It.

FRW Universe

① Our Universe is.

- big ( 7 104pct

- old ( 7 1013yrs )

spatiallyf

- Homogeneous I galaxysurveys

i SDSS l > '

7014pct )

- Isotropic C CMB )

-

Expanding & theexpansion is accelerating I of = - a so )

Ho = look km/s I Mpc ,h ~ 0.7

- Almost flat ; loud cool

- filled with thermal Cosmic Microwave Background radiation

Tomb = 2.72548 it 0.00057 K , Imax n 1mm

No~ 411 fans

- far less baryons

y = they ~ a few x to"

- Most bayous are in H & He

X ~ 75% ,

Y ~ at % ,

Dht - 2. t x lot

- Most matters are dark - Son = 6lb

-

energy Contents is dominated by darkenergy

? Dn - o . 7

- large - scale Structure of the Universe.

② FRW ( Friedmann - Robertson - Walker ) universe

( spatially)

FRW world model :

Homogeneous& Isotropic , Expanding Universe

⑨ FRW metric

Isotropy ⇒ got= o

{homogeneity

⇒ go.it,It = go.ee )

⇒ do = -

gooey'

de' '

t gig

tix) dxidx

( dt-go.ca'

de '' )

=

- de ' t gig Ct,

I ) dxidx

Assumingthat spatial homogeneity &

isotropyat all time

! ⇒

::::::::::::" I

- de '

t atetgycxsdxidx ' ath = scale factor

Note : except for the three maximally symmetric cases,

Minkowski ,ds

,Ads

,

We can un-

ambiguously define

the time coordinate ( constant - time hypersurface )

by usingConstant -

density hypersurface s.

① spatial metric I CII

. lower dimensional example : metric on 5.

2

dssp= dx '

t dy-

t dz ' ( Normal flat 3D space)

④ Constraint not y 't E = R2

Using3D Coordinate Hey ) to describe 5

Icty 't E = R

-

⇒ 2xdxtzydy-zzdz-o.dz = ¥( xdxtydy )

:. dzz=CXdXtYdy#

122×2 - y-

define r'

- Hey-

→ 2rdr= ZxdxtzydyCdr )

'

⇒ dE= -

R2 -

ri.

dsio-dxi-dyi-dzz-drztrzdo.tl?YI---pRIdrY-trzdO

'

r → Rr

⇒def= R'

( IF trader ]

finally ,

define xcri-f.ro#Ff.sYrYgsseedt---s-in-ir=sr=saxdr--GsXdX

: .

dsii,

= R2 [ DX 't sink do' ]

⇒ Balloon example .

• maximally symmetric (Homogeneous

& Isotropic ) space

IR'

: ds¥=o = dr -

t r' [ do 't 5h20 dy

' ]

$3 i def ,

=

dxi-djtdttdwxzty-tz2-wz.AZ

⇒ rztwz -

- a-

i rdr = - wdw

dw = - I ( xdxtydytzdz ) =- I rdr

⇒ dsk , o

=

dxkdyztdzi-az.tl/dxeydytzdz5--a-fdxtdytdz't "¥fI!f¥¥ ]

= a- [ ftp.trz/dO45ln2Odg4 ]

IH'

; dsi ,

= dxidy'

+ de - dw '

XZ tytz'

- wz = - a- ⇒ r

' - W' = - a- ; rdr -

- wdw

dw = Iwcxdxtydytzdz ) = rdtf

⇒ disco= dxkdyztdz.az#z(xdxeydytzdz5=A2fdx4dyr+dzz

-

cxdxtydytzd.DZ/tCxiya+zy

]

= a- [

¥tr ( do 't 5h20 doe ) ]

Note : FRW metric global topology .But

, our Universe does NOT

have to be homogeneous & Isotropic bindthe horizon.

0 FRW metric ,final form

do = - dt2

takes f fi,

+ K ) dxidx '

= - dt2 t act ) ( IIIT tr

- ( do 't 5h20 def ) )

K=

i close C sis ,

i flat UR'

)

i

open CHP )

⇒ Christoffel symbol: Ted,

= kg "( 8ps,

,t Joe

, p

- Fpr . , )

am= f !aiescs.tk

*

D=f !axis

,)

- I

⇒ on -

-

f'

. ya .

