In this review, we shall consider only the SZ effect, since this the only secondary anisotropy to have been observed to date. The SZ effect is in fact made up of two separate effects: One is due to the bulk velocity of the cluster, and the other due to the thermal velocities of the electrons in the cluster gas. These are called the kinematic and thermal SZ effects respectively. The kinematic effect measures the cluster peculiar velocity, whereas the thermal effect can be used, in conjunction with images and spectra of its Xray emission, to study the cluster gas. In particular, for a dynamically relaxed cluster, we can use the thermal SZ effect and Xray data to estimate the physical size of the cluster, and hence its distance. This, in turn, yields an estimate of the Hubble constant
${H}_{0}$
.
If the cluster has a peculiar velocity
${v}_{p}$
along the observer's line of sight, then the temperature of a CMB photon is Dopplershifted by an amount
$$\begin{array}{c}(\Delta T{)}_{kin}={T}_{0}\frac{{v}_{p}}{c}\int {n}_{e}{\sigma}_{T}dl,\end{array}$$ 
(5)

where
${T}_{0}$
is the temperature of the CMB,
${n}_{e}$
is the electron number density,
${\sigma}_{T}$
is the Thomson scattering crosssection and the integral is taken along the line of sight. In terms of the RayleighJeans brightness temperature this becomes
$$\begin{array}{c}(\Delta {T}_{RJ}{)}_{kin}={T}_{0}\frac{{v}_{p}}{c}\int {n}_{e}{\sigma}_{T}dl\left\{\frac{{x}^{2}{e}^{x}}{({e}^{x}1{)}^{2}}\right\},\end{array}$$ 
(6)

or as an intensity
$$\begin{array}{c}(\Delta {I}_{\nu}{)}_{kin}=\frac{2{k}^{3}{T}_{0}^{3}}{{h}^{2}{c}^{2}}\frac{{v}_{p}}{c}\int {n}_{e}{\sigma}_{T}dl\left\{\frac{{x}^{4}{e}^{x}}{({e}^{x}1{)}^{2}}\right\},\end{array}$$ 
(7)

where we have set
$x=h\nu /k{T}_{0}$
.
The effect due to thermal motions of the electrons is secondorder in the electron velocity, and does not preserve the blackbody shape of the CMB spectrum. For the thermal SZ effect the change in the RayleighJeans brightness temperature is given by
$$\begin{array}{c}(\Delta {T}_{RJ}{)}_{thermal}={T}_{0}\int {n}_{e}{\sigma}_{T}\frac{k{T}_{e}}{{m}_{e}{c}^{2}}dl\left\{\frac{{x}^{2}{e}^{x}}{({e}^{x}1{)}^{2}}[xcoth\left(\frac{x}{2}\right)4]\right\},\end{array}$$ 
(8)

where
${T}_{e}$
is the electron temperature and
${m}_{e}$
the electron mass. In terms of intensity this becomes
$$\begin{array}{c}(\Delta {I}_{\nu}{)}_{thermal}=2\frac{{k}^{3}{T}_{0}^{3}}{{h}^{2}{c}^{2}}\int {n}_{e}{\sigma}_{T}\frac{k{T}_{e}}{{m}_{e}{c}^{2}}dl\left\{\frac{{x}^{4}{e}^{x}}{({e}^{x}1{)}^{2}}[xcoth\left(\frac{x}{2}\right)4]\right\}.\end{array}$$ 
(9)

