CMB-slow: How to estimate cosmological

parameters by hand

V. MukhanovSektion Physik, LMU, München

ParametersMetric

After inflation

where13 2

3 2

s

T

nk

nk

k Ak

k h Bk

0.92 0.97sn to

2 2 2(1 2 ) ( )((1 2 ) )TT i kik ikds dt a t h dx dx

gravitational potential gravity waves

Matter

23

8cr

H

G

cr

1Prediction of inflation!

,tot b CDM Q 0 0 0 0 0 0,tot b CDM Q

Main unknown parameters determining the spectrum

075

751)

Hh

kms Mpc

0 0 02) m b CDM 03) b

0,4) Q ampli5) tude A spectral ind6) ex sn

0 0 0,theI nf 1 1tot Q m

Extra parameters influencing the spectrum

number of light neutrinos, gravity waves, reionization

4 2752.3·10eq mz h 0z

3m z

4z

,Q

1200 900recz to

Universe transparent

1z 1z 1/ 20.87 tot

2310recct cm

1050recz

Before recombination

(Im)perfect fluid: coupled baryon-radiation p

( ) viscosity ter

-

ms

lasma

bT p u u p

The fractional density perturbations in radiation / satisfy the equa :tion

/ // // /

2 2 2

3 4 4 12

3S S Sc a c c a

at

viscosity

speed of sound

CDM particles which interact with plasma only gravitat- ionally

Gravitational potential is generated mainly by radiation

before equality and by cold matter after that

After recombinationFree photons cold matter (CDM and ba )- ryons

Gravitational potential

3 3( , , )ijdN f x p d xd p

Conformal t =imedt

a

( )( )0

ij

ij

dpDf f dx f f

d d x d p

0 0 0( , )ix x

i ipn

p

0

2

exp 1( , )i

f

T T x n

2ii

Tn

x T

If dust dominates ( th ) sten =con

0

Tconst

T

along photon's geodesic

0 0( ) ( )i i ix x n

0 0 0 0( , , ) ( , ( ), ) ( , ( )) ( , )j jr

i jr r r

i jT Tx x xn n x

T T

0 0( )jr

ix n in inrec

( , ( ), ) ?jr r

inT

xT

Matching conditions for T

3( )/

2 3/ 2

3( , ( ), ) ( )

4 4 (2 )

jj rik xj mk

r r m kiT i d k

x k eT k

n n

0

/ 3( ))

0 2 3/ 20

3( , , )

4 4 (2 )r

r

ik kk

T d ke

T k

0 nk(x

0 nx

Correlation function

2

1( ) ( ) ( ) (2 1) (cos )

4 lll

T TC l

TC P

T

1 2n n

cos 1 2n •n

where2

/20

00

3 ( ) ( )2( )

4 4 ( )r

k k r ll k l

dj kC j k k dk

k d k

nR oec n-ombi instnation i antas neous

Instanteneous recombination

22( ) 2rke k dk

22 10

characterizes fluctuations on angular sca( 1 les) ll l Cl

Longwave inhomoge (ne ies 1it )rk

1 100l

13 2 / /4 8, ( ) , ( ) 0

3 3S

r

n mrk r r

r m

k A k

0 0

1( , , ) ( , ( ))

4 3 r r

T

T

0 0 nxnx/

for9

( 1) 30 1if100l S

Al l C l n

( 1) ll l C

l30

Shortwave inhomoge (ne ies 1it )rk

1 100l

2/

2 200

0

3 ( ) ( )2( )

4 4 ( )r

r

kk k r ll k l

dj kC j k e k dk

k d k

2

10

2 2 2 220 2( )/

32 200 0

4 9 ( )1 1

16 ( )

( )fo

)r

(rk

l

l

k k lC e dk l

kk k l

4 /

2/ 02

0

11 cos

4 3

r

D

r

k kkk p o S S k

S

T T c k c d ec

2// 3/ 2 0

0

4 sinr

D

r

k kk o S S kT kc k c d e

2 1

3 1 3 / 4S

b

c

Initial spectrum

2 275 75

Silk dissipation scale,

weakly

depends on

and m bh h

275b bh

275bh

b m

1/ 2275eq mh

( )p eqT k

ln eqk

9

10

(1)O ln eqk

( )o eqT k

(1)O

0.4

1.97

Spectrum

30

( 1)

( 1)l

l l

l l CN O

l l C

2

20.3

227 /2

15 .75

4

ln 0.22 2.6200

180( ) ...1 0.65 /

SS l l

S

mb

l

h

ll

N el l

h

275 if increasesbh

275 if incre s s a emh

0.5

1

1.5

2

2.5

3

3.5

200 400 600 800 1000 1200 1400l

275

275

0.04

0.3

b

m

h

h

2/1 2cos cos

42

4Sl lO A l A l e

l

2 27575 ln ,...b mh f h

275' ln ,...mhf

3.1

27

0.16

0

7

50

510.014

1 2.2

r

m

S

mc

h

hd

-1

-0.5

0

0.5

1

1.5

2

200 400 600 800 1000 1200 1400l-1

0

1

2

200 400 600 800 1000 1200 1400l

1

2

3

4

5

6

200 400 600 800 1000 1200 1400l

Determining the cosmological parameters

75 ,, , , , , m b tot m Q sh A n tot

1 =0

tot

The location of the peaks ( 1)tot

275

3.1 0.16

75

1 2.2 b

n

m

hl n

h

21 7540 when increases twicebl h

21 7540 when increases twiceml h

1

2

3

4

5

6

200 400 600 800 1000 1200 1400l

275

0.04

0.3

1

b

m

h

1

2

3

4

5

6

200 400 600 800 1000 1200 1400l

The heights of the peaks

275 if increasesbh

275 if incre s s a emh

275First peak: there is degeneracy, but anyway because

otherwise there is no way to compensate the amplitude

0

in

.

ease

2

crbh

275 if incre s s a emh

275 if increasesbh

The second peak exists 275 0.07bh

2 275 75if0.3 0.03m bh h

, becaus0! e !! 1toQ t

2 275 75for give? n nd aS b mn h h

275 ?h

275peaks location depends on mh

0.163.175peaks heights depends on mh

in loca1% tion 75in7% h