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
