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Comparison of Microwave Background
Radiation models and experiments
By G r a c a R o c h a
e­mail: graca@mrao.cam.ac.uk
Mullard Radio Astronomy Observatory, Cavendish Laboratory, Madingley Road, Cambridge
CB3 0HE, UNITED KINGDOM
We present methods of comparing theoretical predictions for the anisotropy of the Cosmic Mi­
crowave Background Radiation (CMBR), with the results of various experiments. The compari­
sion between experimental results and theory is organized in a plot with temperature fluctuation,
\DeltaT =T , versus full width at half maximum of the beam used in the observations. We describe
the processes of computing the cosmic variance and sample variance and stress their importance
in order to compare properly theory with observations. A brief account of the Monte Carlo
simulations in progress is given.
1. Introduction
Most models of structure formation in the universe predict anisotropies in the CMBR.
Since the COBE detection of large angular scale anisotropies, a great deal of attention
has been devoted to constraining these models. Meanwhile other detections have been
claimed from smaller angular scales. Thus the next step would be comparing experi­
mental results with theoretical predictions and constraining the model parameters (e.g.
the primordial power spectrum of fluctuations, the matter density in our universe, the
baryonic content, the nature of non­luminous matter, the biasing, the process of struc­
ture formation, the statistics of the temperature fluctuations etc.). In this comparision
three different aspects must be considered: (1) what is the dominant anisotropy effect,
(2) theoretical errors, (3) observational strategy. With respect to (1), for the experi­
ments probing large angular scales (larger than the horizon size at recombination) the
anisotropy is mainly due to the so called Sachs--Wolfe effect (due to potential fluctu­
ations in the last scattering surface) whereas on smaller angular scales these are due to
the Doppler effect resulting from peculiar velocities of the electrons scattering off the
CMBR photons during recombination. For (2), the theory can only predict the statistics
of the temperature fluctuations and ensemble average values but not its value in our
universe. Therefore one must take into account not only the theoretically expected value
of the temperature fluctuation but also the so­called cosmic variance, i.e. the theoret­
ical error due to observing only one universe. In smaller scale experiments it is usually
assumed that the cosmic variance is small (i.e. only higher multipoles contribute in the
temperature fluctuations expansion) (Scaramela & Vittorio (1993)) but this only applies
to measurements over the whole sky, otherwise the result is affected by the so called
sample variance, that is the error due to the fact that one is sampling part of the sky
not using all available information to compute, e.g. the \DeltaT rms . For (3), the different
observational strategies may consist of single difference, double differences (beam switch­
ing) or interferometry. To proceed with the comparision one may use two methods: (1)
Monte Carlo simulations (simulation of the observed sky for a given experiment, compu­
tation of \DeltaT rms , cosmic variance, sample variance and the theoretically expected value
of \DeltaT rms ) or (2) an analytical approach (using semi­analytical expression to compute the
cosmic variance for each angular scale and each experiment (Cay'on et al. (1991)). The
1

2 Graca Rocha: Microwave Background Radiation
sample variance is then computed using an approximate relation with cosmic variance,
or directly (Scott et al. (1994)). The models considered are those of: Bond & Efstathiou
(1987) (hereafter BE), Holtzmann (1989), Peebles (1987) and topological defect models
as opposed to the gravitational instabilities.
We have consider the following experiments (with the corresponding FWHM): OVRO,
OVRO/RING (1 0 :8); MSAM ( 28 0 ); MAX3¯Peg, MAX3GUM, MAX4GUM, MAX4ID,
MAX4SH (0:5 ffi \Gamma 0:75 ffi ); SP(ACME/HEMT); MIT (3 ffi :8); Tenerife (5 ffi :1); COBE (10 ffi ).
In this paper we give the predicted expected values of (\DeltaT =T ) rms for BE models in a plot
comparing the predictions of one of the BE models normalised to the COBE detection
and the experimental results or upper limits for some experiments. This gives an overall
idea of what is happening with (\DeltaT =T ) rms versus angular scale (more precisely FWHM).
Work has been done with the other models and it will be published shortly. With respect
to theoretical uncertainty, we give a brief account of the analytical techniques already
existent (Cay'on et al. (1991), Scott et al. (1994)) and the Monte Carlo simulations in
progress (of which the results will be published in the near future).
