{'fraction_non_alphanumeric': 0.03746322893234123, 'fraction_numerical': 0.019661706177539366, 'mean_word_length': 4.3368348147293085, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 3, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
On Observing The Cosmic Microwave Background

The cosmic microwave background (CMB) comprises the oldest photons in the universe and is arguably our most direct cosmological observable. All precise and accurate measurements of its attributes serve to distinguish between cosmological models. Detector technology and observing techniques have advanced to the point where fluctuations in the CMB of order a few microkelvin are measured almost routinely. In these lecture notes, we review recent measurements of both the absolute temperature and the anisotropy of the CMB and discuss the relation between the data and the general theoretical framework. Future directions are indicated and the upcoming satellite experiments are discussed.


Introduction
The CMB is a powerful probe of cosmology because essentially no steps separate what is measured from what is of cosmological import; what you see is what you get. The CMB photons have free-streamed through the cosmos since last scattering off electrons some 100,000 years after the big bang. The spectrum of the CMB is indistinguishable from a Planck function to roughly 0.01% accuracy. This tells us that there were not any highly energetic cosmic processes, that coupled to photons, before z ≈ 10 3 . The near perfect shape of the spectrum is the strongest evidence to date that the hot big-bang model is correct. The pattern of minute spatial variations or anisotropy in the CMB, which are of order δT /T ≈ 10 −5 , is a fossil of the early universe. Furthermore, most models that give rise to cosmic structure affect the CMB, leaving an imprint of a small temperature fluctuation.
There are many review articles on both the spectrum of and anisotropy in the CMB. For the anisotropy see [1]- [6]; for a recent review of both the spectrum and anisotropy results see [7] and [8]; and for reviews of the theory and results on the spectrum see [9], [10], or [11]. In addition, Partridge has written a new book [12] devoted to the subject.
The outline for these notes is as follows. We discuss the microwave/farinfrared sky in Section 2. Next, in Section 3, we discuss the recent results of absolute temperature measurements of the CMB. After a model of the anisotropy is developed in Section 4, Section 5 provides a discussion of the measurement technologies and an overview of the canonical formalism for describing the anisotropy. In Section 6, we review the state of the field. We end with a discussion of the upcoming satellite missions in Section 7.

The Microwave/Far-Infrared Sky
The CMB is the brightest broad-band diffuse emitter in the sky between about 1 and 500 GHz, completely dominating the Galactic foreground emission. At the low frequency end of this range, Galactic synchrotron emission exceeds the CMB and at the high frequency end, interstellar dust emission dominates. Galactic bremsstrahlung, or free-free emission, is the largest foreground near 90 GHz. Figure 1 shows the frequency spectrum of the diffuse Galactic emission between 3 and 3000 GHz near a Galactic latitude of 20 • . There are two sources not shown on this plot. Near 3000 GHz, thermal emission from the interplanetary dust (Zodiacal light) is roughly ten times smaller than the interstellar dust and its brightness scales with frequency as ν 4 . Throughout the plotted range, interstellar molecular line emission is also observed [13].
In addition to the diffuse foreground, galactic and extragalactic pointlike objects such as quasars, blazars, gigahertz-peaked sources, and loud radio sources also emit microwave and far-infrared radiation. The spectrum of most sources falls with increasing frequency, though not of every source. 1 Relatively little is know about sources near 90 GHz. There are no deep unbiased surveys from which to ascertain the ensemble properties with confidence. Also, we know that many of the extragalactic high-frequency sources are variable.
In Figure 1 it is evident that to probe either the spectrum or the anisotropy to a part in 10 3 or 10 5 the foreground emission must be confronted. Before COBE [14], the best full-sky maps were the "Haslam et al." map at 408 MHz [15] and the IRAS dust map, with Zodiacal light subtracted, at 3000 GHz (100 µm) [16]. Extrapolation of these maps to fre- quencies of interest is problematic because the spectral index varies from place to place on the sky and there are components, such as free-free and cold dust emission, that are missing from these maps. A believable CMB spectrum or anisotropy experiment must measure the CMB and foreground emission to similar precision. In addition to the frequency dependence of the foreground emission, there is also a spatial dependence. At the largest angular scales, Galactic emission falls off with Galactic latitude roughly as 1/ sin(b). In other words, its intensity distribution, to first order, resembles a quadrupole: hot around a circle, cold near the poles. Indeed this quadrupole confounded some of the early measurements of the anisotropy and its removal from the COBE/DMR data requires ingenuity [17].
