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4. COSMIC MICROWAVE BACKGROUND ANISOTROPY OBSERVATIONS

Since the COBE DMR detection of CMB anisotropy (Smoot et al., 1992), there have been over thirty additional measurements of anisotropy on angular scales ranging from 7° to 0.°3, and upper limits have been set on smaller scales.

The COBE DMR observations were pixelized into a skymap, from which it is possible to analyze any particular multipole within the resolution of the DMR. Current small angular scale CMB anisotropy observations are insensitive to both high ell and low ell multipoles because they cannot measure features smaller than their resolution and are insensitive to features larger than the size of the patch of sky observed. The next satellite mission, NASA's Microwave Anisotropy Probe (MAP), is scheduled for launch in Fall 2000 and will map angular scales down to 0.°2 with high precision over most of the sky. An even more precise satellite, ESA's Planck, is scheduled for launch in 2007. Because COBE observed such large angles, the DMR data can only constrain the amplitude A and index n of the primordial power spectrum in wave number k, Pp(k) = Akn, and these constraints are not tight enough to rule out very many classes of cosmological models.

Until the next satellite is flown, the promise of microwave background anisotropy measurements to measure cosmological parameters rests with a series of ground-based and balloon-borne anisotropy instruments which have already published results (shown in Figure 4) or will report results in the next few years (MAXIMA, BOOMERANG, TOPHAT, ACE, MAT, VSA, CBI, DASI, see Lee et al., 1999 and Halpern & Scott, 1999). Because they are not satellites, these instruments face the problems of shorter observing times and less sky coverage, although significant progress has been made in those areas. They fall into three categories: high-altitude balloons, interferometers, and other ground-based instruments. Past, present, and future balloon-borne instruments are FIRS, MAX, MSAM, ARGO, BAM, MAXIMA, QMAP, HACME, BOOMERANG, TOPHAT, and ACE. Ground-based interferometers include CAT, JBIAC, SUZIE, BIMA, ATCA, VLA, VSA, CBI, and DASI, and other ground-based instruments are TENERIFE, SP, PYTHON, SK, OVRO/RING, VIPER, MAT/TOCO, IACB, and WD. Taken as a whole, they have the potential to yield very useful measurements of the radiation power spectrum of the CMB on degree and subdegree scales. Ground-based non-interferometers have to discard a large fraction of data and undergo careful further data reduction to eliminate atmospheric contamination. Balloon-based instruments need to keep a careful record of their pointing to reconstruct it during data analysis. Interferometers may be the most promising technique at present but they are the least developed, and most instruments are at radio frequencies and have very narrow frequency coverage, making foreground contamination a major concern. In order to use small-scale CMB anisotropy measurements to constrain cosmological models we need to be confident of their validity and to trust the error bars. This will allow us to discard badly contaminated data and to give greater weight to the more precise measurements in fitting models. Correlated noise is a great concern for instruments which lack a rapid chopping because the 1/f noise causes correlations on scales larger than the beam in a way that can easily mimic CMB anisotropies. Additional issues are sample variance caused by the combination of cosmic variance and limited sky coverage and foreground contamination.

Figure 4

Figure 4. Compilation of CMB Anisotropy observations. Vertical error bars represent 1sigma uncertainties and horizontal error bars show the range from ellmin to ellmax of Table 1. The line thickness is inversely proportional to the variance of each measurement, emphasizing the tighter constraints. All three models are consistent with the upper limits at the far right, but the Open CDM model (dotted) is a poor fit to the data, which prefer models with an acoustic peak near ell = 200 with an amplitude close to that of LambdaCDM (solid).

Table 1. Complete compilation of CMB anisotropy observations 1992-1999, with maximum likelihood DeltaT, upper and lower 1sigma uncertainties (not including calibration uncertainty), the weighted center of the window function, the ell values where the window function falls to e-1/2 of its maximum value, the 1 sigma calibration uncertainty, and references given below.

Instrument Delta T (µK) + 1sigma(µK) - 1sigma(µK) elleff ellmin ellmax 1sigma cal. ref.