It is usual to describe the magnitude of the thermal effect in terms of the
$y$
parameter, which is given by
$$y=\int {n}_{e}{\sigma}_{T}\frac{k{T}_{e}}{{m}_{e}{c}^{2}}dl,$$
The frequency dependencies of the kinetic and thermal effects (i.e. those functions in curly brackets in Equations 6 – 9 ), are shown in Figure 8 .
Figure 8
: The frequency dependences of the thermal and kinetic SZ effects expressed as a brightness temperature change (top) and intensity change (bottom).
Note that at
$\nu =210$
GHz, the maximum change in intensity due to the kinematic effect coincides with the null of the thermal effect. This, in principle, allows one to separate the two effects. The magnitude of the thermal effect for a hot, dense cluster is
$(\Delta {T}_{RJ}{)}_{thermal}\approx 1$
mK, and for reasonable cluster velocities the kinematic effect is an order of magnitude smaller.
Observations of the SZ effect have been made in Cambridge using the Ryle telescope, which is an 8dish interferometer operating at 15 GHz [
75]
, in Caltech using the Owen's Valley 5.5m telescope [
60]
, the Owen's Valley 40m telescope [
40]
and the Owen's Valley Millimeter Array (OVMMI) [
12]
[
11]
, at NASA using the MSAM balloon experiment [
79]
, at the Caltech Submillimeter Observatory using a purpose built instrument called the SunyaevZel'dovich Infrared Experiment (SuZIE) [
38]
and by various other groups. The magnitude of the observed SZ effect in these clusters can be combined with Xray data from ROSAT and ASCA to place limits on
${H}_{0}$
. This has been reviewed in Lasenby & Jones [
53]
.
Another important feature of the SZ effect is that the decrement does not change with redshift.
Therefore, it should be possible to detect clusters out to very high redshift. To test this the Ryle telescope has been making observations towards quasar pairs. Figure
9
Figure 9
: Ryle telescope map of the sky towards quasar pair PC1643+4631 (the positions of which are shown as crosses). The central decrement is about 600
$\mu $
K.
shows the Ryle telescope map of the sky towards the quasar pair PC1643+4631 (Jones et al. 1997) [
47]
. These two quasars are at a redshift of
$\sim 3.8$
and are
$\sim {3}^{\prime}$
apart on the sky. They are a strong candidate for a gravitational lensed object due to the similarity in their spectra. As there is no Xray detection with ROSAT the cluster responsible for the lensing must be at a redshift greater than 2.5 and from modelling of the gravitational lensing, or from fitting for a density profile in the SZ effect, it must have a total mass of about
$2\times {10}^{15}{M}_{\odot}$
.
With observations of distant clusters it is possible to predict the mass density of the Universe. Using the PressSchecter formalism the number of clusters in terms of SZ flux counts can be predicted. Figure
10
Figure 10
: SZ source counts with observational constraints, as a function of SZ flux density expressed at 400 GHz. The two hatched boxes show the 95% onesided confidence limits from the VLA and the RT; due to the uncertain redshift of the clusters, there is a range of possible total SZ flux density, which has for a minimum the value observed in each beam and a maximum chosen here to correspond to
$z>1$
. From the SuZIE blank fields, one can deduce the 95% upper limit shown as the triangle pointing downwards (Church et al. [
18]
). We also plot the predictions of our fiducial open model (
$\Omega =0.2$
) for all clusters (dashed line) and for those clusters with
$z>4$
.
shows the results from Bartlett et al. [
3]
. From this figure it is seen that, if the decrements are really due to the SZ effect and there was no bias in selecting the fields (e.g. the field was chosen because of the magnification of the quasar pair images), then the
$\Omega =1$
model of the Universe is ruled out. An open model is required to be consistent with the data. Confirmation of the detections are needed. Until these follow up observations have been made, it is impossible to say how accurate these findings are.
4.2 Relativistic corrections to the SZ effect
The above treatment of the SZ effect is purely nonrelativistic. In clusters where
${k}_{B}{T}_{e}>10$
keV, relativistic effects become important. The relativistic effect can be included either by extending the equations to include relativistic terms ([
17]
, [
83]
, [
84]
) or by including multiple scattering descriptions of the Comptonization process ([
100]
, [
25]
, [
85]
, [
55]
, [
72]
). Both of these approaches give consistent results. To first order in temperature the correction for the thermal SZ effect is given by
$$\begin{array}{ccc}(\Delta {T}_{RJ}{)}_{thermal}={T}_{0}y[\frac{{x}^{2}{e}^{x}}{({e}^{x}1{)}^{2}}[xcoth\left(\frac{x}{2}\right)4+\frac{k{T}_{e}}{{m}_{e}}(10+\frac{47}{2}xcoth\left(\frac{x}{2}\right)& & \end{array}$$  
$$\begin{array}{ccc}\frac{42}{5}{x}^{2}{coth}^{2}\left(\frac{x}{2}\right)+\frac{7}{10}{x}^{3}{coth}^{3}\left(\frac{x}{2}\right)+\frac{7{x}^{2}}{5{sinh}^{2}\left(\frac{x}{2}\right)}\left(xcoth\left(\frac{x}{2}\right)3\right))]]& & \end{array}$$ 
(10)

and in the RayleighJeans limit (small
$x$
) we find
$$\begin{array}{c}(\Delta {T}_{RJ}{)}_{thermal}=2{T}_{0}y\left[1\frac{17}{10}\frac{k{T}_{e}}{{m}_{e}}+O\left({\left(\frac{k{T}_{e}}{{m}_{e}}\right)}^{2}\right)\right]\end{array}$$ 
(11)

and so it is seen that the inclusion of the relativistic treatment tends to lead to a small decrease in the SZ effect. The Hubble constant is inferred from combined SunyaevZel'dovich and Xray data by a relation of the form [
50]
$$\begin{array}{c}{H}_{\circ}\propto \Delta {T}^{2}.\end{array}$$ 
(12)

The reduction in
$\Delta T$
in the RayleighJeans region, for given cluster parameters, leads to a decrease in the constant of proportionality in Equation 12 , and hence a small reduction in the determined values of
${H}_{\circ}$
. For example, it is found that if the cluster temperature is 8keV the reduction in
${H}_{\circ}$
due to relativistic effects is 5%.
5 Current and future CMB experiments
Since the discovery of a statistical anisotropy four years ago by the COBE satellite, there has been a great increase in the number of ground and balloonbased measurements of CMB anisotropy. Many groups around the world have been engaged in projects which aim to detect individual features in the CMB, and to establish their statistical properties. Table 1 lists all of the major experiments in the field and includes links to their home pages. The focus of these has tended to be towards smaller angular scales than measured by COBE, and this holds with it the exciting prospect of being able to detect structure in the primordial power spectrum.
Space based

COBE[27]


Planck[42]


MAP[30]

