoms
Dark matter
Dark energy
Observations of the CMB have therefore allowed us to determine the parameters
of the cosmological standard model.

Without dark matter, the data would look very diﬀerent:
Figures courtesy of Zhiqi Huang and Dick Bond [ACT collaboration]
Data No Dark Matter
Dark Matter

2 10 30
0
2000
4000
6000
8000
10000
Power
[µK
2
]
500 1000 1500 2000
Multipole
This can also be seen in the power spectrum: ⌦m = 0.32
without dark matter
Planck (2018)

Without dark energy, the data would look very diﬀerent:
Data No Dark Energy
Dark Energy
Figures courtesy of Zhiqi Huang and Dick Bond [ACT collaboration]

2 10 30
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
500 1000 1500 2000
Multipole
This can also be seen in the power spectrum:
Planck (2018)
⌦⇤ = 0.68
without dark energy
with dark energy

with
dark energy
without
dark energy
Riess et al (1998)
Perlmutter et al (1998)
This is consistent with the direct observation of dark energy from supernova
observations:

The peak heights depend on the baryon density: ⌦b = 0.04
The measured baryon density is consistent with BBN.
more baryons
2 10 30
0
1000
2000
3000
4000
5000
6000
7000
8000
Power
[µK
2
]
500 1000 1500 2000
Multipole
Planck (2018)
Baryons

As
✓
k
k0
◆ns 1
The CMB power spectrum also probes the initial conditions:
A B C D
A
B
C
D
Amplitude
scale-dependence
As = 2.20 ⇥ 10 9
evolution
Initial Conditions

ns = 0.96
The primordial power spectrum is close to scale invariant:
WMAP (2009)
Planck (2018)
2 10 30
0
1000
2000
3000
4000
5000
6000
7000
Power
[µK
2
]
500 1000 1500 2000
Multipole
ns = 0.75
ns = 1.25
The observed deviation from scale invariance is significant.

The Standard Model
A simple 5-parameter model fits all observations:
Amount of ordinary matter
Amount of dark matter
Amount of dark energy
Amplitude of density fluctuations
Scale dependence of the fluctuations
⌦b = 0.04
ns = 0.96
109
As = 2.20
⌦⇤ = 0.68
⌦m = 0.32

The Standard Model
A key challenge of modern cosmology is to explain these numbers:
⌦b = 0.04
ns = 0.96
109
As = 2.20
⌦⇤ = 0.68
⌦m = 0.32
Why is there more matter
than antimatter?
What is the dark matter?
What is the dark energy?
What was the origin of
the fluctuations?

Consider a spherical overdensity in a homogeneous universe:
We are interested in the evolution of the density contrast:
R(t)
¯
⇢(t)
⇢(t)
⌘
⇢ ¯
⇢
¯
⇢

Let us first study this overdensity in a static universe.
The acceleration at the sphere’s surface is
R̈ = G
M
R2
=
G
R2
✓
4⇡
3
R3
¯
⇢
◆
R̈
R
=
4⇡G¯
⇢
3
(t)
M =
4⇡
3
R3
(t) ¯
⇢ [1 + (t)] = const
Conservation of mass implies , so that
for | | ⌧ 1 .
R(t) =
✓
3M
4⇡¯
⇢
◆1/3 h
1 + (t)
i 1/3
⇡ R0

1
1
3
(t)
Substituting this into the equation of motion, we get
¨ = (4⇡G ¯
⇢) (t) = Aet/⌧
+ Be t/⌧
Exponential growth

Adding Expansion
Now, consider the same overdensity in an expanding universe with only
pressureless matter:
R(t)
⇢(t)
¯
⇢(t) / a 3
(t)
The acceleration at the sphere’s surface is
R̈ =
GM
R2
=
G
R2
✓
4⇡
3
R3
⇢
◆
=
4⇡G
3
¯
⇢R
4⇡G
3
(¯
⇢ )R

M =
4⇡
3
R3
(t) ¯
⇢(t) [1 + (t)] = const
Adding Expansion
Mass conservation implies , so that
R(t) / ¯
⇢ 1/3
(t)
h
1 + (t)
i 1/3
/ a(t)

1
1
3
(t)
Substituting this into the equation of motion, we get
R̈
R
=
ä
a
1
3
¨ 2
3
ȧ
a
˙ =
4⇡G
3
¯
⇢
4⇡G
3
¯
⇢
¨ + 2H ˙ = (4⇡G¯
⇢)
The density contrast therefore satisfies
Hubble friction
Gravitational force

Clustering of Dark Matter
In a matter-dominated universe, with a / t2/3
H =
2
3t
and , we get
¨ + 2H ˙ =
3
2
H2 ¨ +
4
3t
˙ =
2
3t2
The solution for the density contrast is
Power-law growth
(t) = At2/3
+ B t 1
In a radiation-dominated universe, the growth is only logarithmic:
(t) = A ln t + B
The clustering of matter only begins after matter-radiation equality.