.is , a. inD= - f .taxis' )

& I-

ooo = Too;

= Tio = o

T ; = da I

t÷÷:÷. . . m

.

③ Geodesic equation

III. tried

.EE#--opM=mdxI--( E

, F )ok

⇒ ¥ tilt TILE ⇒

f- o i mode

pot

TIppdpf-oi.mg#po-ciag.gpipJ--o-c*t{gµpMpu

=- Cpg

'

ta.

of , pips = -

m- let peat, Pips

⇒ zpodpo = 2 pdp po=mdI

de

m¥po=m¥Ed¥=m¥ It Emp'a¥i¥

=p DIdt

CH ⇒ potty + Iap . =o ⇒ j+E=o ⇒ Pa Ya

"

cosmological redshift !"

Note : Apply BEI massive & massless particles .

photon : p-

- hh - Ya ⇒ Aaa

⇒ dope = Age = It z

"

redshift"

"

I* . . yaE = hw

ii ) Massive parades

p.

-

- a- of, pips = a I , (mI¥)(m¥÷) a Yaa

.

.

.

dxi

DIA Yaz

⇒ peculiar velocity:

pam, poem

vi.=ad¥=ad¥d¥ -

- CE) a :#fa

Ya

In an expanding universe , peculiar velocity drops as Maces.

Why peculiar velocity= velocity measured by armory

observers.

④ Gmatng

observers are recedingfrom each other

.

÷÷÷:÷: .ua .UGG ft

N theft ) = UH ) - HU ft

⇒ U Cttfe ) - UK )

get→ ¥e -

- - Hv =- III

⇒ vs Ya④

④ Friedmann equation

Einstein Eg . Rm - t Rgm = STIG Tyr

LHS : Rm ← TIP ← gyu

IR " ' ?

Again ,in FRW universe

, Tm must Satisfy the specific form.

① To -1=0 ( isotropic)

② Tig A Fg ( 3×3 tensor Consistent with Hom . & Iso.

)

⇒ Ten= [ Suit Peel ] Up Uo t Put gyu

Note : In FRW Universe, any global vector Un Satisfy

UT = o.

T1 = find

Tau

= ( Stp ) Uduu t Pfk

Ud = ( I,

o,

o,

o ) ; U u= guy Ut = f- I

,o

,o ,

o )

Finest !: If :p; )=

is:p)

from the Hw.# I 342 t 3K/az = 8h45

⇒

cia =- 4 ( St 3P)

j t 3h Cf t p ) = o

i ) H = ate IET -

- I"

Hubble Expansion rate

"

Hee -61 = He = look km/s hype"

Hubbleparameter

"

i closed-

in K -

-

f."

o: flat

- I i

open

Sometimes,

K = Yaz ⇒ 3h 't 3k = 8h45

Recap that act ) is the radius of curvature in k=±l case ! !

on, KCA = Klatt,

is the curvature of the Universe .

iii ) Not all three equations are independent . C Hw.

)

& we only have two Tndep . equations .

BTW ,three unknowns : aces , Scot , poet ? ?

Heuristic,

Newtonian derivation of the Friedmanneq .

[test mass

feel → But =HAIR

yipe EH = - G (¥HRCtP)/E,

Center C ! ) =- 4¥ See , Ray - CH

? ?

[ of .

Ea -

-- 45Gt Costs P ) ]

IntegratingCH R'

'

k=izddI=k¥Cizy= - 4t SRR

=- 4tzG@R31.Piz-ddzf4JG_pR3.fz)

: . Kyra = 8t St Che

I C- - I ! !

[ of Eat -

- 5¥ - Kla . ]

1st law of thermodynamics

dQ=dUtpdV =o ( adiabatic expansion )

U= internalenergy a SRs

{ ⇒ d¥= JR'

t 35 R' R2

Pdhyde = PILE CRY =3 PER-

⇒ It 3ha xp )-

- o! ! ! ! ! ? ? ? ? ?