2. Temperature flucuations
In order to compute the expected temperature fluctuation for a given experiment and
model (i.e. (\DeltaT =T ) rms ) one uses the two­point angular correlation function C(`) which
is defined as being the expected average of the product of the temperature fluctuations
in two directions in the sky separated by an angle `: C(`) = h \DeltaT
T ( ~
n 1 ) \DeltaT
T ( ~
n 2 )i, where
~
n 1 : ~
n 2 = cos `. The expression of (\DeltaT =T ) rms depends on the experimental configuration
(e.g. beam size, switching pattern etc.). With a two­beam experiment one measures the
temperature difference between beams separated by an angle ` on the sky usually called
single difference: \DeltaT =T = (T 1 \Gamma T 2 )=T and its expected variance is written in terms
of the temperature autocorrelation function: ( \DeltaT
T ) 2
rms = h
\Gamma T1 \GammaT 2
T
\Delta 2
i = 2(C(0) \Gamma C(`));
while a three­beam experiment measures the difference between a field point, T f , and
the mean value of the temperatures in two directions which are separated from the field
point by an angle `, usually called double difference: \DeltaT =T = T f \Gamma 1=2(T 1 + T 2 ) and its
expected variance is given by:
D \Gamma (T f \Gamma 1
2 (T 1 + T 2 ))=T
\Delta 2
E
= 3
2 C(0) \Gamma 2C(`) + 1
2 C(2`).
All these expressions are true for an ideal experiment with infinitely narrow beam, for
a real one it is necessary to consider the beam smearing due to the finite resolution of
the antenna. This is usually taken into account by convolving the radiation intensity
with a beam approximated by a Gaussian G(`): G(`) = exp \Gamma ` 2
2oe 2 =(2úoe 2 ) where oe is the
dispersion of the Gaussian. The correlation function of the convolved radiation field is
given by BE:
C(`; oe) '
1
2oe 2
Z 1
0
C(` 0 )e \Gamma ` 2 +` 0 2
4oe 2 I 0
` `` 0
2oe 2
'
` 0 d` 0 (2.1)
where I 0 is the modified Bessel function. So for a single difference one has:
` \DeltaT
T
' 2
rms
= 2(C(0; oe) \Gamma C(`; oe)) (2.2)
and for a double difference: ` \DeltaT
T
' 2
rms
= 3
2 C(0; oe) \Gamma 2C(`; oe) + 1
2 C(2`; oe) (2.3)
where ` is the beamthrow. One is able to compute ( \DeltaT
T ) 2
rms using equations (2.1) and (2.2)
or (2.3) according to the experimental configuration. If, for example, for a given model

Graca Rocha: Microwave Background Radiation 3
an analytical expression (e.g. fitting parameter expression) for the angular correlation
function is given or e.g. a parameterized expression for the transfer function of \DeltaT =T ,
T rad (k), is given to be used in hC(`)i = 1
2ú 2
R 1
0 dk 1
k 2 P (k)T 2
rad (k)[j 0 (kr) \Gamma j 2
0 (kR h ) \Gamma
3j 2
1 (kR h ) cos `] where j 0 and j 1 are spherical Bessel functions, r = 2R h sin(`=2), R h = 2c
is the horizon distance. An alternative method consists of using the angular power
spectrum, C l , which are the coefficients of the Legendre polynomials expansion of C(`).
Commonly one expresses the temperature fluctuations on the celestial sphere using an
expansion in spherical harmonics (completeness property): (\DeltaT =T ) =
P
l;m a m
l Y m
l (ff; OE)
Inflationary models predict Gaussian initial perturbations, in this case the coefficients
a m
l are stochastic variables with random phase, zero mean and variance given by: C l =
hja m
l j 2 i. The expectation value of the temperature correlation function (the ensemble
average) is given by:
C(`) = 1
4ú
X
l
(2l + 1)C l P l (cos `): (2.4)
For a more realistic scenario one must take into account the smearing due to the finite
resolution of the antenna: \DeltaT
T (ff; OE; oe B ) =
P
l;m a m
l Y m
l (ff; OE)W l (oe B ) and
C(`; oe B ) = 1
4ú
X
l
(2l + 1)C l P l (cos `)W 2
l (oe B ) (2.5)
where W l (oe B ) = e \Gamma(2l+1) 2 oe 2
B =8 (Scaramela & Vittorio (1993)). Once given the C l 's one
can compute the correlation using (2.4) or (2.5) and (\DeltaT =T ) according to (2.2) or (2.3) for
