A tutorial I gave at IUCAA Pune

1. Cosmological Microwave Background Radiation
Likelihood Analysis - Part 2
Jayanti Prasad
Inter-University Centre for Astronomy & Astrophysics (IUCAA)
Pune, India (411007)
September 13, 2011

2. Plan of the talk
I Cut-sky likelihood function
I Likelihood of the Polarization field
I Karhunen-Loveve Techniques

3. Likelihood of Correlated Gaussian fields
I CMBR temperature anisotropy field is represented as
T(n̂) =
∞
X
l=0
l
X
m=−l
almYlm(n̂) (1)
where
alm =
Z
dn̂T(n̂)Y ∗
lm(n̂) and < alma∗
l0m0 >= δll0 δmm0 Cl (2)
where the distribution of alm is Guassian with mean zero and
variance given by the angular power spectrum Cl , which
follows χ2 distribution.
f (x; n) =
(
1
2n/2Γ(n/2)
x(n/2)−1
exp[−x/2] for x > 0;
0, otherwise
(3)
note that
χ
2
=
n
X
i=1
"
yi − y(xi )
σi
#2
(4)
which is the “square sum of Gaussian errors”.

4. http://www.iucaa.ernet.in/ jayanti/gsl/chisq.c

5. I CMBR temperature and polarization fields, which individually
follow Gaussian distribution, are correlated.
alm = (aT
lm, aE
lm, aB
lm) (5)
I In this case we have six power spectra, two of these BB and
BE are zero for a “parity-invariant ensemble” so we are left
with CTT
l , CEE
l , CBB
l , CTE
l .
I The covariance matrix Cl and estimator Ĉl are given as:
Cl ≡
D
alm a†
lm
E
and Ĉl ≡
1
2l + 1
X
m
alm a†
lm (6)
I The probability P({alm| Cl )of a set alm at a given l is given
by:
−2ln [P({alm| Cl )] =
m=l
X
m=−l
h
almC−1
l a†
lm + ln|2π Cl |
i
= (2l + 1)
Tr[ Ĉl C−1
l ] + ln| Cl |
+ const (7)

6. I Integrating out all the alm with the same Ĉl (or normalizing
with respect to Ĉl )gives a Wishart distribution for Ĉl :
P(Ĉl | Cl ) ∝
| Ĉl |(2l−n)/2
| Cl |(2l+n)/2
exp
h
−(2l + 1)Tr[ Ĉl C−1
l ]/2
i
∝ L( Cl |Ĉl )
(8)
I It can be shown that the likelihood has a maximum Cl = Ĉl ,
so Ĉl is the maximum likelihood estimator.
I For only temperature (n = 1) the Wishart distribution reduces
to χ2 distribution with (2l + 1) degrees of freedom:
−2lnP(Ĉl | Cl ) = (2l +1)
Ĉl
Cl
+ ln|Cl | −
2l − 1
2l + 1
ln(Ĉl )
#
+const (9)

7. Karhunen-Loveve Techniques
I In any experiment there are always some modes which are
heavily contaminated by noise and somehow if we can identify
and discard those, we can speed up computation. If we can
keep only 10% of the modes, computational can speed up by
a factor of 1000 !
I Karhunen-Loveve (KL) technique provides a prescription for
that.
[Tegmark et al. (1997); Dodelson (2003)]

8. How Karhunen-Loveve Technique ?
We carry out a coordinate transformation on the data vector ∆
∆0
i ≡ Rij ∆j , (10)
and the new covariance matrix
C0
ij = h(R∆)i (R∆)j i or C0
= RCRT
(11)
where the old covariance matrix is given by
C = h∆i ∆j i ≡ CS,ij + CN,ij (12)
Note that CN and CS are real symmetric matrices so can be easily
diagonalized.

9. How Karhunen-Loveve Technique works ?
This technique needs the following three rotations:
I R1 : To diagonalized the noise covariance matrix CN.
I R2 : To make the diagonalized CN, i.e., C0
N unity.
I R3 : To diagonalized the new signal covariance matrix C0
S .
Example:
CN =
σ2
n 0
0 σ2
n
,
CS = σ
2
s
1
1
R1 = I, R2 =
1
σ2
n
I
and
R3 =
1
√
2
1 1
−1 1
which gives
∆
0
1 =
1
√
2σN
(∆1 + ∆2), ∆
0
2 =
1
√
2σN
(∆1 − ∆2)
and
C
0
S =
σ2
s
σ2
n
1 + 0
0 1 −

10. Summary Discussion
I I have discussed the difficulties and approximations used to
compute the CMBR likelihood function.
I My aim was to understand why and how the likelihood
function is computed differently for different values of l and
different type of power spectra.
I I have also discussed various estimators related to angular
power spectrum and the likelihood function.
I I was not able to discuss the actual computation of the
various components of the likelihood function in the WMAP
likelihood code.
I It has been suggested (by Dodelson) that the Nelder Mead
method or downhill simplex method (commonly known
amoeba) can be be used for finding the best fit parameters.
I Another method called NEWUOA which is used for
unconstrained optimization without derivatives, is also
suggested (by Antony Lewis ) to find the best fit parameters,
optimization of the likelihood function.

11. Dodelson, S. 2003, Modern cosmology (San Diego, U.S.A.: Academic
Press)
Tegmark, M., Taylor, A. N., Heavens, A. F. 1997, Astrophys. J. , 480,
22