I.

Introduction

① Cosmology iso

1-object :

,

Universe

- method"

Physical cosmology"

.

mm

⇒

"

physics"

" C special E

-79,

quantum

Adderelativity theory

general ,

- OE -

- ome 3%erelativity

-

-→ parade creatimfannihilation

Gru-

- 849k¥ • ⇐¥Y!I

'

%GM I s Htt > = EH >

gut quantum<

! quaffing ,g{LQG ? mechanics 0×010342

String theory?oEot3h42

xquantumheld theory in curved space-time

"

need to describe the beginning of the Universe"

Inflation

Strong nuclear fro

) short -

range* four fundamental Interactions

{Weak nuclear force

electro -

magnetic force

) long-rangegravitational force

( Standard model of particle physics )

fermions Bosons"

Leptons fuel (1) WEH

Jian"If"Early universe

.

of ,( It

, -

. .

i 8)

Quarks (4) (5) Eb) y

The most important

interaction in

Cosmology

?Gravity

! !

Why ? mass always accumulates !

② Units : Natural unit Cparticle physics ) & Astronomical

unit ( Oom version )

C re 3×105 km/s = 3×10"

ants dyn = g.am/sz

K I 10-34

Is = 10-27ergs

{erg

= dyn.cm\KB

II.4×10-23

Jfk = 1.4×10't erglk

② - I.

natural unit ⇒ C .

- K - KB -

- I : Note : let1,6×0

- ' 'J

. 6×1512erg

.

= If Not'

erg

⇒ erg-

- 6.3×10" eV

Is = 3 X to"

cm = 1027ergt

= 1.6×10's

eV- '

= 1.6×1024 GeV"

{ ik = , !!, ?:t " " er'

=

s-xw.am/g=erg-er8-= 6.3×10 "eV I 6.3×10

"

GeVam -152

=

to"

erg-ywsye.ge= to

"

erg

Useful Conversion : Kc = I = 3×15"

erg.

an= 18.9×10-6 eV - an

I 200 MeV .fm-

femtometer C- 15km )

2.7K = 2.4×10-4 eV I 12 an"

x ( 0.08am )- I

= ( o . 8mm )"

Peak of Planck fu. n 343T @ K = 0.23mm → D= 1.4mm

run me

2. 4×10-4 eV ~

3-8×10"

5'

r

380GHz3193T @ w =1140GHz

⇒ f =180GHz

mm

② - 2. Planck unit

Gn = 6.7×10-8 omfg

15=6/7×10-8x ( txio"

GeV- '

13¥10"

GeV )/( 1.6×1024 GeV-

Y-

~ 13¥ × 10-8+39-23- 48

gey- z

~ b- X 10-39 GeV -2=1 i GeV = 7×10-20

⇒ Is = 1.6×1024 Get'

= 2×1043 ⇒ tpenfx 15445

Im = b- Xlo"

GeV"

= 7×10"

⇒ Ipe n I X to- 33

m

Ig = 6.3×1023 GeV = 4×104 ⇒ wipe n 2×10-5of\

, , , g. , , ,,µµ , , , ,,

⇒

q ,, my ,

In natural unit,

Mpl u to " GeV

side note : Relativist 's unit sets 8IGn=I ,instead of Gn .

" Reduced Planck mass

"

Mpe = ~ Mg ~ Zxiocevm

② -3 . Astronomical Unit

Mo . = 2x 1033g ,Ro. = 8×10

"

an → So.

=

= # x to'

{Lo .

= 4×1033 erg Is n I glans

6×1027- Me,

= 6×1027g ,

Rot = 6×108 an → So,

=

¢¥p= ¥ X 103

~ 6 glans

I All = too light second = too x 3×10 "

an = tfxlo "

an

Iparsec

= I"

parallax = tf = Itu-r 216000 AU = 3×10' '

an

"

1/80/3600It

( o 8000000

216000 All = 216000 x too light second = alyr = 3.26 Lynmm

to the center of the Galaxy = 8- tkpc = 2. b- x 1022cm

to M 31 ~ 0.78 Mpc~ 2. 3X 1024am{ tothe

edgeof the Universe ~ 14 Gpc - 4×1028 an

③ Review of General theory of relativity

key quantity of dynamics : Tat ) from ICE ) & J CE )

* Newtonian theory ofgravity

eat . of motion : I .

- ima =MDI

dt2

{gravitational held

equation for the

grau. potential :

otcxie)

( Poisson 'sequation ) 024C I , -4 = 4Th Gns CI , -4 & I = - m 0/0

Equivalently ,

motion takes to extrcmize the action :

tfS [ ICED =/ de Ll the ,

nice , ie ]

ti

S C Tues t faith ] - Scotch ]

= f dtflckthtfxth ,nicest foie , se ) - LC Its

,niceties )

=

-2,4¥,

+

ftp..fi/toC8x4=-.fE.ise..+¥C¥÷s÷ ) . ¥E¥

. ) toes ,

= I at El :# . ¥ sci -

- o

⇒ 3¥.

-

- ¥E⇒)⇒ mic -

-

- mat

L = T - U = tzmxz - my

* Einstein 's theory ofgravity

igravity

= Curvature of spacetime .