Ground based

Tenerife[43]


South Pole[82]


Saskatoon[96]


CAT[58]


ATCA[1]


Python[13]


OVRO[9]


SuZIE[10]


Jodrell Bank[44]


IAC Bartol[41]


White Dish[15]


Viper[14]


COBRA[92]


MAT[94]


DASI[93]


VSA[59]


CBI[8]


POLAR


Brown/Wisc polarization[97]

Balloon Borne

FIRS[68]


ARGO[57]


MAX[88]


BEAST[91]


MSAM[28]


BAM[87]


ACE[90]


QMAP[95]


Boomerang[16]


MAXIMA[89]


Top Hat[29]



Table 1
: The main CMB experiments with links to their WWW home pages where available
A summary of the properties and results of several experiments, covering the period through to 1996, is given in Lasenby & Hobson [
51]
. Since that time, i.e. in the last year, there have been significant developments with regard to a number of experiments, and we concentrate on these new results here. Table 2 shows in a convenient form the main parameters of several recent experiments.
Experiment

Type

Beamwidth/

$\nu $
(GHz)



Beamthrow (
${}^{\circ}$
)


DMRCOBE

Satellite

7

31, 53, 90

Tenerife

Ground

5/8

10, 15, 33

MIT/FIRS

Balloon

3.8

180 + 3 higher

ACME/HEMT

Ground

1.5/2.1

30/40

MAX

Balloon

0.5/1.0

110, 180, 270, 360

MSAM

Balloon

0.5/0.6

180 + 3 higher

White dish

Ground

0.18/0.47

90

Python

Ground

0.75/2.75

90

Saskatoon

Ground

1.5/2.45

$2636$

ARGO

Balloon

0.9/1.8

150 + 3 higher

CAT

Ground

0.25

$1316$

OVRO

Ground

0.10.4

14.5 + 32

IACBartol

Ground

2

91 272



Table 2
: Some recent CMB anisotropy measurements.
5.1 The COBE satellite
The Cosmic Background Explorer satellite (COBE) was launched on 18th November 1989.
Primordial anisotropy measurements are made using the DMR experiment, which consists of six differential microwave radiometers, two at each of 31.5 GHz, 53.0 GHz and 90.0 GHz. The firstyear COBE observations provided convincing statistical evidence for the existence of CMB fluctuations.
It was not however, possible to see individual CMB features, on the scale of the beam size, in the DMR maps, because even combining all of the maps together, the noise level per beam area was
$\sim 45\mu $
K and the signal to noise remained less than one.
The results of the analysis of all four years of DMR data have now become available. A convenient summary of all the results is given in Bennett
et al. (1996) [
5]
. On a statistical level, the results can be used to constrain the normalisation of a power law primordial spectrum. For a given slope
$n$
, normalisation is usually expressed via the implied amplitude of the quadrupole component of the power spectrum,
${C}_{2}$
, as
$$\begin{array}{c}{Q}_{\text{rmsps}}=T\sqrt{\frac{5{C}_{2}}{4\pi}},\end{array}$$ 
(13)

