Chia-Min Lin 林家民

Institute of Physics

Academia Sinica

Kobe University (from May 2012)

Lectures given in Chuo U/ Ochanomizu U April 10/2012

Plan of my lectures

• Lecture 1: from basic cosmology to inflation

• Lecture 2: primordial density perturbation and experimental tests

General remarks

• This lecture is aimed at 1st year master degree students so I will try to make things as simple as possible.

• This means some explanation is not very rigorous but I hope you can get the concept.

• Please ask questions.

Suggested reading

• Kinney arXiv: 0902.1529

• Baumann arXiv: 0907.5424

• Lyth and Riotto arXiv: hep-ph/9807278

• Linde arXiv: 0705.0164

• “The early universe”, by Kolb and Turner

• “Primordial Density Perturbation”, by Lyth and Liddle

Lecture I

Hot Big Bang

Why we believe in Hot Big Bang?

• Hubble expansion

• Relic temperature and Cosmic Microwave Background (CMB)

• Big Bang Nucleosynthesis (BBN)

Hubble expansion

a

aH

a: scale factor

Edwin Hubble

My biggest blunder?

Hrv

axr

x: comoving coordinate

H: Hubble parameter (Hubble constant?)

Relic Temperature

Wilson and Penzias

Nobel Prize in Physics (1978)

George Gamow

Big Bang theory John Mather George Smoot

Nobel Priza in Physics (2006)

Cosmic Microwave Background (CMB)

510~

73.2~

T

T

KT

COBE

WMAP A baby picture of our universe (when it was 300000 years old. Now it is more than 10000000000 years old.)

Big Bang Nucleosynthesis (BBN)

1 n : 7 p (freeze out)

16 nucleons: 2 n and 14 p 4 of 16 (25 %) combined into One helium-4 nucleus

BBN happened when the Universe is 3-20 minutes old.

Alpher-Bethe-Gamow paper (known as αβγ) 1948

Friedmann Equation

2

3

2

13

4

ama

ma

G

Potential energy + Kinetic energy = const.

223 PMH

28

1

PMG

GeVMP18104.2

The equations do not depend on the size of the sphere, therefore GR exactly reduces to Newtonian theory.

+𝑐𝑜𝑛𝑠𝑡.

+𝑐𝑜𝑛𝑠𝑡.

𝑎2

Friedmann Equation

22

2

3 a

k

MH

P

223 Pc MH 𝑘 = 0

2)(1

aH

k

c

Friedmann

Friedmann Equation

ama

ma

G

2

3

3

4

a

a

M P

26

amF

a

a

M

p

P

26

3

In general relativity presure gravitates!

Continuity equation

)(3 pH

2

2

22 33 PP Ma

aMH

a

a

M

p

P

26

3

Continuity equation (II)

There is another way to derive continuity equation:

𝐸 = 𝜌𝑉

𝑑𝐸 = −𝑝𝑑𝑉 = 𝑉𝑑𝜌 + 𝜌𝑑𝑉

𝑑𝜌 +𝑑𝑉

𝑉𝜌 + 𝑝 = 0 𝑉 ∝ 𝑎3

𝑑𝑉

𝑉=3

𝑑𝑎

𝑎

Devided by dt

𝜌 +3𝐻 𝜌 + 𝑝 = 0

Matter (or dust) For matter: 0p

Haa

a

a

33 3

3

The relation between p and 𝜌 is called the equation of state

𝜌 +3𝐻 𝜌 + 𝑝 = 0

Matter

𝐻2 ∝ 𝜌

3

2

2

1

2

1

2

3

3

ta

dtdaa

aa

aa

a

aa

a

0

0

a

t

Radiation

1 ahc

hE

3 an

4 a

Radiation For radiation:

H

a

4

4

31

p

𝜌~𝑇4

𝑎~1

𝑇

3TT

ps

.~ 3 constsaS

𝜌 +3𝐻 𝜌 + 𝑝 = 0

Radiation

2

1

1

2

4

ta

dtada

aa

aa

a

aa

a

𝐻2 ∝ 𝜌

0

0

a

t

T

Hot big bang singularity

Cosmological constant (dark energy?)

