CMB Data Analysis
Maximum Likelihood Analysis
Given an N_{p}pixel map of the sky temperature \Delta_{p}
and a measure of the pixelpixel correlations in the noise N_{pq}
we want to find the most likely underlying cosmological model and parameters
that would produce the signal observed in the map. For a given model and
set of parameters we calculate the associated pixelpixel correlations
in the CMB signal S_{pq}. Assuming that the signal and the noise
are uncorrelated, the pixelpixel correlations in the map  M_{pq}
 are simply the sum of those in the signal and the noise. Assuming that
the CMB fluctuations are Gaussian the likelihood of the particular model
and parameters is
Our goal is to find the bestestimated set of cosmological
parameters. That is those cosmological parameters which maximizes the likelihood
function for that class of cosmological models. A short paper outlining
the implementation of algorithms for locating the peak of the likelihood
function for a general sky temperature map can be found here
.
Timing
For a map with N_{p} pixels, and a target cosmological
model with N_{b} parameters to be determined, the quadratic
estimator algorithm requires

(2 x N_{b} + 2) x N_{p}^{2}
x 4 bytes of disc storage.

2 x N_{p}^{2}
x 8 bytes of RAM.

(2 x N_{b } + ^{2}/_{3}
) x N_{p}^{3}
floating point operations.
assuming that

all the necessary matrices are simultaneously stored on disc
in single (4byte) precision.

matrices are loaded into memory no more than two at a time
in double (8byte) precision.
If the cosmological model has 10 parameters, then the computational
requirements for the current MAXIMA and BOOMERanG balloon experiments for
a single iteration of the algorithm on (i) a 600 MHz workstation and (ii)
the NERSC Cray T3E900 (using the specified number of processors and running
at 2/3 peak) are
Dataset 
Map Size 
Disk 
RAM 
Flops 
Serial CPU Time 
T3E Time 
BOOMERanG N.America 
26,000 
55 Gb 
11 Gb 
3.6 x 10^{14} 
7 days 
2 hours
(x 64) 
MAXIMA1 
32,000 
85 Gb 
17 Gb 
0.6 x 10^{15} 
12 days 
5.6 hours
(x 64) 
MAXIMA2 
80,000_{ }^{*} 
0.5 Tb 
100 Gb 
1 x 10^{16} 
7 months 
9 hours
(x 512) 
BOOMERanG Antarctica 
450,000_{ }^{*} 
15 Tb 
3 Tb 
2 x 10^{18} 
100 years 
70 days
(x 512) 
(* projected)
If we project further and consider MAP and Planck data
sets, we find even larger numbers.
Assume that the cosmological model has 20 parameters
(some cosmological & some astrophysical/experiments) parameters,
then the computational requirements for the current MAP and PLANCK missions
for a single iteration of the algorithm on (i) a 600 MHz workstation and
(ii) the NERSC Cray T3E900 (using the specified number of processors and
running at 2/3 peak) are
Dataset 
Map Size 
Disk 
RAM 
Flops 
Serial CPU Time 
T3E Time 
MAP
(single frequency) 
10^{6} 
160 Tb 
16 Tb 
4 x 10^{19} 
2x10^{3} years 
4 years
(x 512) 
MAP 
10^{5}10^{6} 
1.610^{3} Tb 
0.116 Tb 
4 x 10^{22} 
2x 10^{1}10^{4} years 
0.440 years
(x 512) 
PLANCK
(LFI) 
10^{6}10^{7} 
10^{2}10^{3} Tb 
161600 Tb 
4 x 10^{24} 
2x10^{6 }years 
200 years
(x 1024) 
PLANCK
(HFI) 
10^{7} 
10^{4} Tb 
1600 Tb 
4 x 10^{24} 
2x10^{6} years 
200 years
(x 1024) 
(* projected)
Note that a single map containing N_{p
}the correlation matrix is N_{p }by N_{p }and requires
storage size 4 N_{p}^{2} in single
precision
(not exploiting symmetry or same in double precision
using the symmetry). Thus the correlation matrix for a single map with
N_{p }= 10^{6}10^{7}, will be 4400 Terabytes
in size.
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