The spatial distribution of celestial sources is commonly quantified with the angular spectrum. This formalism is used to describe the anisotropy, radio sources, and the diffuse foreground. It allows a direct comparison of the contribution from each. To illustrate it, the relatively simple case of foreground emission by extra-galactic radio sources is considered. We assume radio sources are randomly placed on the sky and so the angular spectrum should be that of "white noise." If we measure the total source emission temperature in each sky pixel, then that distribution may be represented as an expansion in spherical harmonics, wherex is the unit vector in some direction. At this time we will ignore the effects of finite measurement resolution; this will be considered later. The correlation function (or two-point function) of the temperatures is defined as and where the diagonal bracket indicates an ensemble average over many universes. For small angular scales, the ensemble average can be replaced by an average over positions in our universe. For a Gaussian random field, all the information is contained in the two-point correlation function (eq. 2). We do not know if the CMB is a Gaussian field, and we know that Galactic emission is certainly not. However, we still use the two-point function; a more complete description would use the higher-point correlation functions. Because C(θ) depends only on the angular separation between two directions, it may be expanded in Legendre polynomials as Here the brackets denote averaging over the 2l + 1 values of m. The variance of the pixel temperatures is given by C(0). Because P l (0) = 1, the variance in each "mode" l is just (2l + 1)C l /4π. For "white noise," C l = const. We may see this by considering that the magnitude squared of the Fourier transform of a uniform distribution of sources will also be a uniform distribution in 2-D Fourier space; there is no preferred scale. Call the uniform value U . For small angles, the conjugate variable to angular separation is l + 1/2, where l + 1/2 ≈ l = 1/δθ where δθ is the angular scale. Now, the variance for a band of modes of width δl at a radius l is given by the area in Fourier space or U 2πlδl. In other words, the C l ∝ U = const.
To visualize the angular spectrum ("power spectrum" is usually reserved for the C l ), one generally plots (l(2l + 1)C l /4π) 1/2 = δT l versus l. The Figure 2. Spatial spectrum of the foregrounds from Netterfield et al. [18] and references therein. This is for the region near the North Celestial Pole at a frequency of 40 GHz. At higher frequencies, the flux from dust increases and that from synchrotron and radio sources decreases. The curved line is for standard CDM (from [19]). extra factor of l means that δT l is the root mean squared fluctuation per logarithmic interval, δl/l. The results from such an analysis for radio sources and Galactic emission at 40 GHz are shown in Figure 2. From Figures 1 and 2, it is clear that the foreground emission must be well understood before the absolute CMB spectrum is known to 0.01% near 3 GHz or before the angular spectrum is known to the few percent level. One mitigating factor for the anisotropy is that the foreground fluctuations add to the signal in quadrature in the approximation that they are random fields. The absolute measurements do not enjoy this benefit. We are a long way from understanding the Galactic emission and radio sources. For a thorough up-to-date assessment, see Tegmark and Efstathiou [20].

Measurements of the Spectrum
The spectrum of the CMB is as close to that of a blackbody as can be measured; no distortions have been detected. This is our best evidence that the universe went through a hot dense phase when everything that interacts with photons was in thermal equilibrium. In current models, the epoch of photon production ended at z ≈ 3×10 6 , when the universe was roughly two months old. In the subsequent expansion, an injection of energy mediated by baryons would distort the spectrum, though the injection would have to be large (or efficient) to be detectable today because there are roughly 10 9 photons per baryon. Thus a study of the spectrum is a study of the history of cosmic energetics.
The FIRAS experiment [21] aboard the COBE [14] satellite measured the flux from the sky between 2 and 96 cm −1 (60 − 2880 GHz). Fixsen and colleagues give the most recent results [22] and discuss the exhaustive program of systematic checks and instrument calibration [23]. The FIRAS team finds that the flux is described by the Planck function with a temperature of One doubts that short of another satellite-based experiment this result will be matched or bettered at frequencies above 100 GHz. It is comforting that the UBC rocket experiment [24] gives a consistent result.
The error on the FIRAS result, 4 mK, is due entirely to systematic effects and Fixsen et al. interpret it as a 95% confidence limit. (The statistical error is 7 µK.) The error exists to tell the readers how confident the authors are in the results. It should not be interpreted in the sense that "if one hundred experiments were performed only ten would lie outside the error bounds." When interpreting this result, one must bear in mind that it comes from a model of the data. If the model is not correct, the results must be re-interpreted. To be more specific, the analyzed data come from maps of the sky in multiple frequency bands between 2 and 21 cm −1 . A four parameter fit is made to the maps at |b| > 5 • at each frequency, The fit parameters scale the spatial distributions of a uniform background, the CMB dipole, channel 9 from the DIRBE experiment (D9, 72 cm −1 ) and channel 10 from the DIRBE experiment (D10, 42 cm −1 ). The last two maps are found to be good measures of the interstellar dust distribution. Two maps are needed because there may be multiple dust components or, in an explanation that Fixsen [25] prefers, the dust temperature may be a function of position.
The set of coefficients of the uniform component, α 0 (ν), is then fit to a combination of four frequency distributions. They are a) a blackbody, b) the derivative of a blackbody (to fit an error to the temperature scale), c) the Figure 3. Residuals of the fit to α 0 (ν) in eq. 5 from [22]. If one starts with the α 0 (ν) in eq. 5 and subtracts a blackbody (curve not shown), a small calibration correction term (0.0001 dB/dT, where B(ν) is the Planck function), and a model of the Galactic spectrum, the solid line is obtained. The Galactic spectrum is shown scaled to 1/4 its value at the galactic poles. In other words, if a Galactic-type spectrum were not subtracted, the spectrum of the uniform sky map would rise with frequency. The peak of the CMB is near ν = 5.5 cm −1 = 165 GHz. The intensity measured there is roughly 385 MJy/sr. The µ and y distortions are shown at the 95% CL values in eq. 6. The units on the abscissa are converted to GHz by multiplying by 30.
spectrum of the Galaxy (that accounts for residual Galactic signal in the monopole!), and d) a spectral distortion. The distortion is parameterized by either a chemical potential µ, or a Compton y−factor. Only one distortion is fit at a time because the spectral signatures of the two are anti-correlated. The residuals to this fit and some of the basis functions are shown in Figure  3.