So far, we have described the evolution of fluctuations in the hot Big Bang
and the formation of the large-scale structure of the Universe:
Gravitational
clustering
Primordial
fluctuations
CMB
fluctuations
Galaxies
Cosmic
sound waves
What created the primordial fluctuations?
We now want to ask:

90◦
18◦
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
2◦
0.2◦
0.1◦
0.07◦
Angular separation
An important clue is the fact that the CMB fluctuations are correlated over
the whole sky:
Superhorizon

distance light travelled
since the Big Bang
In the standard hot Big Bang theory, this is impossible:
2
Big Bang
Observable
universe

The correlations must have been created before the hot Big Bang:
Big Bang
?
Photon
decoupling
distance light travelled
since the Big Bang
CMB

Inflation solves the problem by invoking a period of superluminal expansion:
Guth (1980)
Linde (1982)
Albrecht and Steinhardt (1982)
Inflation
Big Bang
Photon
decoupling
CMB
a(t) = eHt
Inflation

In less than
The entire observable universe then originated from a microscopic, causally
connected region of space.
a(t) = eHt
Inflation
10 32
seconds, the universe doubled in size at least 80 times:

Inflation
To achieve inflation requires a substance with a nearly constant energy density
(like dark energy):
V ( )

Inflation
To end inflation, this substance must decay (like a radioactive material):
The product of this decay is the hot Big Bang. One of these bubbles
is our universe.

Inflation
In quantum mechanics, the end of inflation is probabilistic and varies
throughout space:
This creates the primordial density ﬂuctuations.

Quantum Fluctuations
In quantum mechanics, empty space is full of violent fluctuations:
x p ~

Lamb shift
These quantum fluctuations are real, but usually have small eﬀects:

During inflation, these quantum fluctuations get amplified and stretched:
After inflation, these fluctuations become the large-scale density fluctuations.

The nearly constant inflationary vacuum energy leads to an approximately
scale-invariant power spectrum:
Primordial Correlations
The slow decay of the inflationary energy predicts slightly more power on
large scales:
P(k)
k
Askns 1
ns < 1

✓
C(✓)
The well-understood physics of the photon-baryon fluid turns these primordial
correlations into correlations of the CMB anisotropies:
CMB Anisotropies

The predicted correlations are in remarkable agreement with the data:
90◦
18◦
0
1000
2000
3000
4000
5000
6000
Power
[µK
2
]
2◦
0.2◦
0.1◦
0.07◦
Angular separation
CMB Anisotropies

How can inﬂation become part of the standard history of the
universe with the same level of conﬁdence as BBN ?
“Extraordinary claims require extraordinary evidence”
Carl Sagan
10 μs 1 s 3 min
QCD phase
transition
Neutrino
decoupling
BBN
10-32 s
Inflation?

Besides density fluctuations, inflation predicts gravitational waves:
Primordial Gravitational Waves
The strength of the signal depends on the energy scale of inflation, which
may be as high as 1016 GeV.

These gravitational waves would produce a characteristic swirl pattern
(called B-modes) in the polarization of the CMB:
B-modes
Detecting these B-modes is a central goal of observational cosmology.

Ongoing Experiments
EBEX
SPIDER
POLARBEAR
BICEP
SPTPol
ACTPol
ABS
CLASS
CMB Stage III

CMB Stage III.5
(2021-2028)
Atacama Desert
Planned Experiments

Atacama Desert
South Pole
Planned Experiments
CMB Stage IV
(2028-2035)

Selected by JAXA in 2019
Launch in 2028
LiteBIRD
Planned Experiments

A B-mode detection would be a milestone towards a complete
understanding of the origin of all structure in the universe
10 32

10 μs 370 000 yrs
1 s 3 min
e
-
1 billion yrs
We have a remarkably consistent picture of the history of the Universe from
fractions of a second after the Big Bang until today:
We also have tantalizing evidence that the primordial seed fluctuations for the
formation of structure were created during a period of inflation:

⌦b
Yet, many fundamental questions remain:
• What is dark matter and dark energy?
• Did inflation really occur? And what was driving it?
• What is the origin of the matter-antimatter asymmetry?
• …
We hope that future observations will shed light on these questions.
Observations of the CMB have revolutionized cosmology:
⌦m
⌦⇤
As , ns

Introduction to Cosmology

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