COBE1 8.5 16.0 8.5 2.1 2 2.5 0.7 1
COBE2 28.0 7.4 10.4 3.1 2.5 3.7 0.7 1
COBE3 34.0 5.9 7.2 4.1 3.4 4.8 0.7 1
COBE4 25.1 5.2 6.6 5.6 4.7 6.6 0.7 1
COBE5 29.4 3.6 4.1 8.0 6.8 9.3 0.7 1
COBE6 27.7 3.9 4.5 10.9 9.7 12.2 0.7 1
COBE7 26.1 4.4 5.3 14.3 12.8 15.7 0.7 1
COBE8 33.0 4.6 5.4 19.4 16.6 22.1 0.7 1
FIRS 29.4 7.8 7.7 10 3 30 -a 2
TENERIFE 30 15 11 20 13 31 -a 3
IACB1 111.9 49.1 43.7 33 20 57  20 4
IACB2 57.3 16.4 16.4 53 38 75  20 4
SP91 30.2 8.9 5.5 57 31 106  15 5
SP94 36.3 13.6 6.1 57 31 106 15 5
BAM 55.6 27.4 9.8 74 28 97 20 6
ARGO94 33 5 5 98 60 168 5 7
ARGO96 48 7 6 109 53 179 10 8
JBIAC 43 13 12 109 90 128 6.6 9
QMAP(Ka1) 47.0 6 7 80 60 101 12 10
QMAP(Ka2) 59.0 6 7 126 99 153 12 10
QMAP(Q) 52.0 5 5 111 79 143 12 10
MAX234 46 7 7 120 73 205 10 11
MAX5 43 8 4 135 81 227 10 12
MSAMI 34.8 15 11 84 39 130 5 13
MSAMII 49.3 10 8 201 131 283 5 13
MSAMIII 47.0 7 6 407 284 453 5 13

PYTHON123 60 9 5 87 49 105 20 14
PYTHON3S 66 11 9 170 120 239 20 14
PYTHONV1 23 3 3 50 21 94 17b 15
PYTHONV2 26 4 4 74 35 130 17 15
PYTHONV3 31 5 4 108 67 157 17 15
PYTHONV4 28 8 9 140 99 185 17 15
PYTHONV5 54 10 11 172 132 215 17 15
PYTHONV6 96 15 15 203 164 244 17 15
PYTHONV7 91 32 38 233 195 273 17 15
PYTHONV8 0 91 0 264 227 303 17 15
SK1c 50.5 8.4 5.3 87 58 126 11 16
SK2 71.1 7.4 6.3 166 123 196 11 16
SK3 87.6 10.5 8.4 237 196 266 11 16
SK4 88.6 12.6 10.5 286 248 310 11 16
SK5 71.1 20.0 29.4 349 308 393 11 16
TOCO971 40 10 9 63 45 81 10 17
TOCO972 45 7 6 86 64 102 10 17
TOCO973 70 6 6 114 90 134 10 17
TOCO974 89 7 7 158 135 180 10 17
TOCO975 85 8 8 199 170 237 10 17
TOCO981 55 18 17 128 102 161 8 18
TOCO982 82 11 11 152 126 190 8 18
TOCO983 83 7 8 226 189 282 8 18
TOCO984 70 10 11 306 262 365 8 18
TOCO985 24.5 26.5 24.5 409 367 474 8 18
VIPER1 61.6 31.1 21.3 108 30 229 8 19
VIPER2 77.6 26.8 19.1 173 72 287 8 19
VIPER3 66.0 24.4 17.2 237 126 336 8 19
VIPER4 80.4 18.0 14.2 263 150 448 8 19
VIPER5 30.6 13.6 13.2 422 291 604 8 19
VIPER6 65.8 25.7 24.9 589 448 796 8 19