where
$T$
is the mean CMB temperature. (Note that this
$\text{}{Q}_{rmsps}\text{}$
value need not be the same as the actual quadrupole component. The fit is to a whole power spectrum as parameterised by a given
$n$
). For an assumed value of
$n=1$
, (the HarrisonZeldovich value), Bennett et al.
quote
$\text{}{Q}_{rmsps}\text{}=18\pm 1.6\mu K$
. The joint best fit values of
$n$
and
$\text{}{Q}_{rmsps}\text{}$
are
$n=1.2\pm 0.3$
and
$\text{}{Q}_{rmsps}\text{}=15.3\mu K$
. This restriction on the value of
$n$
is of course of great interest in the context of inflationary predictions that
$n=1$
. It is also of interest that inflation predicts Gaussian fluctuations, and while this is much harder to test for than finding the amplitude and slope of the spectrum, the data are also consistent with this prediction. Specifically, Bennett et al. state `statistical tests prefer Gaussian over other toy statistical models by a factor of
$\sim 5$
'.
With the accumulation of four years of data, the individual anisotopy features within the maps on the scale of the beam size are now becoming statistically significant. Figure
11 shows the allsky maps at each frequency taken from Bennett et al. [
5]
. Some of the features in these maps away from the Galactic plane are expected to be real CMB fluctuations, since the signal to noise in these regions is now about 2 sigma per 10 degree sky patch. Indeed, features which repeat well between the different frequencies are now clearly visible.
Figure 11
: The COBE DMR 4 year data displayed as allsky maps.
5.2 The Tenerife experiments
[Project collaborators: R. Davies (Jodrell Bank Principal Investigator); R. Rebolo, C. Gutierrez, R. Watson, R. Hoyland (IAC, Tenerife); A. Lasenby, A. Jones, G. Rocha, S. Hancock (MRAO, Cambridge).] A description of these experiments was given in Lasenby & Hancock [
49]
. Here we briefly summarise the relevant features. The Tenerife experiments constitute a suite of three instruments, working at 10, 15 and 33 GHz, designed and built at Jodrell Bank (Davies et al. 1992 [
21]
, 1996 [
20]
) and operated by the IAC insitu on Tenerife island. Data are taken by drift scanning in right ascension at fixed declination and by sampling at
${2.5}^{\circ}$
intervals in declination with a beamwidth of
${5.1}^{\circ}$
(FWHM) it is possible to build up a fully sampled twodimensional map of the sky at each frequency [
70]
. Initially, observations were concentrated on the strip of sky at declination
$+{40.0}^{\circ}$
where the low frequency surveys of Haslam et al. (408 MHz) [
35]
and Reich and Reich (1420 MHz) [
71]
indicate a minimum in Galactic foreground emission.
Analysis of the data at declination
$+{40.0}^{\circ}$
has been made and reported in Hancock et al.
(1994) [
32]
. Results from the Dec.
${35}^{\circ}$
data scans at 10 and 15 GHz have recently been reported in Gutierrez et al. [
31]
and are consistent with the Dec.
${40}^{\circ}$
data. A level of
$\text{}{Q}_{rmsps}\text{}=20\pm 6$
$\mu $
K was found in a high Galactic latitude region at Dec.
${35}^{\circ}$
at 15 GHz. The COBE data was used in Bunn et al. (1996) to make a prediction for the Tenerife data. The comparison between this prediction and the Tenerife 15 GHz data is shown in Figure 12 , and it is seen that there is very good agreement with the main features in the COBE scan. This is evidence for features that are constant in amplitude over a frequency range of 15 GHz to 90 GHz and is a very strong candidate for a real CMB anisotropy.
Figure 12
: Comparison between the maximum entropy reconstruction of the Tenerife Dec.
${35}^{\circ}$
data at 15 GHz (solid line) and the COBE DMR predictions of Bunn et al. (1996) (dashed line) at 53 and 90 GHz.
The Tenerife programme is continuing, with the objective of mapping some 4000 square degrees of sky at 10, 15 and 33 GHz. In conjunction with the COBE fouryear data set, these experiments will continue to offer a useful source of largescale CMB anisotropy measurements and hence to directly probe cosmological theories. In terms of power spectrum results, a revised estimate of the fluctuation amplitude for
$\ell $
values near
$\sim 20$
has been constructed from the Dec
${40}^{\circ}$
data, making an allowance for an atmospheric contribution (Hancock et al. 1997 [
33]
) that was not subtracted in Hancock et al. [
32]
. This is used below in comparison with theoretical curves.
Preliminary analysis of the full 15 GHz data set (Dec.
$+{30}^{\circ}$
to
${45}^{\circ}$
) has been made and will be presented in a future paper. For an assumed value of
$n=1$
a full two dimensional likelihood analysis gives
$\text{}{Q}_{rmsps}\text{}=22\pm 4$
$\mu $
K.
5.3 Saskatoon
This experiment is a groundbased telescope located in Saskatoon, Canada. A cooled HEMT receiver with six channels is used to span the frequency range 26 to 46 GHz. The chopping strategy is quite complex, and can be used to synthesise `window functions' appropriate to a range of angular scales. The beam sizes used range from
$\sim {1.5}^{\circ}$
at the lowest frequency to
$\sim {0.5}^{\circ}$
at the highest.
The analysis of observations of a 24 hour RA strip at declination
$+{85.1}^{\circ}$
is described in Wollack et al. (1993) [
99]
and Netterfield et al. (1995) [
63]
, indicating a detection of primordial anisotropy.
More recently in Netterfield
et al. (1997) [
62]
, exciting results have been presented which show not just a detection, but for the first time for a switchedbeam instrument, evidence for the form of the power spectrum itself on the angular scales probed. The final map after 3 years of data from this experiment is shown in Figure 13
Figure 13
: Saskatoon 3 year map showing region analysed as compared to the COBE full sky coverage.
and the power spectrum from this map will be used below, in comparison with theoretical predictions.
We note here that there is currently an overall scaling uncertainty in the Saskatoon results of
$\pm 14$
%, due to calibration uncertainties. Recent analysis of the Saskatoon data (Knox, in prep.) appear to show that the previous calibration is an underestimate of the true level of the Saskatoon data.
This would make the Doppler peak even higher and lower the value of
${H}_{\circ}$
found below.
5.4 The Owen's Valley Radio Observatory
The Owen's Valley Radio Observatory (OVRO) is a groundbased, 40m single dish, telescope.
A HEMT reciever operating at two frequencies (14.5 GHz and 32 GHz) has recently been added to improve the sensitivity of the telescope. The resolution is
${0.12}^{\circ}$
. The original experiment put very strong constraints on Galaxy formation scenarios [
61]
[
69]
. Recent data on 36 pointed observations at Dec.
${88}^{\circ}$
give
$\delta T={56}_{6.5}^{+8.6}\mu $
K at
$\ell \sim 589$
(Leitch, private communication).
5.5 Python
Python is located at the AmundsenScott South Pole Station in Antartica. It consists of a single bolometer on a 75cm telescope, operating at a frequency of 90 GHz with a
${0.75}^{\circ}$
FWHM beam. Three seasons of observations have already been analysed (Python I [
23]
, Python II [
74]
and Python III [
67]
). The Python results can be separated into two distinct bins in
$\ell $
space (one centred at
$\ell \sim 87$
and the other at
$\ell \sim 170$
). The results for these two bins are
$\delta T={60}_{13}^{+15}\mu $
K and
${66}_{16}^{+17}\mu $
K respectively. Python IV and V data have already been taken and the analysis should be published very shortly. Details of the IV
${}^{th}$
season can be found in Kovac et al. (1997) [
48]
.
5.6 The Cosmic Anisotropy Telescope
[Project collaborators: J. Baker, P.J. DuffettSmith, M. Hobson, M. Jones, A. Lasenby, C. O'Sullivan, G. Pooley, R. Saunders and P. Scott.] The Cosmic Anisotropy Telescope (CAT) is a three element, groundbased interferometer telescope, of novel design [
73]
. Hornreflector antennas mounted on a rotating turntable, track the sky, providing maps at four (nonsimultaneous) frequencies of 13.5, 14.5, 15.5 and 16.5 GHz. The interferometric technique ensures high sensitivity to CMB fluctuations on scales of
${0.5}^{\circ}$