the switching experiments. These C l 's may be obtained directly using the solutions to the
equations describing temperature fluctuations evolution by C l = Vx
8ú
R 1
0 k 2 dkj\Delta T l (k; Ü 0 )j 2
where \Delta T l (k; Ü 0 ) are the coefficients of the Legendre polynomials expansion of, \Delta T (k; Ü 0 ),
the radiation intensity fluctuations and Ü is the conformal time (Efstathiou (1987)). In
particular in the case of large angular scales
for\Omega = 1 and an initial power­law form of
the density fluctuations power spectrum P(k)/ k n an expression for the C l (n) may be
used (Efstathiou (1987), Scaramela & Vittorio (1988)):
C l = C 2
\Gamma(l + (n \Gamma 1)=2)\Gamma((9 \Gamma n)=2)
\Gamma(l + (5 \Gamma n)=2)\Gamma((3 + n)=2) (2.6)
with n ! 3, l – 2, l ! 40; this is important mainly because one is able to predict
(\DeltaT =T ) rms and its error bands as function of n providing a direct test of the Harrison--
Zeldovich spectrum (scale­invariant, n = 1) although this is only adequate for experi­
ments probing angular scales larger than the horizon size at recombination (– 1 ffi ) where
only Sachs--Wolfe or isocurvature effects are dominant. (As an example it was applied
to put constraints on n by comparing a simulated Tenerife experiment normalised to
the COBE detection with the actual experiment (Hancock et al. (1994)). An alternat­
ive method consists of using the expression of the expected temperature autocorrelation
function and the orthogonality property of the Legendre polynomials:
C l = 2ú
Z ú
0
C(`)P l (cos `) sin `d`: (2.7)
These methods were applied to BE models using the fitting parameter expression;
C(`) = A exp(\Gamma(`=` 1 ) 2 )
(1 + (`=` 2 ) 2 + (`=` 3 ) 4 ) ü + 3a 2
2
2ú
`
ln(1 + 4=` 2
4 ) \Gamma 1 \Gamma
ln(1 + (¯!=` 4 ) 8 )
4 \Gamma
3
2 cos `
'
(2.8)
where ¯
! = 2 sin(`=2) and A, a 2 , ` 1 , ` 2 , ` 3 , ` 4 are tabulated for each model (BE). Inserting
(2.8) in (2.7) and computing the integral one obtains the C l . These are in good agreement

4 Graca Rocha: Microwave Background Radiation
experiment model (\DeltaT =T )=10 \Gamma5
h0 = 0:75 h0 = 0:5 h0 = 0:5 h0 = 1 h0 = 0:75
\Omega B =
0:03\Omega B =
0:03\Omega b =
0:1\Omega B =
0:03\Omega B = 0:03
adiabatic adiabatic adiabatic adiabatic isocurvature
COBE, 10 ffi : 0.72 1.10 1.25 0.57 8.06
Tenerife, 5:6 ffi
fi = 8:1 ffi , triple beam: 0.55 0.80 0.95 0.47 5.79
MIT, 3:8 ffi : 0.92 1.38 1.59 0.75 10.02
South­Pole, 1:5 ffi
fi = 2:1 ffi , double beam: 0.92 1.30 1.68 0.80 6.16
MAX, 0:5 ffi
fi = 1:0 ffi , double beam: 1.32 2.12 2.85 1.12 5.51
MSAM, 0:5 ffi ,
double beam: 1.01 1.62 2.19 0.86 3.85
fi = 0:6 ffi , triple beam: 0.72 1.22 1.61 0.62 2.32
OVRO, 1:6 0
fi = 7:15 0 , triple beam: 0.53 0.85 1.23 0.43 0.65
Table 1. (\DeltaT =T )rms for BE models (fi is beamthrow)
with those plotted in BE. The calculations of (\DeltaT =T ) rms using both (2.1) and the C l
methods are in good agreement, the results are in Table 1. Figure 1 shows both one
BE model prediction (normalised to COBE) and observational results (detection and
instrumental error or upper limits) via (\DeltaT =T ) rms versus FWHM. The same method is
to be applied to other models and experiments. As most of the experimental results are
given as the result of a likelihood analysis, they are a measure of intrinsic anisotropies
i.e. values before beam convolution. In order to compare properly with theoretical
predictions the observed values (convolved with the beam and single or double difference
as appropriate) must be computed and used in Figure 1. The actual figure uses the
values as published by each experiment.
As previously mentioned this comparision is not complete until one compares the
predicted mean values with the theoretical uncertainties. Once this is accomplished we
may look for intersections of the experimental and theoretical bands for each model and
experiment and decide upon consistency. The methods to compute theoretical errors are
to be described in next section.