BOOM971 29 13 11 58 25 75 8.1 20
BOOM972 49 9 9 102 76 125 8.1 20
BOOM973 67 10 9 153 126 175 8.1 20
BOOM974 72 10 10 204 176 225 8.1 20
BOOM975 61 11 12 255 226 275 8.1 20
BOOM976 55 14 15 305 276 325 8.1 20
BOOM977 32 13 22 403 326 475 8.1 20
BOOM978 0 130 0 729 476 1125 8.1 20
CAT96I 51.9 13.7 13.7 410 330 500 10 21
CAT96II 49.1 19.1 13.7 590 500 680 10 21
CAT99I 57.3 10.9 13.7 422 330 500 10 22
CAT99II 0. 54.6 0. 615 500 680 10 22
OVRO/RING 56.0 7.7 6.5 589 361 756 4.3 23
HACME 0. 38.5 0. 38 18 63 -a 29
WD 0. 75.0 0. 477 297 825 30 24
SuZIE 16 12 16 2340 1330 3070 8 25
VLA 0. 27.3 0. 3677 2090 5761 -a 26
ATCA 0. 37.2 0. 4520 3500 5780 -a 27
BIMA 8.7 4.6 8.7 5470 3900 7900 -a 28

REFERENCES: 1- Kogut et al. (1996a); Tegmark & Hamilton (1997) 2- Ganga et al. (1994) 3- Gutierrez et al. (1999) 4- Femenia et al. (1998) 5- Ganga et al. (1997b); Gundersen et al. (1995) 6- Tucker et al. (1997) 7- Ratra et al. (1999) 8- Masi et al. (1996) 9- Dicker et al. (1999) 10- De Oliveira-Costa et al. (1998) 11- Clapp et al. (1994); Tanaka et al. (1996) 12- Ganga et al. (1998) 13- Wilson et al. (1999) 14- Platt et al. (1997) 15- Coble et al. (1999) 16- Netterfield et al. (1997) 17- Torbet et al. (1999) 18- Miller et al. (1999) 19- Peterson et al. (1999) 20- Mauskopf et al. (1999) 21- Scott et al. (1996) 22- Baker et al. (1999) 23- Leitch et al. (1998) 24- Ratra et al. (1998) 25- Church et al. (1997); Ganga et al. (1997a) 26- Partridge et al. (1997) 27- Subrahmanyan et al. (1993) 28- Holzapfel et al. (1999) 29- Staren et al. (1999)
a Could not be determined from the literature.
b Results from combining the +15% and -12% calibration uncertainty with the 3µK beamwidth uncertainty. The non-calibration errors on the PYTHONV datapoints are highly correlated.
c The SK DeltaT and error bars have been re-calibrated according to the 5% increase recommended by Mason et al. (1999) and the 2% decrease in DeltaT due to foreground contamination found by De Oliveira-Costa et al. (1997).

Figure 4 shows our compilation of CMB anisotropy observations without adding any theoretical curves to bias the eye 2. It is clear that a straight line is a poor but not implausible fit to the data. There is a clear rise around ell = 100 and then a drop by ell = 1000. This is not yet good enough to give a clear determination of the curvature of the universe, let alone fit several cosmological parameters. However, the current data prefer adiabatic structure formation models over isocurvature models (Gawiser & Silk, 1998). If analysis is restricted to adiabatic CDM models, a value of the total density near critical is preferred (Dodelson & Knox, 1999).

4.1. Window Functions

The sensitivity of these instruments to various multipoles is called their window function. These window functions are important in analyzing anisotropy measurements because the small-scale experiments do not measure enough of the sky to produce skymaps like COBE. Rather they yield a few "band-power" measurements of rms temperature anisotropy which reflect a convolution over the range of multipoles contained in the window function of each band. Some instruments can produce limited skymaps (White & Bunn, 1995). The window function Well shows how the total power observed is sensitive to the anisotropy on the sky as a function of angular scale:

Equation 13 (13)

where the COBE normalization is DeltaT = 27.9 µK and TCMB = 2.73 K (Bennett et al., 1996). This allows the observations of broad-band power to be reported as observations of DeltaT, and knowing the window function of an instrument one can turn the predicted Cell spectrum of a model into the corresponding prediction for DeltaT. This "band-power" measurement is based on the standard definition that for a "flat" power spectrum, DeltaT = (ell(ell + 1) Cell)1/2 TCMB / (2pi) (flat actually means that ell(ell + 1) Cell is constant).