, (baselines
$\sim 1$
m) whilst providing an excellent level of rejection to atmospheric fluctuations. Despite being located at a relatively poor observing site in Cambridge, the data is receiver noise limited for about 60% of the time, proving the effectiveness of the interferometer strategy. The first observations were concentrated on a blank field (called the CAT1 field), centred on RA
${08}^{h}$
${20}^{m}$
, Dec.
$+{68}^{\circ}$
${59}^{\prime}$
, selected from the Green Bank 5 GHz surveys under the constraints of minimal discrete source contamination and low Galactic foreground. The data from the CAT1 field were presented in O'Sullivan et al. (1995) [
64]
and Scott et al. (1996) [
77]
.
Recently observations of a new blank field (called the CAT2 field), centred on RA
${17}^{h}$
${00}^{m}$
, Dec.
$+{64}^{\circ}$
${30}^{\prime}$
, have been taken. Accurate information on the point source contribution to the CAT2 field maps, which contain sources at much lower levels, has been obtained by surveying the fields with the Ryle Telescope at Cambridge, and the multifrequency nature of the CAT data can be used to separate the remaining CMB and Galactic components. Some preliminary results from CAT2 have been presented in Baker (1997) [
2]
and the 16.5 GHz map is shown in Figure 14 . Clear structure is visible in the central region of this map, and is thought to be actual structure, on scales of about
$1/{4}^{\circ}$
, in the surface of last scattering.
Figure 14
: 16.5 GHz CAT image of
${6}^{\circ}\times {6}^{\circ}$
area centred on the CAT2 field, after discrete sources have been subtracted. Excess power can be seen in the central
${2}^{\circ}\times {2}^{\circ}$
primary beam (because the sensitivity drops sharply outside this area, the outer regions are a good indicator of the noise level on the map). The flux density range scale spans
$\pm 40$
mJy per beam.
When interpreting this map, however, it should remembered that for an interferometer with just three horns, the `synthesised' beam of the telescope has large sidelobes, and it is these sidelobes that cause the regular features seen in the map. In the full analysis of the data, these sidelobes must be carefully taken into account.
For an interferometer, `visibility space' correlates directly with the space of spherical harmonic coefficients
$\ell $
discussed earlier, and the data may be used to place constraints directly on the CMB power spectrum in two independent bins in
$\ell $
. These constraints, along with those from the other experiments, are shown in Figure 15 .
Figure 15
: Recent results from various CMB experiments. The solid line is the prediction (normalised to COBE) for standard CDM with
${\Omega}_{\circ}=1.0$
,
${\Omega}_{b}=0.1$
and
${H}_{\circ}=45$
km s
${}^{1}$
Mpc
${}^{1}$
. The Saskatoon points have a 14% calibration error.
5.7 Future experiments
The next few years should bring great improvements in the quality of CMB data available, both as regards to accuracy, and to range of coverage of angular scales. For balloon experiments, these improvements will come through the use of longduration balloon flights, launched e.g. in Antarctica, and circling the pole, and from the use of arrays of detectors covering several frequency bands. Specific projects already in the pipeline using these techniques are TOPHAT, BEAST, ACE and Boomerang. On the ground, interferometers will play an increasingly important rôle, with three new arrays now under development, the Very Small Array (VSA), the Degree Angular Scale Interferometer (DASI), the Cosmic Background Interferometer (CBI) and the Millimeter Anisotropy eXperiment Imaging Array (MAXIMA). We will concentrate here briefly on a new interferometer array to be built by Cambridge and Jodrell Bank in the U.K., and to be sited in Tenerife (the design of the three new interferometer arrays are similar), and then discuss the two new satellite projects recently selected.
5.8 The Very Small Array
Although the CAT has already provided maps of CMB anisotropy on scales
$\sim {0.25}^{\circ}$
, these are relatively poor as images due to the limited number of baseline lengths and pixels available. In fact, the CAT is a prototype for a considerably more advanced instrument, the Very Small Array (VSA).
The objectives of the VSA are to obtain detailed maps of the CMB with a sensitivity approaching
$5\mu $
K and covering a range of angular scales from
${10}^{\prime}$
to
${2}^{\circ}$
. An artists impression of the VSA is shown in Figure 16 .
Figure 16
: Artist's impression of the completed VSA array.
Using two interchangeable
$T$
shaped configurations of 10–15 horn elements, simulations have shown that it is possible to obtain maps of suitable sensitivity over the desired range of angular scales. The planned instrument would be sited at the Teide Observatory, Tenerife, at an altitude of 2400m and would make observations between 28 and 38 GHz, to enable the Galactic component to be estimated and removed. The good accuracy available over a scale range that is wellmatched to the positions of the first and secondary Doppler peaks in the power spectrum, should enable measurements of
$\Omega $
and
${H}_{0}$
to be made to an accuracy of better than 10% after 12 months of observations. We believe that this will be refined somewhat by better array configuration design and the use of proper models and secondary peak information (all work in progress). In addition, simulations have shown that the proposed observing strategy will be quite sensitive to the nonGaussian features expected on these angular scales if (e.g.) textures or monopoles are the seed perturbations for galaxy formation (Maisinger, Hobson, Lasenby & Turok [
56]
). The instrument is currently under construction at Cambridge and Jodrell Bank in the U.K., and is hoped it will be operational by the year 2000.
5.9 Future satellite experiments
Two new satellite experiments to study the CMB have recently been selected as future missions.
These are MAP, or Microwave Anisotropy Probe, which has been selected by NASA as a Midex mission, for launch in August 2000, and the Planck Surveyor, which has been selected by ESA as an M3 mission, and will be launched hopefully soon after 2005. An artist's impression of the MAP satellite, which has five frequency channels from 30 GHz to 100 GHz, is shown in Figure
17 .
Figure 17
: Artist's impression of the MAP Satellite.
An artist's impression of the Planck Surveyor satellite, which combines both HEMT and bolometer technology in 10 frequency channels covering the range 30 GHz to 850 GHz, is shown in Figure 18 .
Figure 18
: Artist's impression of the Planck Surveyor Satellite (formally COBRAS/SAMBA)
A crucial feature of a satellite experiment is the potential allsky coverage that it affords, and the ability to map features on large angular scales (
$>{10}^{\circ}$
).
Figure 19
: Expected capability of a satellite experiment as a function of resolution. The percentage error in recovering cosmological parameters from the CMB power spectrum is shown versus the resolution available. This figure is taken from Bersanelli et al. 1996 [
6]
.
6 Analysis of CMB data
6.1 Maximum Entropy analysis
With such a large increase in the amount of data available from CMB experiments it is becoming increasingly important to improve the analysis techniques used. With multifrequency observations of the same patch of sky to high precision (the satellite will cover the full sky) it should be possible to extract information on the foregrounds and leave a `clean' map of the CMB. One technique that has been shown to be robust in the analysis of a number of different experiments is Maximum Entropy ([
37]
, [
45]
and [
56]
).
If we start with Bayes' theorem which states, given a hypothesis
$H$
and some data
$D$
the posterior probability
$pr\left(H\rightD)$
is the product of the likelihood
$pr\left(D\rightH)$
and the prior probability
$pr\left(H\right)$
, normalised by the evidence
$pr\left(D\right)$
,
$$\begin{array}{c}pr\left(H\rightD)=\frac{pr\left(H\right)pr\left(D\rightH)}{pr\left(D\right)}.\end{array}$$ 
(14)