Before z ≈ 3 × 10 6 , double Compton scattering and free-free emission maintain the thermal equilibrium of the CMB with the surroundings by creating photons. An energy input simply results in a hotter CMB temperature. Between 10 5 < z < 3 × 10 6 , single Compton scattering, which conserves the number of photons, is the dominant scattering mechanism over most of the frequency spectrum. In this epoch, the CMB is in statistical equilibrium with its surroundings and the distribution is characterized by a chemical potential µ. (The quoted numbers are for the unitless chemical potential; the flux is S ν (T, µ) = 2hν 3 /[exp(hν/kT CM B + µ) − 1]). At long wavelengths, free-free emission is still effective and "fills in the tail" of the distribution. For z < 10 5 , hot electrons, which are neither in statistical nor in thermal equilibrium with the CMB, can inverse Compton scatter the CMB photons to produce a Compton y distortion. When there are relatively few scattering events, one may think of y as the average fractional energy change per scattering event times the average number of scatterings [26], or y = 1/m e c 2 [k(T e − T CM B )]dτ e [9] where T e is the electron temperature and τ e is the optical depth due to scattering. Finally, if the universe is ionized at z < 10 3 then there may be enough free-free emission from the plasma to increase the photon occupation number at large λ and thus the temperature there. This distortion is parameterized by with T ef f the plasma temperature. These and other distortions, along with their interpretation, are discussed in [9], [11], [27], [4].
The best limits on y and µ come from FIRAS [22]. From these, Wright et al. [27] constrain energy injection in the early universe as shown in Figure  4. The limit on Y f f [28] comes from a fit of the low frequency data. The limits are: The y distortion is manifest at high frequencies and it will be a long time before the FIRAS limit is improved. The limit on y also strongly constrains alternative models of the origin of the CMB. One may try to mimic a Planck spectrum with a superposition of multiple grey bodies. At long wavelengths, the Rayleigh-Jeans region, the results cannot be distinguished. However, near the peak of the spectrum, such a superposition will result in a ydistortion. We also note that if the universe is inhomogenous on the largest scales, and we are not at a preferred center, a distortion will result [29].
The signatures of any µ and Y f f distortions are evident at low frequencies. While the current generation of experiments will just barely, if at all, improve on the FIRAS limits, they are paving the way for the next genera- tion which may detect a distortion. Figure 5 shows a plot of the spectrum along with the µ and Y f f distortion limits. Two groups [30], [31] are pursuing long-wavelength measurements of the spectrum. Outside of the precise instrumentation necessary to perform absolute measurements between 0.1% and 1%, one must contend with Galactic and atmospheric emission. At 600 MHz, the temperature of the Galaxy is roughly 5.8 K [34] and falls as ν −2.7 . In Figure 1, this is roughly where the "synchrotron" and "CMB" lines cross. Observing from the ground, the atmosphere emits at roughly 1.5 K between 0.6 GHz and 1.4 GHz [34], [35]. Ground-based experiments thus require precise modeling and sky dips to subtract the atmospheric signal. Compact experiments may be flown from balloons to rise above it. Finally, in the not-too-distant future, narrow-band measurements will be limited by the FIRAS error when characterizing distortions. Figure 5. Plot of the long wavelength (low frequency) distortions to the CMB. The flat line is for a perfect blackbody. The curve with a pronounced minimum near 700 MHz shows a µ distortion with µ = 9 × 10 −5 , the COBE limit. The curve that begins to rise near 10 GHz is for a Y f f distortion with Y f f = 1.5 × 10 −5 . All data with error bars small enough to fit on this plot are shown. The measurement at 10.7 GHz comes from Staggs et al. [30]. Note the three FIRAS data points near 100 GHz. This plot was adapted from a similar plot made by Al Kogut.

The Anisotropy
The photons that end their lives in our detectors were last scattered off electrons when the universe had a temperature of 5000 K and was evolving from a plasma to a state of neutral hydrogen and helium plus a thermal background 2 . This era is called the epoch of decoupling. The plasma was responding to the gravitational potential wells that would eventually foster galaxies and clusters of galaxies. The photons, now decoupled, bring to us an imprint of the potential wells and a signature of the dynamics of the plasma's response to the wells. From the angular spectrum of the fluctuations, one can distinguish among various possible mechanisms of structure formation. In recent years, it has become evident that for a certain class of models (eg. standard CDM 3 ), a measurement of the detailed shape of the power spectrum will yield values for many of the cosmological parameters 4 such as Ω 0 , Ω B , H 0 , and Λ [36]. Figure 6 shows the angular spectra for a few of the many models of structure formation. The point of the plot is to indicate that the model predictions are rather different and that given measurements with uncertainties of order the width of the plot line, the best model could be identified.