3. Variances
The temperature correlation function, C(`; oe), is a random variable with ensemble
average given by (2.5). It can be expressed as C(`; oe) = 1
4ú
P
l–2 Q 2
l (x)P l (cos `)W 2
l (oe),
where Q 2
l (x) =
P l
m=\Gammal ja m
l j 2 , with a ü 2
2l+1 distribution and hQ 2
l (x)i = (2l + 1)a 2
l ; a 2
l =
C l , variance var(Q 2
l ) = 2hQ 2
l i 2 =(2l + 1). To compute the cosmic variance we use two
approaches: (1) analytical and (2) Monte Carlo simulations. The first method uses the
probability density function of the temperature autocorrelation given in a semi­analytical
form (Cay'on et al. (1991)):
f z (t) = 1
2ú
Z +1
\Gamma1
/ Y
l=2
(1 \Gamma 2ix ¯
oe l
! \Gamma(2l+1)=2
exp(\Gammaitx) dx (3.9)
where Z = C(`; oe), ¯
oe l = (oe 2
l =4ú)P l (cos `) exp(\Gamma(l +1=2) 2 oe 2 ) with oe 2
l = (4A)=(úl(l + 1)),
P(k) = Ak. This expression applies to a single beam experiment, for beam­switching

Graca Rocha: Microwave Background Radiation 5
Figure 1. \Delta T/T versus FWHM: predictions and experimental values
experiments new variables Z 2 , Z 3 are defined. The distribution function is defined us­
ing the same expression with the new ¯
oe l defined according to the switching pattern.
The sample variance (oe 2
sam ) is obtained using an approximate relation with the cosmic
variance (Scott et al. (1994)):
oe 2
sam '
` 4ú
A
'
oe 2
cos (3.10)
where A is the solid angle the experiment covers; or directly using the two­point correl­
ation function C(cos `) (Scott et al. (1994)):
oe 2
sam = 2
A 2
Z
A
d\Omega 1
d\Omega 2 C 2 ( ~
n 1 \Delta ~
n 2 ) (3.11)
These methods may be applied to other experiments considering e.g. their scan strategy,
appropriate solid angle and using (3.11) or computing the cosmic variance via (3.9) and
then using the approximate relation (3.10). Both approaches may then be compared
with the results obtained using method (2). This method involves producing Monte
Carlo simulations of the temperature fluctuations according to specific characteristics of
each experiment such as the beamwidth, the sky coverage, the switching pattern, the
beamthrow, etc. then computing the correlation at zero lag, its distribution, variance
and confidence intervals. Proceeding in this way one computes the overall theoretical
error for a given experiment. These simulations follow different strategies according to
angular scale and experimental configuration. For large angular scales one uses the stand­
ard spherical harmonics expansion of \DeltaT =T , considering the coefficients as stochastic
variables with a Gaussian distribution, zero mean and as variance the angular power
spectrum, C l , (as appropriate to Gaussian random fields). The interval of l's to be used
for each experiment is found using its filter function, F l , and a standard criterion. These
scales are contributed to by low order multipoles only, the relation between the angular
scale ` and the multipole l is given by l ' 1=`, with ` in radians, used to compute the l
corresponding to the dispersion of the beam. This relation gives an idea of the significant
values of l contributing to the expansion. The C l 's are obtained using equations (2.6) or
(2.7), the former suiting the aim of testing the primordial spectrum directly. This was

6 Graca Rocha: Microwave Background Radiation
applied to COBE and Tenerife for the BE models and simulations of other experiments
are in progress. For small angular scales one considers the CMB radiation pattern as
a Gaussian field in a flat 2D space (BE) with \DeltaT =T given by a Fourier expansion as
follows: \DeltaT =T =
P
nx;ny D(~n) exp(i 2ú
L ~n \Delta ~ `) where L is the size of the small patch of
the sky, D(~n), is a variable with a Gaussian distribution with zero mean and variance
hjD(~n)j 2 i = C l with l ¸ 2ú
L n and with a random phase in the interval (0; 2ú). \DeltaT =T is
obtained by evaluating a Fast Fourier Transform (FFT) of D(~n). FFT's for L = 10 ffi on a
512 \Theta 512 grid were evaluated and grey scale pictures produced. For each experiment one
must consider the sky coverage and use the selected points to do statistics according to
experimental configuration. The C l 's used were computed using equation (2.7). Mean­
while other methods may be applied for instance the process used by G'orsky et al. (1993)
for the SP91 experiment. Simulations are in progress and its results will be published
in the near future. These will be organized in a plot with (\DeltaT =T ) rms versus FWHM
with both the theoretical expected value \Sigma theoretical error and the detected value \Sigma
instrumental error or observational upper limits as applied to most of the experiments
and more relevant models.
4. Conclusion
We have considered methods to compute the theoretical predictions for different ex­
perimental strategies and presented results for the BE models for some experiments and
a plot comparing these with the experimental results. This will be extended to other
models and experiments. We have stressed the importance of assigning the theoretical
uncertainties to the theoretical expected values in order to make a fair comparision of
models and experiments. We have described processes to compute the cosmic and sample
variances, proceeding with the Monte Carlo simulations. Once all these calculations are
complete a proper plot comparing theory and observations can then be produced.
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