If the instrumental noise on each frequency channel is Gaussiandistributed, then the probability distribution of the noise is a multivariate Gaussian. Assuming the expectation value of the noise to be zero at each observing frequency, the likelihood is therefore given by
$$\begin{array}{c}pr\left(D\rightH)\propto exp[{\chi}^{2}\left(H\right)]\end{array}$$ 
(15)

where the
${\chi}^{2}$
misfit statistic has been introduced.
We now have to decide the form of the prior probability. Let us consider a discretised image
${h}_{j}$
consisting of
$L$
cells, so that
$j=1,\dots ,L$
; we may consider the
${h}_{j}$
as the components of an image vector
$H$
. If we base the derivation of the prior on purely information theoretic considerations (subset independence, coordinate invariance and system independence) we are naturally led to the Maximum Entropy Method (MEM). It may be shown [
80]
the prior probability takes the form
$$\begin{array}{c}pr\left(H\right)\propto exp\left[\alpha S\right(H,M\left)\right],\end{array}$$ 
(16)

where the dimensional constant
$\alpha $
depends on the scaling of the problem and may be considered as a regularising parameter, and
$M$
is a model vector to which
$H$
defaults in the absence of any data.
In standard applications of the maximum entropy method, the image
$H$
is taken to be a positive additive distribution (PAD). Nevertheless, the MEM approach can be extended to images that take both positive and negative values by considering them to be the difference of two PADS, so that
$$\begin{array}{c}H=UV\end{array}$$ 
(17)