The autocorrelation function for measured temperature anisotropies is a convolution of the true expectation values for the anisotropies and the window function. Thus we have (White & Srednicki, 1995)

Equation 14 (14)

where the symmetric beam shape that is typically assumed makes Well a function of separation angle only. In general, the window function results from a combination of the directional response of the antenna, the beam position as a function of time, and the weighting of each part of the beam trajectory in producing a temperature measurement (White & Srednicki, 1995). Strictly speaking, Well is the diagonal part of a filter function Well ell' that reflects the coupling of various multipoles due to the non-orthogonality of the spherical harmonics on a cut sky and the observing strategy of the instrument (Knox, 1999). It is standard to assume a Gaussian beam response of width sigma, leading to a window function

Equation 15 (15)

The low-ell cutoff introduced by a 2-beam differencing setup comes from the window function (White et al., 1994)

Equation 16 (16)

4.2. Sample and Cosmic Variance

The multipoles Cell can be related to the expected value of the spherical harmonic coefficients by

Equation 17 (17)

since there are (2ell + 1) aellm for each ell and each has an expected autocorrelation of Cell. In a theory such as inflation, the temperature fluctuations follow a Gaussian distribution about these expected ensemble averages. This makes the aellm Gaussian random variables, resulting in a chi22ell+1 distribution for summ aellm2. The width of this distribution leads to a cosmic variance in the estimated Cell of sigmacv2 = (ell + 1/2)-1/2 Cell, which is much greater for small ell than for large ell (unless Cell is rising in a manner highly inconsistent with theoretical expectations). So, although cosmic variance is an unavoidable source of error for anisotropy measurements, it is much less of a problem for small scales than for COBE.

Despite our conclusion that cosmic variance is a greater concern on large angular scales, Figure 4 shows a tremendous variation in the level of anisotropy measured by small-scale experiments. Is this evidence for a non-Gaussian cosmological model such as topological defects? Does it mean we cannot trust the data? Neither conclusion is justified (although both could be correct) because we do in fact expect a wide variation among these measurements due to their coverage of a very small portion of the sky. Just as it is difficult to measure the Cell with only a few aellm, it is challenging to use a small piece of the sky to measure multipoles whose spherical harmonics cover the sphere. It turns out that limited sky coverage leads to a sample variance for a particular multipole related to the cosmic variance for any value of ell by the simple formula

Equation 18 (18)

where Omega is the solid angle observed (Scott et al., 1994). One caveat: in testing cosmological models, this cosmic and sample variance should be derived from the Cell of the model, not the observed value of the data. The difference is typically small but will bias the analysis of forthcoming high-precision observations if cosmic and sample variance are not handled properly.

4.3. Binning CMB data

Because there are so many measurements and the most important ones have the smallest error bars, it is preferable to plot the data in some way that avoids having the least precise measurements dominate the plot. Quantitative analyses should weight each datapoint by the inverse of its variance. Binning the data can be useful for display purposes but is dangerous for analysis, because a statistical analysis performed on the binned datapoints will give different results from one performed on the raw data. The distribution of the binned errors is non-Gaussian even if the original points had Gaussian errors. Binning might improve a quantitative analysis if the points at a particular angular scale showed a scatter larger than is consistent with their error bars, leading one to suspect that the errors have been underestimated. In this case, one could use the scatter to create a reasonable uncertainty on the binned average. For the current CMB data there is no clear indication of scatter inconsistent with the errors so this is unnecessary.

If one wishes to perform a model-dependent analysis of the data, the simplest reasonable approach is to compare the observations with the broad-band power estimates that should have been produced given a particular theory (the theory's Cell are not constant so the window functions must be used for this). Combining full raw datasets is superior but computationally intensive (see Bond et al. 1998a). A first-order correction for the non-gaussianity of the likelihood function of the band-powers has been calculated by Bond et al. (1998b) and is available at http://www.cita.utoronto.ca/~knox/radical.html.



2 This figure and our compilation of CMB anisotropy observations are available at http://mamacass.ucsd.edu/people/gawiser/cmb.html; CMB observations have also been compiled by Smoot & Scott (1998) and at http://space.mit.edu/home/tegmark/cmb/experiments.html and
http://www.cita.utoronto.ca/~knox/radical.html Back.

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