① Spacetime : ( I -13 ) dimension

go =L

KM = ( t, X. y ,

Z ) I nd = K

,K2

=L ,x

' = Z

curved spacetime is described by the metric tensor :

gyu

⇒ defines the inner product of any giventwo Vectors

.

e.

g . spacetime intervalof

Einstein Convention

do = ( dxn , dxu ) = % gyu dxndxu =

gyu dxmdxu

nowIll

Minkowski Htt.

no gravity )spacetime

:

des;:[If:!!!;' oh -

-

f'

!;)Inner product of energy

-

momentum 4 - vector i P' = C E, p )

p'

= ( pm, Po ) =

yµ pmpu

= -

E- 2+11512

= -

m'

("

rest"

mass )

C at the local inertial observer )

⇒ gyu pnpu = - m- C general observer )

. NoteJust like the "

rest"

mass , space-timescalar muse be the same for all observers

.

mum

o

"

spacetime tells mass how to move" C

geodesic

equation)

=

i geodesic minimizes the spacetimeinterval along the path

⇒ Action S = - mfdt ( de = -

dsa = - gyudxmdxo )

! SR limit

=- mf def

'

- mfdtC I - Kia ) = fdtfmtkm.li)

S -

-

- mfde = - mf.dk/-8mdIITYI-

⇐-

Leg . ) 3¥ =¥(f¥ ) with he - mf-gmdffdf.IT =- md¥

¥ L'lxncxl ) = - Guiche ⇒ 2L }÷,

= -

magmarinse

12L3¥ = -

mega. Shiv-

magnify= - m

.

( gain

-1quasi )

⇒ f¥= -Fi( game + grain )

n-

-It ) ( gait + senior ]

=ILg. to + Had'¥ ]2

: ¥t⇒=¥¥d¥,

= - E . #I s . "e¥I¥¥+zaed¥.de#+g..dIIetsmdII.]

I 2

= ¥ .= -

EsmaKI¥l=- Yasmin :¥I¥

-

=t⇒'

⇒

( s . "e¥Id¥+%e¥¥¥¥ + g.it#.-ismd:I3--smald¥1 I

zg.io#.=gm.aIEoiI-s...e.dfIdEI-q.pdoEIE

Here, only

"

a"

is the real index,

and all others are dummy .

28*917.

= ( % . a

- gaap -

gear ) III 1¥

④ tzgdm ( where ganga = SI )~

↳ inverse metric

g.97¥ = = - Yasir ( g + spa . .-

spa . HII IET

= - t ;i¥i¥÷

⇒ DIET t TIP III III -

- o i

geodesic equation

-

↳ Christoffel symbol.

" "

. Christoffel Symbol : Tae

't= Tag leg

of; covariant derivative of the

-

-

#

coordinatebasis

.

velocity vector i V

'= WEE = doffs of

⇒ acceleration : a'

= doY = dat ee + uao¥eE

UP Tag eg"

-

= Has EE -- u

'

Effie:

= ( II t Tofu ez

: geodesic equation ⇐ a o ( Parallel transportation)

mm

o Examples on surface

C X )FfO Newtonian vs

.

Relativistic Motion due to Gravity

"÷÷

Central - force,

two - body problem"

Spacetime geodesic"

.

O"

mass tellsspacetime

how to Curve"

.

gyu ⇒ e. am .

What determines gyu?

distribution of mass

Einstein Equation"

Einstein tensor

" "

energy- momentum tensor

"

Gyu =

STIGTru

i ) Einstein tensor

Gm = Ru - 42 gyu R

-

H IRico

,Scalar R -

-

gyu RN =gmRµ=RMµRicci tensor

Ryu = Rdr an

run

I

Riemann tensor ( Tra - Qq ) Ud = Rdppu UP

Ramu = IIe - III. t to .

- TETE

Ii ) Energy - momentum tensor

Tmo =µ

- component of momentum flow through he = Gust

. hypersurface .

~

Jpg III fcp , 8D C pkm )

Too n

energy density

Tot ~

energyflow i To - momentum density{

Ty - Stress tensor ( ~ minus )

( perfect fluid >

Tm= ( Stp ) Unh t Pgm ( un

-

-

gu w )

Note : Tru = Tvr C symmetric )

Continuity equation: Tn Tmo = o

⇐

8¥ + y ,Levi ) =o Tn SE

Ito ,csuiui , = o

I ( Hatspacetime )

⇐Or -11=0

( RTI = 2T 's t FukTT - EETI )

⇒ anti = dit + Tfa Tf -TITI= o

a-

Again ,

"

connection"

!

* Action for the Einsteinequation

SCgyu

] = SEH t Sm

= ¥qJd4x Fg R + fd4x Fg Lm

I Ts matter Lagrangian density

Ricci scalar

Fgu=¥a #

Egg::*= fttqf -

'

kissing Rap tf Ryu t Fg go SgRg÷ )me

-

→ surface term.

= I÷g[ Rm - Ya Rgm ]quo

→ o.

to = - EFF.

⇒ Rm - Yim R = 8 TG Tru

CH uses Jacobi formula : I eat = e*A forany

nxn matrix A

⇒ IAI = etrh A

⇒ SIGI = lgltrlg 's g )

⇒ ffg = Yzfggmsgru

= - Ifs gaggert Kit

"-

-4

Example of matter action ( scalar field : 0 thx ) )

Sm C ¢ ] = folk Ff L ( 0, 24 ; x )

= folk Fg ( -

'

kg"

On 4 a ¢ - V coli )

⇒ Ssn = forex Fg Sgm ( %gr f

Lay04 - Veon ) -

'

k 9404 ]-

=- Yasu

= forex Fg 8g " [ Ya 8in ('

k Tay pay tuco , ) - Ya 74 Or of ]mm

=- Ya Tm