We divide the spectrum into three regions. At l < 80, or large angular scales (δθ > 2 • , l ≈ 180/δθ with δθ in degrees), separate regions of the sky are not causally connected at decoupling. Indeed, the relative isotropy at these scales was one of the motivations for the inflation model. If inflation is correct, these potential wells and hills are the manifestation of quantum fluctuations that were superluminally expanded beyond the Hubble radius, grew with the expansion of the universe, and then re-entered the Hubble radius at a later time. The largest scales entered the Hubble radius most recently. In 10 billion years, a new CMB quadrupole will come into view as the universe expands and what we now call the quadrupole will be distributed among the higher moments. At these large scales, the anisotropy is produced by photons climbing out of the gravitational potential wells or sliding down the hills, just after their last scattering event, as discovered by Sachs and Wolfe. The anisotropy in this region reflects the primordial power spectrum of fluctuations, P(k). One of the strengths of inflation is that it predicts the shape of this spectrum 5 . At large angular scales, C l ∝ 1/l(l + 1), in other words, a nearly flat line in Figure 6.
At smaller angular scales, greater than l ≈ 80, there was time for the primordial plasma to communicate. Hu and colleagues [6], [5] have presented an intuitive physical picture of the mechanisms behind the anisotropy although models date back to Silk [39], Sunyaev and Zel'dovich [40], Peebles & Yu [41] and others. To first order, we may think of the plasma as a photon-baryon fluid that acoustically oscillates in response to fluctuations in the gravitational potential produced by the dark matter. As the fluid flows into a potential well, it heats up (the phase is opposite to that of Figure 6. Angular spectrum for a sample of anisotropy models. The curves grouped on the left hand side are the predictions of isocurvature models from Peebles [37]. The curves that peak on the right are for the standard and open bubble inflation models (Ratra and Sugiyama [38], [19]) with Ω 0 = 1.0, 0.4, 0.3, 0.2 and 0.1 for peaks going from left to right.
the Sachs-Wolfe effect). This is responsible for the large peak in the power spectrum which occurs at roughly the angular scale of the largest potential that can support plasma oscillations. Oscillations of the fluid in response to fluctuations at smaller scales give rise to the other peaks. A full description accounts for the doppler effect, the self gravity of the photon-baryon fluid, the inertia of the baryons, and the evolution of the fluid with time.
Because decoupling happened so fast, δz/z ≈ 0.1, the thickness of the surface of last scattering is less than the horizon size at z ≈ 1400. The scale of the fluctuations near the first peak is of order the horizon size, about 100 Mpc in comoving units 6 , and we observe the full signature of the potential wells there. On smaller angular scales, corresponding to smaller physical scales, there are more fluctuations contained within the horizon and their effects average out. In addition, diffusion of photons out of the potential wells diminishes the temperature fluctuations. The combination of these effects leads to the suppressed anisotropy near l = 2000 as can be seen in Figure 6.
Cosmological parameters are extracted from the shape of the power spectrum. For standard CDM type models this is done, for instance, by noting that the distance between the peaks depends on the sound speed at decoupling. This in turn depends on h 2 Ω B . Also, as Ω B increases, the photon-baryon fluid has more inertia (the sound speed decreases) and the compressional peaks (the odd ones starting at l = 200) get bigger. On the other hand, if the universe goes through a period of reionization, all the peaks in the anisotropy can be wiped out. Extracting generic cosmological information, regardless of model, is an active area of research [42].
One particularly nice demonstration of what the anisotropy can tell us was noted by Kamionkowski et al. [43]. The location in l of the first peak is a good indicator of Ω 0 + Ω Lambda . This happens because a universe of any geometry, in its early stages, evolves as though Ω 0 = 1. Because the physical size of a fluctuation depends on the sound horizon at decoupling, it is a "standard yardstick." The angular size of the standard yardstick, as viewed today, depends on the overall geometry of the universe. If the universe is flat, it will appear at 2 • , if it is open it will appear at a smaller angle; this last is because there is "more space" far away. This effect is seen in the CDM models in Figure 6. The lower Ω 0 , the further to the right the peak moves. The scaling is roughly l peak ≈ 200/Ω 1/2 0 . In models of the formation of large scale structure, the same potential fluctuations that produce the anisotropy also produce the large scale clustering of galaxies. We still do not know what type of mass comprises galaxies, or how it couples to gravitational fluctuations, or even what produced the fluctuations. In the past few years, a number of large galaxy surveys have been undertaken and older surveys have been re-analyzed. Over a region of l-space between l = 30 and l = 600, there is overlap between the two probes of the fluctuations: galactic surveys and CMB anisotropy data. There is a nice plot of this in White, Scott, & Silk's review [3]. Unfortunately, the data are not sufficiently good, from either probe, to draw firm conclusions. However, the indications are that the simplest models for the formation of structure are incorrect.

