CMB Temperature Field

The basic observable of the CMB is its intensity as a function of frequency and direction on the sky $\hat {\bf n}$. Since the CMB spectrum is an extremely good blackbody [Fixsen et al, 1996] with a nearly constant temperature across the sky $T$, we generally describe this observable in terms of a temperature fluctuation $\Theta(\hat {\bf n}) = \Delta T/T$.

If these fluctuations are Gaussian, then the multipole moments of the temperature field

$$\Theta_{\ell m} = \int d{\hat {\bf n}} Y_{\ell m}^*(\hat {\bf n}) \Theta(\hat {\bf n})$$ (1)

are fully characterized by their power spectrum

$$\left< \Theta_{\ell m}^{*} \Theta_{\ell'm'} \right> = \delta_{\ell \ell'}\delta_{m m'} C_{\ell}\,,$$ (2)

whose values as a function of $\ell$ are independent in a given realization. For this reason predictions and analyses are typically performed in harmonic space. On small sections of the sky where its curvature can be neglected, the spherical harmonic analysis becomes ordinary Fourier analysis in two dimensions. In this limit $\ell$ becomes the Fourier wavenumber. Since the angular wavelength $\theta = 2\pi/\ell$, large multipole moments corresponds to small angular scales with $\ell \sim 10^2$ representing degree scale separations. Likewise, since in this limit the variance of the field is $\int d^2 \ell C_\ell/(2\pi)^2$, the power spectrum is usually displayed as

$$\Delta_T^2 \equiv {\ell(\ell+1) \over 2\pi} C_\ell T^2\,,$$ (3)

the power per logarithmic interval in wavenumber for $\ell \gg 1$.

Plate 1 (top) shows observations of $\Delta_T$ along with the prediction of the working cosmological model, complete with the acoustic peaks mentioned in §1 and discussed extensively in §3. While COBE first detected anisotropy on the largest scales (inset), observations in the last decade have pushed the frontier to smaller and smaller scales (left to right in the figure). The MAP satellite, launched in June 2001, will go out to $\ell\sim 1000$, while the European satellite, Planck, scheduled for launch in 2007, will go a factor of two higher (see Plate 1 bottom).

The power spectra shown in Plate 1 all begin at $\ell=2$ and exhibit large errors at low multipoles. The reason is that the predicted power spectrum is the average power in the multipole moment $\ell$ an observer would see in an ensemble of universes. However a real observer is limited to one Universe and one sky with its one set of $\Theta_{\ell m}$'s, $2\ell+1$ numbers for each $\ell$. This is particularly problematic for the monopole and dipole ($\ell=0,1$). If the monopole were larger in our vicinity than its average value, we would have no way of knowing it. Likewise for the dipole, we have no way of distinguishing a cosmological dipole from our own peculiar motion with respect to the CMB rest frame. Nonetheless, the monopole and dipole - which we will often call simply $\Theta$ and $v_\gamma$ - are of the utmost significance in the early Universe. It is precisely the spatial and temporal variation of these quantities, especially the monopole, which determines the pattern of anisotropies we observe today. A distant observer sees spatial variations in the local temperature or monopole, at a distance given by the lookback time, as a fine-scale angular anisotropy. Similarly, local dipoles appear as a Doppler shifted temperature which is viewed analogously. In the jargon of the field, this simple projection is referred to as the freestreaming of power from the monopole and dipole to higher multipole moments.

Table 1: CMB experiments shown in Plate 1 and references.