where
$U$
and
$V$
are the positive and negative parts of
$H$
respectively. In this case, the cross entropy is given by
$$\begin{array}{c}S(H,{M}_{u},{M}_{v})={\sum}_{j=1}^{L}\left\{{\psi}_{j}{{m}_{u}}_{j}{{m}_{v}}_{j}{h}_{j}ln\left[\frac{{\psi}_{j}+{h}_{j}}{2{{m}_{u}}_{j}}\right]\right\},\end{array}$$ 
(18)

where
${\psi}_{j}=[{h}_{j}^{2}+4{{m}_{u}}_{j}{{m}_{v}}_{j}{]}^{1/2}$
and
${M}_{u}$
and
${M}_{v}$
are separate models for each PAD. The global maximum of the cross entropy occurs at
$H={M}_{u}{M}_{v}$
. The most probable image
$H$
is then just the result from finding the maximum probability or, equivalently, the minimum of
${\chi}^{2}\alpha S$
. This can be done by using any known minimising routine.
It can be shown [
37]
that the Lagrange multiplier
$\alpha $
is completely defined in a Bayesian way and any prior correlation information can also be incorporated into the analysis. Also, the assignment of errors is straightforward in the Fourier domain where all the pixels in the discretised image will be independent.
Hobson
et al. [
37]
simulated data taken by the Planck Surveyor satellite and used MEM to reconstruct the underlying CMB and foregrounds. They used six input maps (the CMB, thermal and kinetic SZ, dust emission, freefree emission and synchrotron emission) to make up the data and then added Gaussian noise to each frequency. After using MEM with the Bayesian value for
$\alpha $
and giving the algorithm the average power spectra of each channel, it was found that features in all six maps were recovered. Without any prior power spectrum information it was found that only the kinetic SZ was not recovered and all others were recovered to some degree (the CMB and dust were almost indistinguishable from the input maps with residual errors of
$6\mu $
K and
$2\mu $
K per pixel respectively). Figure 20 shows the results from MEM as compared to the input maps for the case with assumed average power spectrum. It is easily seen that MEM reconstructs both the Gaussian CMB and the nonGaussian thermal SZ effect very well.
Figure 20
: The left hand side shows the input maps used in the Planck simulations for a CDM simulation of the CMB and the thermal SZ effect. The right hand side shows the reconstructions obtained by MEM. It is easily seen that MEM does a very good job at reconstructed these two components. For comparison, the grey scales on the input maps are the same as on the reconstructed maps.
6.2 Wiener filtering
In the absence of nonGaussian sources it is possible to simplify the Maximum Entropy method using the quadratic approximation to the entropy (Hobson et al. [
37]
). This has the same affect as chosing a Gaussian prior so that Equation 16 becomes
$$\begin{array}{c}pr\left(H\right)\propto exp[\frac{1}{2}{H}^{\u2020}{C}^{1}H]\end{array}$$ 
(19)

where
$C$
is the covariance matrix of the image vector
$H$
given by
$$\begin{array}{c}C={H}^{\u2020}H.\end{array}$$ 
(20)

The solution to the analysis with this Bayesian prior is called Wiener filtering (see, for example, [
7]
and [
86]
) and has been applied to many data sets in the past when nonGaussianity could be ignored. The data
$D$
can be written as
$$\begin{array}{c}D=B\u25c6H+\epsilon \end{array}$$ 
(21)

where the convolution of the image vector
$H$
is with
$B$
, the beam response of the instrument and frequency dependance of
$H$
.
$\epsilon $
is the noise vector. In this case, the best reconstructed image vector,
$\hat{H}$
, is given by
$$\begin{array}{c}\hat{H}=WD\end{array}$$ 
(22)

where the Wiener filter,
$W$
, is given by
$$\begin{array}{c}W=C{B}^{\u2020}(BC{B}^{\u2020}+N{)}^{1}\end{array}$$ 
(23)

and
$N$
is the noise covariance matrix given by
$N={\epsilon}^{\u2020}\epsilon $
. It is important to note that not only the CMB signal but all the foregrounds are implicitly assumed to be nonGaussian in this method ([
37] [
46]
and [
36]
).
6.3 The CMB power spectrum versus experimental points
It will have become apparent in the preceding sections that the CMB data are approaching the point where meaningful comparison between theory and prediction, as regards the shape and normalisation of the power spectrum, can be made. This is particularly the case with the new availability of the recent CAT and Saskatoon results, where the combination of scales they provide is exactly right to begin tracing out the shape of the first Doppler peak. (If this exists, and if
${\Omega}_{tot}=1$
.) Before embarking on this exercise, some proper cautions ought to be given. First, the current CMB data is not only noisy, with in some cases uncertain calibration, but will still have present within it residual contamination, either from the Galaxy, or from discrete radio sources, or both. Experimenters make their best efforts to remove these effects, or to choose observing strategies that minimise them, but the process of getting really `clean' CMB results, free of these effects to some guaranteed level of accuracy, is still only in its infancy. Secondly, in any comparison of theory and data where parameters are to be estimated, the results for the parameters are only as good as the underlying theoretical models and assumptions that went into them. If CDM turns out not to be a viable theory for example, then the bounds on
$\Omega $
derived below will have to be recomputed for whatever theory replaces it. Many of the ingredients which go into the form of the power spectrum are not totally theoryspecific (this includes the physics of recombination, which involves only wellunderstood atomic physics), so that one can hope that at least some of the results found will not change too radically.
Bearing these caveats in mind, it is certainly of interest to begin this process of quantitative comparison of CMB data with theoretical curves. Figure
21 shows
Figure 21
: Analytic fit to power spectrum versus experimental points. (From Hancock et al. [
34]
, 1997.)
a set of recent data points, many of them discussed above, put on a common scale (which may effectively be treated as
$\sqrt{\ell (\ell +1){C}_{\ell}}$
), and compared with an analytical representation of the first Doppler peak in a CDM model. The work required to convert the data to this common framework is substantial, and is discussed in Hancock et al. (1997) [
34]
, from where this figure was taken. The analytical version of the power spectrum is parameterised by its location in height and left/right position, and enables one to construct a likelihood surface for the parameters
$\Omega $
and
${A}_{peak}$
, where
${A}_{peak}$
is the height of the peak, and is related to a combination of
${\Omega}_{b}$
and
${H}_{0}$
, as discussed above.
The dotted and dashed extreme curves in Figure
21 indicate the best fit curves corresponding to varying the Saskatoon calibration by
$\pm 14\%$
. The central fit yields a
$68\%$
confidence interval of
$$\begin{array}{c}0.2<\Omega <1.5,\end{array}$$ 
(24)