TECHNOLOGIES
The desire to characterize the anisotropy has led to improvements in detectors and instrument technology. For the anisotropy, three classes of detec-tor are now in use: HEMT based amplifiers , superconductorinsulator-superconductor (SIS) based mixers , and a variety of types of bolometers (90-1000 GHz). Anisotropy measurements require very stable observing conditions. This is especially true for "configuration-space" measurements, as opposed to interferometric measurements. To overcome atmospheric fluctuations, the primary culprit, experiments are performed at stable or high sites (Mauna Kea, South Pole, Owens Valley, Saskatoon, & Tenerife) and from balloon platforms. Plans are underway for long duration balloon flights that circumnavigate the Antarctic and for balloons that can stay aloft for 100 days. Interferometers, which currently operate mostly below 30 GHz, are intrinsically less sensitive to the atmosphere.
The HEMT amplifiers for most of the HEMT-based experiments were designed by Marian Pospieszalski at the National Radio Astronomy Observatory (NRAO) electronic research lab. One couples celestial radiation to the amplifiers with waveguide. The incident electric field is simply amplified to a reasonable level and then detected with a diode. The amplifiers are special because they are rugged, require only modest cooling (≈ 20 K), and have a 30% bandwidth. A good wide-band (10 GHz) sensitivity for an optimized system near 40 GHz is roughly 500 µKs 1/2 . In other words, one can detect 1/2 mK signal with a signal-to-noise of one with one second of integration. A number of semi-conductor groups are pushing to make better high-frequency transistors (for instance TRW and Hughes) and to make entire radiometers on a single chip (TRW and Lockheed-Martin).
SIS-based systems have been used for a number of years (Timbie [44], Meinhold [45], Robertson [46]) though the anisotropy has not yet been detected with them. The currently favored designs, and devices, come from Anthony Kerr and S-K Pan at NRAO. The SIS is a mixer. It is simultaneously illuminated with celestial radiation and with a "local oscillator." The output of the SIS is a signal containing the sum and difference frequencies of the LO and sky. The signal is low-pass filtered, amplified with HEMTs, and detected. For a system with a LO at 144 GHz and a 4 GHz IF bandwidth, a reasonable sensitivity is 400 µKs 1/2 . These devices must operate below the superconducting transition of niobium, or below roughly 4.2 K. They are more difficult than HEMTs to operate but are still straightforward.
Both HEMT and SIS systems are coherent; in other words, one works with the electric field right up until the final detection. This allows possibilities for phase sensitive techniques such as correlation receivers and interferometers. In addition they both have a very large audio bandwidth. Signals at many megahertz are easily detected.
The current bolometers are essentially thermistors held at 0.3 K or below. When they are illuminated, they heat up and change resistance, and the change in resistance is electronically read out. This is called incoher-ent detection because the phase information is lost. The great advantage to bolometers is that they are very sensitive. A decade ago, they were near 400 µKs 1/2 [47], and some current devices (the "spider" bolometers, Bock et al. [48]) achieve better than 100 µKs 1/2 . The disadvantage is that they are more difficult to use than HEMTs and SISs and their intrinsic time constants are longer, but both of these problems are actively being worked on. There are different types of bolometers in various stages of development. They include frequency sensitive bolometers [49], hot electron bolometers [50], monolithic silicon bolometers [51], spider web composite bolometers [48], and transition edge bolometers [52].
The bolometer's advantage is not its intrinsic sensitivity per photon, which is comparable to that of HEMTs or SIS, but rather the fact that one can detect in multiple electromagnetic modes (in conventional use, waveguide supports just one mode) and almost arbitrary bandwidth. In other words, bolometers detect more photons than single mode waveguide systems. The power on a device is where S ν (T ) is the flux from some source, ν is the RF frequency, A is the area of the detector (or antenna), and Ω is the solid angle incident on that detector (or antenna). The quantity dAdΩ is called the throughput oŕ etendue. Generally, AΩ = nλ 2 where λ is the wavelength at the passband center and n is the number of modes. For a single mode system, n = 1, and we get the quantity on the right of the arrow, where T (ν) is the effective temperature of the source.

TECHNIQUES
The experimental challenge is to measure a variance of a random field, which is of order 30 µK, with a noisy detector and a background signal of 300 K. As much effort has gone into determining robust ways to do this as has gone into understanding the detector systems. The configurationspace techniques are the better developed so we will focus on those. The interferometric techniques are rapidly maturing and have a lot of promise; the first detection of the anisotropy with an interferometer was just reported in Scott et al. [53].
