Please Note: the e-mail address(es) and any
external links in this paper were correct when it was written in 1995,
but may no longer be valid.
Go to: Front page, Participants, Group Photograph, Preface, Contributions, Acknowledgements
Comparison of Microwave Background
Radiation models and experiments
Graca
Rocha
e-mail:
graca@mrao.cam.ac.uk
Mullard Radio Astronomy
Observatory,
Cavendish Laboratory, Madingley Road, Cambridge CB3 0HE, UNITED KINGDOM
Abstract:
We present methods of comparing theoretical predictions for the anisotropy of
the Cosmic Microwave Background Radiation (CMBR), with the results of various
experiments. The comparision between experimental results and theory is
organized in a plot with temperature fluctuation, 
, 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.
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 experimental 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 non-luminous matter, the biasing, the process of structure 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 experiments 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 fluctuations 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
so-called cosmic variance, i.e. the theoretical 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
. For (3), the different observational strategies may
consist of single difference, double differences (beam switching) 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,
computation of 
, cosmic variance, sample variance and the
theoretically expected value of 
) or (2) an analytical
approach (using semi-analytical expression to compute the cosmic variance for
each angular scale and each experiment (Cayón et al. (1991)). The 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 (
); MSAM ( 
); MAX3
Peg, MAX3GUM, MAX4GUM,
MAX4ID, MAX4SH (
); SP(ACME/HEMT); MIT
(
); Tenerife (
); COBE (
). In this
paper we give the predicted expected values of (
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 (
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 (Cayón 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).
In order to compute the expected temperature fluctuation for a given experiment
and model (i.e. (
) one uses the two-point 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 
: 
, where
. The expression of 
depends on the experimental configuration (e.g. beam size, switching
pattern etc.). With a two-beam experiment one measures the temperature
difference between beams separated by an angle 
on the sky usually
called single difference: 
and its expected
variance is written in terms of the temperature autocorrelation function:
;
while a three-beam experiment measures the difference between a field point,
, and the mean value of the temperatures in two directions which are
separated from the field point by an angle 
, usually called double
difference: 
) and its expected variance is
given by: 
. 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 
:
where 
is the dispersion of the Gaussian. The correlation function of
the convolved radiation field is given by BE:
where 
is the modified Bessel function. So for a single difference one
has:
and for a double difference:
where 
is the beamthrow. One is able to compute 
using equations (2.1) and (2.2) or
(2.3) according to the experimental configuration. If, for example, for
a given model 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 
, 
, is given to be
used in 
where 
and 
are
spherical Bessel functions, 
, 
is the
horizon distance. An alternative method consists of using the angular power
spectrum, 
, which are the coefficients of the Legendre polynomials
expansion of 
. Commonly one expresses the temperature fluctuations
on the celestial sphere using an expansion in spherical harmonics (completeness
property): 
Inflationary models predict Gaussian initial perturbations, in this case the
coefficients 
are stochastic variables with random phase, zero mean
and variance given by: 
. The expectation
value of the temperature correlation function (the ensemble average) is given
by:
For a more realistic scenario one must take into account the smearing due to
the finite resolution of the antenna: 
and
where 
(Scaramela & Vittorio (1993)).
Once given the 
's one can compute the correlation using (2.4)
or (2.5) and 
according to (2.2) or
(2.3) for the switching experiments. These 
's may be obtained
directly using the solutions to the equations describing temperature
fluctuations evolution by 
where 
are the
coefficients of the Legendre polynomials expansion of,
, the radiation intensity fluctuations and 
is
the conformal time (Efstathiou (1987)). In particular in the case of large angular
scales for 
and an initial power-law form of the density
fluctuations power spectrum P(k)
an expression for the
may be used (Efstathiou (1987),Scaramela & Vittorio (1988)):
with 
, 
, 
; this is important mainly because one is able to
predict 
and its error bands as function of 
providing a direct test of the Harrison-Zeldovich spectrum (scale-invariant,
) although this is only adequate for experiments probing angular scales
larger than the horizon size at recombination (
) where only
Sachs-Wolfe or isocurvature effects are dominant. (As an example it was
applied to put constraints on 
by comparing a simulated Tenerife experiment
normalised to the COBE detection with the actual experiment (Hancock et al. (1994)). An
alternative method consists of using the expression of the expected temperature
autocorrelation function and the orthogonality property of the Legendre
polynomials:
These methods were applied to BE models using the fitting parameter expression;
where 
and A, 
, 
,
, 
, 
are tabulated for each model (BE).