with a maximum likelihood point of
$\Omega =0.7$
after marginalisation over the value of
${A}_{peak}$
.
Incorporating nucleosynthesis information as well, as sketched above (specifically the Copi
et al.
[
19]
bounds of
$0.009\le {\Omega}_{b}{h}^{2}\le 0.02$
are assumed), a
$68\%$
confidence interval for
${H}_{0}$
of
$$\begin{array}{c}30km{s}^{1}Mp{c}^{1}<{H}_{0}<50km{s}^{1}Mp{c}^{1}\end{array}$$ 
(25)

is obtained. This range ignores the Saskatoon calibration uncertainty. Generally, in the range of parameters of current interest, increasing
${H}_{0}$
lowers the height of the peak. Thus taking the Saskatoon calibration to be lower than nominal, for example by the 14% figure quoted as the onesigma error, enables us to raise the allowed range for
${H}_{0}$
. By this means, an upper limit closer to
$70km{s}^{1}Mp{c}^{1}$
is obtained.
The best angular resolution offered by MAP is 12 arcmin, in its highest frequency channel at 90 GHz, and the median resolution of its channels is more like 30 arcmin. This means that it may have difficulty in pining down the full shape of the first and certainly secondary Doppler peaks in the power spectrum. On the other hand, the angular resolution of the Planck Surveyor extends down to 5 arcmin, with a median (across the six channels most useful for CMB work) of about 10 arcmin. This means that it will be able to determine the power spectrum to good accuracy, all the way into the secondary peaks, and that consequently very good accuracy in determining cosmological parameters will be possible. Figure
19 , taken from the Planck Surveyor Phase A study document, shows the accuracy to which
$\Omega $
,
${H}_{0}$
and
${\Omega}_{b}$
can be recovered, given coverage of 1/3 of the sky with sensitivity
$2\times {10}^{6}$
in
$\Delta T/T$
per pixel. The horizontal scale represents the resolution of the satellite. From this we can see that the good angular resolution of the Planck Surveyor should mean a joint determination of
$\Omega $
and
${H}_{0}$
to
$\sim 1\%$
accuracy is possible in principle. Figure 22 show the likelihood contours for two experiments with different resolutions.
Figure 22
: The contours show 50, 5, 2 and 0.1 percentile likelihood contours for pairs of parameters determined from fits to the CMB power spectrum. The figures to the left show results for an experiment with resolution
${\theta}_{FWHM}={1}^{\circ}$
. Those to the right for a higher resolution experiment with
${\theta}_{FWHM}={10}^{\prime}$
plotted on the same scale (central column) and with expanded scales (rightmost column). This figure is taken from Bersanelli et al. 1996 [
6]
.
These figures do not, however, take into account any reduction in sensitivity as a result of the need to separate Galactic foregrounds from the CMB. Nevertheless, simulations using a maximum entropy separation algorithm (Hobson, Jones, Lasenby & Bouchet, in press) suggest that for the Planck Surveyor the reduction in the final sensitivity to the CMB is very small indeed, and that the accuracy of the cosmological parameters estimates indicated in Figure 19 may be attainable.
One additional problem is that of degeneracy. It is possible to formulate two models with similar power spectra, but different underlying physics. For example, standard CDM and a model with a non zero cosmological component and a gravity wave component can have almost identical power spectra (to within the accuracy of the MAP satellite). To break the degeneracy more accuracy is required (like the Planck Surveyor) or information about the polarisation of the CMB photons can be used. This extra information on polarisation is very good at discriminating between theories but requires very sensitive polarimeters.
Acknowledgements The authors thank all those at MRAO and Jodrell Bank, who are involved in the Tenerife, Ryle Telescope, CAT and VSA projects. Aled Jones wishes to acknowledge a Research Fellowship at King's College, Cambridge, U.K. References

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