A typical telescope has a beam described by P (θ) ∝ exp(−θ 2 /2σ 2 ). If the telescope observes the sky which has temperature T (x), it measures t(x) = P (θ)T (x)dx. If two measurements made near each other, t(x 1 ) and t(x 2 ) are subtracted, the common atmospheric signal drops out. One is then sensitive to only the gradient in the atmosphere and to the CMB temperature difference. If two single differences, with one position in common, are performed and then subtracted the "double difference" is s i = 2t(x 1 )−t(x 2 )−t(x 3 ). The double difference is still sensitive to the CMB but is insensitive to atmospheric gradients as well as the atmospheric temperature. This extra difference helps because the fluctuation spectrum of the atmosphere drops with both increasing spatial and spectral frequency. Let us call the effective beam profile for the double difference measurement H i (x); it will have one large positive central lobe and two negative lobes as shown in Figure 7. We can write We will consider this a single measurement at a pixelx. In practice, we make a set of N similar measurements over a patch of sky. For a first order estimate, we may find the intrinsic variance of the sky from where σ 2 data is the square of the average statistical error per measurement and σ meas is the variance of the N data points. This answer is usually only correct to 25% so it is used as a sanity check. Also, it does not give the correct error. Note too that this method ignores all intrinsic correlations in the data.
What variance does one expect from such a measurement? After working through the math, we find that the inclusion of finite beams modifies eq. 3 to the following: (10) This is the full theoretical covariance matrix for the observing pattern for measurements i and j. 7 W l is called the window function, it tells us the portion of l-space being examined. Generally, just the diagonal elements are plotted. For the beam in Figure 7, the window function is shown in Figure 8. When i = j, we get a prediction for σ 2 sky . The most frequently used analyses follow Bond's work [4]. A complete analysis requires knowledge of the covariance matrix of the data; we call this C D . The full theory-plus-data covariance matrix is given by M = C D + C T . Our goal is to determine the probability of a theory given the data, P (T |D).
To do this, we use Bayes's theorem with a uniform prior, P(T) = 1, and set the probability of getting the data, P(D), equal to 1: where L is the likelihood function and t is a vector of the data. The argument of the exponent is proportional to χ 2 . This boils down to saying that the likelihood of the data plotted as a function of some parametrization, for instance σ sky , is the probability of obtaining σ sky with a given set of data.
When the signal to noise is high, the likelihood is fairly Gaussian. To get an error, we find bounds symmetric around the maximum, that contain 68% of the area under the likelihood curve. To estimate the angular spectrum, we take the most likely value of σ sky with its error, convert it into a "band-temperature" 8 and plot it at the l-weighted center of the window function. A horizontal error bar is often plotted indicating the width of the window function. Any one experiment observes in multiple windows and so the spectrum may be mapped out. One example is given in Figure 9 from the Saskatoon experiment [18].

Anisotropy Measurements: The Current Results and Immediate Future
The anisotropy of the CMB was first unambiguously measured by the DMR experiment aboard the COBE satellite [57] using 7 • resolution full-sky maps at 30, 53, and 90 GHz. To date, these are still the cleanest and best checked data. All indications are that the fluctuations are thermal, though I am not aware of any formal limits on, say, the Compton y parameter of the anisotropy. When smoothed to a 10 • resolution, the rms of the temperature fluctuations is about 30 µK. This is the canonically quoted value. However, Figure 9. Results from the Saskatoon experiment [71], [18]. The spectra from six theories are also shown. From top to bottom at l = 160 they are a flat Λ+CDM model with Ω Λ = 0.7 [38], Standard CDM [19], a PPI model [62], an Ω 0 = 0.4 open bubble model [19], a texture model [63], and a model with reionization [64]. There is a 14% overall calibration uncertainty that is not included in the error bars. This affects the normalization of the spectrum, but not the shape.
with a 1/2 • resolution map, the rms is closer to 90 µK. The final DMR results (the satellite is now turned off) are published in Ap.J Vol 464, 1996 [58], [17], [59], [60], & [61]. The lasting contribution will be the maps of the sky at 30, 53, and 90 GHz. A combination of these maps, optimized to give the anisotropy, has a signal-to-noise of two per 10 • × 10 • pixel. The two primary results derived from these maps are: 1. From a fit of the data to a power spectrum parameterized by the spatial index and the quadrupole amplitude, the DMR team finds n DM R = 1.21 ± 0.3 and Q rms−P S = 15.3 +3.8 −2.8 µK. The quadrupole of the raw maps is slightly below Q rms−P S , but not by a statistically significant amount. Note that for "standard CDM" one expects n DM R = 1.1 because DMR probes the low-l tail of the acoustic peak.
In this notation, the C l (eq. 10) are given by [66]: with n DM R = 1 this reduces to C l ∝ 1/l(l + 1). 2. The data appear best described by Gaussian statistics [65]. At these large angular scales this is not surprising because even non-Gaussian processes at small scales, when averaged over a large enough volume, appear Gaussian. However, it is reassuring that the statistics we all assume have some basis in reality.
My favorite way of quantifying the DMR maps is to show the power spectrum over compact regions of l-space. A direct computation of the power spectrum is hampered by the unevenly-weighted non-uniform sky coverage imposed by the scan pattern and the elimination of data contaminated by Galactic emission. There are a number of ways around this problem ( [69], [70]). The most recent approaches ( [68], [67]) work to minimize the width of a representative bin in l-space. The results are shown in Figure 10. Górski's method in particular shows the power within ∆l = 0. Though one should expect to get exactly the same results from two different methods applied to the same data set, we can be pleased by the general concordance.