𝑝 = −𝜌 If 𝜌 = 𝑐𝑜𝑛𝑠𝑡.

𝐸 = 𝜌𝑉

𝑑𝐸 = 𝜌𝑑𝑉 = −𝑝𝑑𝑉

𝜌 = 0

Vacuum energy

Energy Density vs Scale Factor

figure from 0708.2865

Units 𝑐 = ℏ = 𝑘 = 1

time

lengthc [Length]=[Time]

2mcE [Energy]=[Mass]

hcE [Energy]=[Length]-1

kTE [Energy]=[Temperature]

[Energy]=[Mass]=[Temperature]=[Length]-1=[Time]-1

Dimensional analysis

• We set

• the Hubble parameter: dimension of mass

• Energy density of the universe:

• Size of the universe:

• Age of the universe:

• Light:

• heavy:

• Decay:

Hl /1~

Ht /1~

Hm

Hm

H~ Htdec /1~/1~

Hm /1/1~

223 PMH

Compton wavelengh:

𝑐 = ℏ = 𝑘 = 1 Sometimes : 𝑀𝑃 = 1

Planck scale and Hubble parameter

)10/(1)105/(11010 35442832 mseVKM P

Planck scale

Hubble parameter

MpcskmH /)/(700

eVs

161051

kmMpc 19103Megaparsec yearslight 26.3parsec

eVs

H 33180

10~10

1~

KeV 410~ eVK 410~

Age and Size of the universe

m

years

s

eVH

26

10

17

33

0

10

10~

105

110

1

seV

161051

Energy density

DEDM

P

eV

eV

MH

~~)(103

)(10103

3

411

45566

22

441544 10~)(10~)3(~ eVKT

10105 n

nB 134

1010

~ eV

GeV

E

Eproton

04.0~B

Decoupling and recombination

DM410~

10 a310eqa

eVKTeq 1.0~3000

124 10~~ eqeq T 0

510~ HH years10~1

10~1 5

0

5

HH

But our observable universe was 310 times smaller. years710

Recombination and last scattering surface

eVKTeq 1.0~3000

This means CMB contains many causally disconnected regions!

Last scattering surface

Picture from 0907.5424

Cosmology and high energy physics

Higgs mechanism and symmetry breaking

)(V

cTT

CML and Peter Higgs

History of the Universe

Topological defects

Domain wall:

+

-

+

- +

-

+

Topological defects

Cosmic string

monopole

Problems of hot big bang

• Flatness problem

• Horizon problem

• Unwanted relics problem(monopole, gravitino, Polonyi field etc.)

• Galaxy formation problem(Primordial desity perturbation)

The flatness problem

Radiation domination:

42 aH R

2

42

11 a

aa

64

2

2

0

2

0

2

10|1|

|1|

0

P

P

TT

TT

T

T

a

aP

Entropy and lifetime of the universe

87312

3

9933 10)(10)(

110~ eV

eVTaS

sec10~ 433/2 Stc

The horizon problem

LS

HLS

HLSHT

TtR

a

atRt 00

0

0 )()()(

332 TaH M

2/3

0

0

1 )(

T

TtRH LSHLS

6

2/3

0

3

3

10)(

LSLS

LSH

T

T

H

T

The horizon problem

Monopole problem

cc

P

TT

M

H

110~

1 32

GeVTc1510~

nTH

n cM939

31010

)/1(

1

Similar to number density of baryon!

proton

GUT

GUTM mGeV

mm 1616 1010

Baryone occupy 4% of the energy density of our universe.

Hot big bang + grand unified theory (+ supersymmetry) =disaster !?

Appendix

If you like general relativity

Robertson-Walker metric

If you like general relativity

Christoffel symbol

Riemann tensor

Ricci tensor

Thermal history of the universe

The birth of cosmic inflation

CML and A. Starobinsky

CML and K. Sato

Alan Guth

“...the standard big bang theory says nothing about what banged, why it banged, or what happened before it banged. The inflationary universe is a theory of the “bang” of the big bang.”