Name	Authors	Journal Reference
ARGO	Masi S et al. 1993	Ap. J. Lett. 463:L47-L50
ATCA	Subrahmanyan R et al. 2000	MNRAS 315:808-822
BAM	Tucker GS et al. 1997	Ap. J. Lett. 475:L73-L76
BIMA	Dawson KS et al. 2001	Ap. J. Lett. 553:L1-L4
BOOM97	Mauskopf PD et al. 2000	Ap. J. Lett. 536:L59-L62
BOOM98	Netterfield CB et al. 2001	Ap. J. In press
CAT99	Baker JC et al. 1999	MNRAS 308:1173-1178
CAT96	Scott PF et al. 1996	Ap. J. Lett. 461:L1-L4
CBI	Padin S et al. 2001	Ap. J. Lett. 549:L1-L5
COBE	Hinshaw G, et al. 1996	Ap. J. 464:L17-L20
DASI	Halverson NW et al. 2001	Ap. J. In press
FIRS	Ganga K, et al. 1994.	Ap. J. Lett. 432:L15-L18
IAC	Dicker SR et al. 1999	Ap. J. Lett. 309:750-760
IACB	Femenia B, et al. 1998	Ap. J. 498:117-136
QMAP	de Oliveira-Costa A et al. 1998	Ap. J. Lett. 509:L77-L80
MAT	Torbet E et al. 1999	Ap. J. Lett. 521:L79-L82
MAX	Tanaka ST et al. 1996	Ap. J. Lett. 468:L81-L84
MAXIMA1	Lee AT et al. 2001	Ap. J. In press
MSAM	Wilson GW et al. 2000	Ap. J. 532:57-64
OVRO	Readhead ACS et al. 1989	Ap. J. 346:566-587
PYTH	Platt SR et al. 1997	Ap. J. Lett. 475:L1-L4
PYTH5	Coble K et al. 1999	Ap. J. Lett. 519:L5-L8
RING	Leitch EM et al. 2000	Ap. J. 532:37-56
SASK	Netterfield CB et al. 1997	Ap. J. Lett. 477:47-66
SP94	Gunderson JO, et al. 1995	Ap. J. Lett. 443:L57-L60
SP91	Schuster J et al. 1991	Ap. J. Lett. 412:L47-L50
SUZIE	Church SE et al. 1997	Ap. J. 484:523-537
TEN	Gutiérrez CM, et al. 2000	Ap. J. Lett. 529:47-55
TOCO	Miller AD et al. 1999	Ap. J. Lett. 524:L1-L4
VIPER	Peterson JB et al. 2000	Ap. J. Lett. 532:L83-L86
VLA	Partridge RB et al. 1997	Ap. J. 483:38-50
WD	Tucker GS et al. 1993	Ap. J. Lett. 419:L45-L49
MAP	http://map.nasa.gsfc.gov
Planck	http://astro.estec.esa.nl/Planck

How accurately can the spectra ultimately be measured? As alluded to above, the fundamental limitation is set by ``cosmic variance'' the fact that there are only $2\ell+1$ $m$-samples of the power in each multipole moment. This leads to an inevitable error of

$$\Delta C_\ell = \sqrt{2 \over 2 \ell +1} C_\ell \,.$$ (4)

Allowing for further averaging over $\ell$ in bands of $\Delta \ell \approx \ell$, we see that the precision in the power spectrum determination scales as $\ell^{-1}$, i.e. $\sim 1\%$ at $\ell=100$ and $\sim 0.1\%$ at $\ell=1000$. It is the combination of precision predictions and prospects for precision measurements that gives CMB anisotropies their unique stature.

There are two general caveats to these scalings. The first is that any source of noise, instrumental or astrophysical, increases the errors. If the noise is also Gaussian and has a known power spectrum, one simply replaces the power spectrum on the rhs of Equation (4) with the sum of the signal and noise power spectra [Knox, 1995]. This is the reason that the errors for the Planck satellite increase near its resolution scale in Plate 1 (bottom). Because astrophysical foregrounds are typically non-Gaussian it is usually also necessary to remove heavily contaminated regions, e.g. the galaxy. If the fraction of sky covered is $f_{\rm sky}$, then the errors increase by a factor of $f_{\rm sky}^{-1/2}$ and the resulting variance is usually dubbed ``sample variance'' [Scott et al, 1994]. An $f_{\rm sky}=0.65$ was chosen for the Planck satellite.