Many groups are working to measure the anisotropy. Though some are focussing on large angular scales and frequencies not observed with DMR, most concentrate on smaller angular scales. Table 1 contains a list for recent, current and planned experiments. It does not include the satellite experiments nor does it claim to be comprehensive. I apologize for any omissions or misrepresentations.
Unlike measurements of the absolute temperature of the CMB, where the final result is completely dominated by one's control of subtle systematic errors, anisotropy measurements require a combination of high sensitivity and immunity to systematic effects. The state-of-the-art in absolute measurements, excluding FIRAS, is 1% [30]; the anisotropy has yet to be measured to 15% accuracy.
To give a broad and almost un-biased sense of what the anisotropy data are telling us, we take the compilation from Ratra [97] (which I believe is the most comprehensive and thoroughly checked compilation to date) and bin the data according to the following: − Select logarithmically spaced bins in l with four bins per decade. − Ignore the widths of window functions and add data to a bin according to the value of the weighted mean of the window, l e in [97]. Many of the data have ∆l/l e < 1/4 so this is not the sin it may at first appear.  [80] 0.6 • 30 GHz HEMT C/G UCSB IAB(f) [81] 0.83 • 150 GHz Bol C/G Bartol MAT(p) [82] 0.2 • 30-150 GHz HEMT/SIS C/G Penn, Princeton MAX(f) [83] 0.5 • 90-420 GHz Bol C/B UCB, UCSB MAXIMA(p) [84] 0.2 • 90-420 GHz Bol C/B UCB, Caltech MSAM(c) [85] 0.4 • 40-680 GHz Bol C/B Chicago, Brown, Princeton, NASA/GSFC OVRO 40/5(c) [86] 0.033 • , 0.12 • 15-35 GHz HEMT C/G Caltech, Penn PYTHON(c) [87] 0.75 •

35-90 GHz
Bol/HEMT C/G Carnegie Mellon Chicago, UCSB QMAP(f) [88] 0.2 • 20-150 GHz HEMT/SIS C/B Princeton, Penn SASK(f) [89] 0.5 • 20-45 GHz HEMT C/G Princeton SuZIE(c) [90] 0.017 • 150-300 GHz Bol C/G Caltech TopHat(p) [91] 0.33 • 150-700 GHz Bol C/B Bartol, Brown, DSRI,Chicago, NASA/GSFC Tenerife(c) [92] 6.0 • 10-33 GHz HEMT C/G NRAL, Cambridge VCA(p) [93] 0.33 • 30 GHz HEMT I/G Chicago VLA(c) [94] 0  − Compute the weighted mean of the data that fall into each bin and call that the band-powerδT . Use the inverse root of the total weight as the error bar. Use all the unique data in [97]. Be aware that many of the data points are unconfirmed! − Compute the arithmetic mean valuel, of the l e that fall in a given bin. − Plotl versusδT and connect the ends of the error bars. − Ignore the intrinsic calibration uncertainties and the relative calibration uncertainties. − Ignore upper limits for l < 500. For some cases, (eg SASK [89]), the data can be combined to give a detection [18]. − In addition to the Ratra compilation, add the Tegmark results [68], ATCA [75], and the new CAT [53] results. Figure 11. Angular spectrum of the anisotropy. The thick lines (diamonds) were obtained following the prescription given in the text. Bharat Ratra has used somewhat different criteria and obtained the results indicated by the thin lines. The asterisks near l = 1000 are upper limits. Clearly, the data indicate a rise in the spectrum from l = 10 to l ≈ 200. For l > 20, the quantity on the y-axis is the same as in Figure 3.
The principle conclusion one should draw from Figure 11 is that there is a general rise in δT l as one moves from the COBE scales to smaller angular scales. This is a stunning observation that was predicted long before the anisotropy was discovered. Has a peak to the spectrum been detected? Possibly, but it is still too early to say this with any confidence. All indications are that the power spectrum at l ≈ 10 is lower than at l ≈ 200 but we cannot say where the spectrum turns over above l ≈ 200. The errors are simply too large and there are too many systematic effects hidden in the data. Also, there are plenty of examples where 95% upper limits have become detections at a higher level and examples where detections have become upper limits. However, it is somewhat reassuring that the χ 2 /(ν − 1) for most of the individual bins is not too different from one.
The measurements have come a long way in the past two years. In three cases multiple experiments have observed the same region of sky and seen the same thing. They are DMR and FIRS [98], DMR and Tenerife [92], and MSAM ( [99], [100] & [101]) and SASK [18], [8]. The spectrum of the fluctuations for MSAM and SASK is thermal from 25 to 200 GHz. However, one should still view the data with some caution. In the analyses that give rise to Figure 11, an entire data set is reduced to give one measurement and one statistical error bar. When this is done, there is simply not enough signal-to-noise to quantify systematic effects that are lurking at the 1σ to 2σ level. On top of this, the analysis of these experiments is tricky; "new" effects are still being discovered by many groups. Finally, the intercalibration of the experiments is uncertain to the 10%-15% level.