Old inflation

)(V

What is inflation?

23~.~~ HconstV 1PM

Hdta

da

adt

da

a

aHconst

.

NHdt eea N: number of e-folds

Note that: 0a Anti-gravity!

This is called de Sitter phase

a

a

M

p

P

26

3

Remember?

New inflation (slow-roll inflation)

Coleman-Weinberg theory

Nowaday most inflation models are slow-roll inflation

Scalar field in cosmological background

)(2

1

)(2

1

2

2

Vp

V

)(3 pH

03 VH

For 222

1mV 022 mand 0H

Klein-Gordon Equation

The “friction term” is from the expading background

Scalar field

)(V

cTT

Particle physicists usually are interested in those points:

Symmetry restored

Symmetry broken Cosmologists are interested in this part

Slow-roll approximation

03 VH 03 VH

3

2 VH

Neglect this term

H

V

3

1PM

1

9

2

2

2

2

2

V

V

VV

V

VH

V

VAssume:

Slow-roll conditions

1

19

33

||

2

V

V

H

V

HH

V

Slow-roll parameters

1PM

V

V

V

V

2

2

1During inflation:

1||

1

The end of inflation

end is determined by 1~ or 1~

Number of e-folds

ddV

V

dV

Hd

HHdtN

begin

end

end

begin

2

1

3 2

NHdt eea

Scalar field is strange

Slow-rolling

Oscillating

Vacuum

matter

p 0

02

1

2

1 222 mp

Oscillating scalar field as matter

22

2

1)( mV

22

2

1ampm

2

2

1ampm

mn

The physical meaning of oscillating amplitude is particle number density!

Klein-Gordon equation is not a single-particle equation!

is proportional to the number of particles present. ---p.127 “Quantum Field Theory” Ryder

2||

Reheating

42222 TMMH PP

Pr MT 2.0

H~

Reheating happens when

Scale of everything

during inflation the horizon is almost a constant but the scale factor grows exponentially:

ahorizon

factor scale

after inflation: 42 TH

therefore 2/1~/1~horizon TH

but the scale factor goes like: Ta /1

afactor scale

horizon

Let’s say after inflation we have:

KGeVT 2815 10~10~

But now

KT 1~

Therefore grows by a factor of a 2810 after inflation T

a1

This means during inflation we need: 2810~Ne

60~N

How much time does inflation take?

Assuming inflation is around GUT scale:

sec10~10

~1

sec10~1

10~~

~

3410

44

102

422

P

P

P

P

GUT

GUTP

MH

M

MM

MH

MMH

sec10~60

~

60~~

32

Ht

tHN

Now we may understand better about

• Flatness problem

• Horizon problem

• Unwanted relics problem(monopole, gravitino, Polonyi field etc.)

• Galaxy formation problem(Primordial desity perturbation)

This is the topic for my next lecture.

Inflation solves those problems because our observable universe was not outside the horizon, it was inside the horizon!

Chia-Min Lin 林家民

Institute of Physics

Academia Sinica

Kobe University (from May 2012)

Lectures given in Chuo U/ Ochanomizu U April 17/2012

Lecture II

Inflation

Structure formation

Cosmic Microwave Background

510~

73.2~

T

T

KT

COBE

WMAP

Primordial Density Perturbation

The question is: can we explain the primordial density perturbation in the universe?