Inserting (2.8) in (2.7) and computing the integral one
obtains the 
. These are in good agreement with those plotted in BE. The
calculations of 
using both (2.1) and the
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
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.
Table 1: (
for BE models (
is beamthrow)
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.
Figure 1: 
T/T versus FWHM: predictions and experimental values
The temperature correlation function, 
, is a random variable
with ensemble average given by (2.5). It can be expressed as
, where 
, with a 
distribution and 
; 
, variance 
. 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 semi-analytical form (Cayón et al. (1991)):
where 
, 
with 
, P(k) = Ak. This expression applies to a single beam
experiment, for beam-switching experiments new variables 
, 
are
defined. The distribution function is defined using the same expression with
the new 
defined according to the switching pattern. The
sample variance (
) is obtained using an approximate
relation with the cosmic variance (Scott et al. (1994)):
where A is the solid angle the experiment covers; or directly using the
two-point correlation function C(
) (Scott et al. (1994)):
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 standard spherical harmonics expansion of
, considering the coefficients as stochastic variables with a
Gaussian distribution, zero mean and as variance the angular power spectrum,
, (as appropriate to Gaussian random fields). The interval of 
's to
be used for each experiment is found using its filter function, 
, and a
standard criterion. These scales are contributed to by low order multipoles
only, the relation between the angular scale 
and the multipole 
is
given by 
, with 
in radians, used to compute the 
corresponding to the dispersion of the beam. This relation gives an idea of the
significant values of 
contributing to the expansion. The 
's are
obtained using equations (2.6) or (2.7), the former suiting
the aim of testing the primordial spectrum directly. This was 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 
given by a Fourier
expansion as follows: 
where L is the size of the
small patch of the sky, 
, is a variable with a Gaussian
distribution with zero mean and variance 
=
with 
and with a random phase in the interval
. 
is obtained by evaluating a Fast Fourier Transform
(FFT) of 
. FFT's for L = 
on a 
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 
's used were computed using
equation (2.7). Meanwhile other methods may be applied for instance
the process used by Górsky 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 
versus FWHM with both the
theoretical expected value 
theoretical error and the detected value 
instrumental error or observational upper limits as applied to most of the
experiments and more relevant models.
We have considered methods to compute the theoretical predictions for different
experimental 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.
- Bond J.R. & Efstahiou G., 1987,
MNRAS, 226, 655.
- Cayón L., Martinez-González E. & Sanz J.L., 1991,
MNRAS, 253, 599.
- Cheng E.S. et al., 1994,
ApJ, 422, L37.
- Clapp A.C. et al., 1994,
ApJ, 433, 57.
- Ganga K. et al., 1993,
ApJ, 410, L57.
- Myers S.T., 1993,
ApJ, 405, 8.
- Schuster J. et al., 1993,
ApJ, 412, L47.
- Efstathiou G., 1987,
in Physics of the Early Universe, eds Peacock et al.
- Górsky K.M., Stompor R. & Juszkiewicz R., 1993,
ApJ, 410, L.
- Hancock S. et al., 1994,
Nature, 367, 333.
- Holtzmann J.A., 1989,
ApJS, 71, 1.
- Peebles P.J.E., 1987,
ApJ, 315, L73.
- Scaramela R. & Vittorio N., 1988,
ApJ, 331, L53.
- Scaramela R. & Vittorio N., 1993,
ApJ, 411, 1.
- Scott D., Srednicki M. & White M., 1994
ApJ, 421, L5.
- Smoot G.F. et al., 1992
ApJ, 396, L1.
YERAC 94 Account
Wed Feb 22 21:50:22 GMT 1995