To improve on these results, a number of experimental and observational challenges must be met. The results that went into Figure 11 are primarily from difference measurements. Eventually we will want maps of the sky so that experiments are easily compared, foreground contamination is more easily identified, and powerful statistical tests can be performed. Interferometers offer one proven way to do this at low frequencies (and eventually at higher frequencies) but other techniques and strategies are needed. The calibration of the experiments must be better than 10% in order to distinguish between the various models. Currently, the best reference is the intrinsic dipole in the CMB. Finally, to distinguish features near l ≈ 1000, high resolution will be needed. One desires at least δl/l ≈ 1/10. Lower resolutions smear the features in the power spectrum.
A measurement of the polarization in the CMB is now within grasp. In the standard CDM models, the signal is predicted to be at 1% to 5% of the anisotropy [66]. John Ruhl at Santa Barbara, Suzanne Staggs at Princeton, and Peter Timbie at Wisconsin are actively working on these measurements. The polarization is caused by Thompson scattering. At angular scales of order ten degrees, the polarization may be used to identify any primordial gravity waves (tensor modes) though the signal is expected to be largest at degree scales. Crittenden and Turok [102] point out that there is a correlation between the polarization and the anisotropy that is different for the scalar and tensor modes. This correlation should also aid in separating the CMB polarization from polarized foreground emission, about which very little is known.

The New Satellite Experiments
Two satellite missions are planned that will endeavor to map the CMB anisotropy over the entire sky. The ESA mission is called PLANCK (originally COBRAS/SAMBA). The NASA mission, which is in the final design and definition phase, is called MAP. Because I am part of the MAP team, I will focus on it.
PLANCK 9 uses both bolometers and HEMT-based amplifiers. It will carry cryogens so that the bolometers may be operated at 0.1 K. At this stage in the design, the instrument is planned to span the frequencies between roughly 30 and 900 GHz with an angular resolution between 30 ′ and 4.4 ′ . The lower frequency channels will be polarization-sensitive. The planned launch date is 2005.
MAP is based on the HEMT amplifiers developed by Marian Pospieszalski. The radiometers are intrinsically polarization-sensitive and differential, similar in some regards to the successful COBE/DMR design. The instrument will span from 20 to 106 GHz in five frequency bands with an angular resolution ranging between 54 ′ and 15 ′ . The sensitivity per 0.3 • × 0.3 • pixel (of which there are roughly 400,000 in the sky) will be about 35 µK. Because the instrument is passively cooled, it can in principle observe longer than the 15 month design life.
The primary goal of MAP is to make multi-frequency, high-fidelity, highsensitivity maps of the sky. This requires extreme control of systematic effects. We believe the best vantage for these observations is L2, the Earth-Sun Lagrange point. At L2, the Sun, Earth, and moon are ≈ 90 • out of the beams and the environment is essentially isothermal. From the work on DMR and balloons, the team has found that successful map production requires reference of one pixel to another over many directions and over many time scales. MAP plans to do this with the scan strategy shown in Figure 12.
From a high-quality map, one may not only obtain the power spectrum, but may also compare the data to the results of other CMB experiments and to maps of the foreground emission at different frequencies. Also, a map gives the best data set for testing the underlying statistics of the fluctuations. For instance, we will be able to tell from the MAP data if the CMB is a Gaussian random field. Finally, with its high sensitivity and large scale coverage, the time-line data from MAP will be ideal for searching for transient radio emission.
The question of how well one can determine the parameters of cosmological models is still an active area of research. The most recent published work on parameter estimation for inflation-based models is in [36] but one must remember that there are other classes of promising models. At this school, Dick Bond discussed an approach where one works in an eigenparameter space to circumvent the strong covariance between many of the standard parameters such as Ω B , Λ, h, etc. At any rate, if the anisotropy is show the path for one side of a differential pair. The other pair member follows a similar path, only delayed by 1.1 min. There are four principal time scales for the observations. The phase of the difference signal is switched by 180 • at 2.5 KHz. The spacecraft spins around its symmetry axis with a 2.2 min period (bold circle) with cone opening angle of roughly 135 • . This pattern precesses about the Earth-Sun line with a period of 60 minutes. Thus, in about 1 hour, over 30% of the sky is covered. Every six months, the whole sky is observed. Note that any pixel is differenced to another pixel in many directions.
normally distributed, the MAP CMB data will be cosmic variance limited up to l ≈ 600 (assuming the foreground/radio source emission is successfully removed) and will probe multipoles up to l ≈ 1000.
In the current schedule, the satellite design and definition will be com- I would like to thank Roberta Bernstein, Venya Berezinsky, Piero Gallotti, David Schramm and the Ettore Majorana Center staff for organizing a wonderful school. Marsala will never taste the same. Conversations with many colleagues were helpful in preparing these notes. I would especially like to thank Tom Herbig, Gary Hinshaw, Bharat Ratra, Suzanne Staggs, and Ned Wright. Ned gave me the computer code to produce Figure 4. This work was supported by the US National Science Foundation and the David and Lucile Packard Foundation.