The answer of inflationary cosmology: it is from quantum fluctuation

From quantum to classical

CML and S. Hawking

2~

H

Hawking temperature

Scale of fluctuation

NV

VHH

t

t

Ht

t

H

~~2

~~~

~

~

2

1

2

Primordial density perturbation

Curvature perturbation

22

22

222

222

)21)((

)21)((

)(

)(

dxta

dxNta

dxeta

dxxadl

N

N

roughly speaking, this is called delta-N formalism. Interestingly, it is true even including higher order perturbations.

Spectrum

32

2

V

VV

H

V

V

V

VN

25

2

3

2

2 )105(12

1~

V

VP

We call it CMB normalization T

T~

Sachs-Wolfe effect

CMB Anisotropy (“see the sound”)

The spectral index

1 s

nkP knkP s

ns ln)1(lnln1

1ln

ln sn

kd

Pd

Nek

1~

1~

dV

VdNkd

ln

62231

lnln

ln

ln1

3

32

3

2

V

VV

V

V

V

V

V

V

d

dP

PV

V

d

Pd

V

V

dN

Pd

kd

Pdns

Tensor to scalar ratio

2

28

HPT

16P

Pr T

Primordial gravity waves

r determines the scale of inflation

GeV1001.0

~ 164/1

4/1

rV

Chaotic inflation

Why chaotic inflation?

• Both old and new inflation have problems.

• It is good if inflation can start at planck scale without a state of thermal equilibrium from the beginning.

CML and Linde

Linde PLB 129, 177 (1983)

Chaotic Inflation

2

2

22

2

1

mV

mV

mV

2

2

2

2

2

2

1

V

V

V

V

1~

1~2

end

end

The simplest chaotic inflation model can be realized by just a mass term.

Chaotic inflation

4~

442

222

enddd

V

VN

60N 15~

252

2

42

22

3

)105(619612

mm

V

VP

GeV10~10~ 135m

Chaotic inflation

97.0

30

11

81

621

2

sn 13.016 r

1001.4538

Chaotic Inflation

Linde PLB 129, 177 (1983)

Initial condition and total number of e-folds

Natural initial condition is inflation should begin when

5

5

422

10~

10~

~2

1

m

MmV P

This means “total” number of e-folds is

102 10~~NAmazingly large number!

m

1~

Classical motion and quantum fluctuation

H

V

3||

Classical motion:

H

1After a characteristic time scale:

1~~

1

3~

V

V

HH

Vclassical

mH

quantum ~2

~

m

1 Fluctuation > classical motion !

“quantum” fluctuation (which acutally is classical)

Eternal Inflation

Spacetime foam

What I am really interested in is knowing whether God could have created the world in a different way---A. Einstein

Inflation Models

New inflation

)(V

cTT 40

4

1VV

1210~

Hybrid Inflation

Andrei Linde Phys. Lett. B259, 38 (1991) Phys.Rev.D49(1994)748-754 astro-ph/9307002

Hybrid Inflation

V

φ

During inflation:

V0

Hilltop Inflation

φ

V

CML, David Lyth, Kaz Kohri

More hilltop inflation models Kazunori Kohri, Chia-Min Lin, David H. Lyth JCAP 0712 (2007) 004 0707.3826

Hilltop inflation Lotfi Boubekeur, David H. Lyth JCAP 0507 (2005) 010 hep-ph/0502047

Hilltop Inflation and Spectral index

1

1

21~

2

1~

s

s

s

n

n

V

V

n

N

Red spectrum

Blue spectrum

This also implies negligible tensor to scalar ratio

For small field model, epsilon is small.

Hilltop Inflation

Hill Top—home of Beatrix Potter

Hilltop Inflation

How do we test inflation?

Observables

• The spectral index

• Non-Gaussianity

• Gravity waves

• Cosmic strings

• etc.

The spectral index

Non-Gaussianity

2

2

5

3

)(2

1

gNLg f

NNN

26

5

N

NfNL

Non-Gaussianity

Non-Gaussianity

Current observation (WMAP) bound (95% C. L.):

7410 NLf

Roughly speaking this means:

85255 1010~)10(10010~

T

T

Precision cosmology!

100~NLf(by using )

Primordial Gravity Waves

http://www.kek.jp/intra-e/feature/2009/CMB.html

http://www.kek.jp/intra-e/feature/2009/CMB.htmlhttp://www.kek.jp/intra-e/feature/2009/CMB.htmlhttp://www.kek.jp/intra-e/feature/2009/CMB.html

Cosmic strings

PLANCK satellite

PLANCK May 14, 2009

01.0 sn

5 NLf

001.0r