3.
Note on conventions d[T 2 (θ)] ( + 1)c = d[ln[ ] 2π The natural units for a temperature/polarization map are µK 2 . In the ﬂat sky approximation (after a CMB map has been subjected to a top-hat bandpass ﬁlter in harmonic space), d2 c 1 max d 2 c( ) T 2 (Ω) = (4π) = ( 2c ) = ∆ ln[ ] S2 (2π)2 4π 2π min 2π Scale invariance implies that integral is logarithmically convergent. Curvature correction appear at higher order in an expansion in powers of 1/ , but no natural way to extend curvature correction to scale invariance.
4.
Reduction to a power spectrum ∞ + T (Ω) = a m Y m (Ω), Ω = (θ, φ) ∈ S 2 =2 m=− In standard inﬂation ﬂuctuations are very nearly Gaussian and the probability of obtaining a sky map given a predicted theoretical power spectrum (depending on a number of cosmological parameters θ) is (th) 1 |a m |2 P({a m }|c ) = (constant) exp − 1 2 (th) ,m c (th) c or we may write χ2 = −2 ln[P] (obs) (obs) c c χ2 = (2 + 1) (th) − 1 − ln (th) + (constant) c c (obs) where c = (2 + 1)−1 m |a m |2 . The above is usually called likelihood because we are interested in how it changes as we varying the parameters of the theoretical model, rather than predicting the outcome of the experiment.
5.
Predicted temperature power spectrum CMB scalar anisotropies 1e+04 1e+02 l*(l+1)*CXX/2*pi [muK**2] 1e+00 1e−02 2 5 10 20 50 100 200 500 1000 Multipole number (l) At low- the amplitude of the temperature ﬂuctuation is ≈ 33µK (i.e., ∆T /T ≈ 1. × 10−5 ) As√ increases to ≈ 220, sees the rise to the ﬁrst acoustic peak, by a factor of ≈ 6 in temperature (and 6 in the power spectrum). Most of the power seen in the CMB maps arises from the Doppler peak. The size of the spots is visible to the eye and this is the salient feature (and not the scale invariance.) At larger one observes a sequence of secondary Doppler peaks (and corresponding troughs) as well as Silk dampening, due to viscosity in the photon electron-plasma as well as the ﬁnite width of the last scattering surface.
6.
Including instrument noise (and incomplete skycoverage) χ2 = Tsky (C + N)−1 Tsky + log[det(Cth + N)] T In general, noise is assumed (to a ﬁrst approximation) uncorrelated between pixels but not necessarily uniform on the sky. Therefore, the N no longer takes the simple form N m; m =N δ δmm √ √ N m; m = (constant)(1/ t) m (1/ t) m or even a less restricted form when non-whiteness (i.e., redness) of the noise is allowed. Warning : Although the above deﬁnes an exact representation of the likelihood, given the large number of pixels present, evaluating the above is not feasible, at least on all scales, and much work has gone into developing good approximations to the likelihood that are fast to compute. (If Npix = 106 for example, Npix = 1018 operations are required to invert a matrix.) 3
7.
Polarization Percentage linear polarization Correlation coefficient [cte/sqrt(ctt*cee)] 25 0.6 0.4 20 Percent polarization [sqrt(cee/ctt)] TE correlation coefficient 0.2 15 0.0 10 −0.2 −0.4 5 −0.6 0 0 500 1000 1500 2000 2500 3000 0 500 1000 1500 2000 2500 3000 Multipole number Multipole number The CMB is also predicted to be mildly polarized. This polarization encodes complementary information. Polarization is represented by a double-headed vector or second-rank symmetric tensor on the celestial spehere. Pij (Ω), in [µK ] δT (Ω, n) = δT (Ω) + Pij (Ω)ni nj , n ≡ (direction of linear polarizer)
8.
Scalar and Tensor CMB Anisotropies with PLANCK Red=scalar (top to bottom) TT, TE, EE, BB (lensing) Blue=tensor (T/S=0.1) = TT, TE, EE, BB, dotted BB (T/S=0.01, 0.001) Green PLANCK capabilities = top single alm, bottom agressive binning
11.
E and B Mode Polarization E mode B mode (E) 2 1 Y m,ab = a b − δab Y m (Ω) ( − 1) ( + 1)( + 2) 2 (B) 2 1 Y m,ab = ac c b+ a bc c Y m (Ω) ( − 1) ( + 1)( + 2) 2
12.
Technologies for the detecting CMB ﬂuctuations
13.
Microwave horns ACBAR horns Idea is to admit only a single mode (in the transverse direction) so that the microwaves entering can be can be regarded as a two-component scalar ﬁeld theory—that could either be detected directly (as in Planck) or put onto a microstrip transmission line and modulated electronically. The horn is to produce as Gaussian a beam as possible with the fastest falling side-loabs. The telescope in underiluminated to prevent the sideloabs from seeing the instrument.
14.
Schematic of single-mode detection hν ν (6.6 × 10−34 J · s) ν = = kB TCMB (1.38 × 10−23 J K −1 )(2.73 K ) 57 GHz Single−mode detection Flux from microwave sky Matched impedance Transmission line resistor Microwave feed horn Return flux of resistor noise P = (power onto a single-mode transmission line) ∆ν = 2 pW 50 GHz
15.
Review of photon counting statistics 1 Z = 1 + x + x2 + . . . = , x = exp[− ω/kB T ] 1−x ∂ x 1 N = Z −1 x Z = = ∂x 1−x exp[ ω/kB T ] − 1 2 ∂ 2x 2 x N 2 = Z −1 x Z = + =2 N 2 + N ∂x (1 − x)2 (1 − x) ¯ Two limits and an intermediate regime (N = N ) : ¯ N 1 highly correlated arrival times, as if photons arrive in bunches of ¯ N photons. ¯ N 1, nearly Poissonian (uncorrelated) arrival times. ¯ N ≈ 1, moderate “bunching”, postive correlation in arrival times.
16.
Noise for a noiseless (perfect) detector operating thein Rayleigh-Jeans (classical) regime δI = (∆ν)t I t ≡ [Integration time (in sec) (∆ν) ≡ [Bandwidth (in Hz) Basically, (∆ν)−1 is the rate of independent realizations of an independent Gaussian stochastic process. In the presence of detector noise, which may be characterized as a “system temperature” Tsys one has Tdetector = Tsignal + Tsystem and δI Tsys = 1+ (∆ν)t I Tsig Robert Dicke, 1947 Mirrors and other optical elements can be very hot and if they are also very reﬂective do not add much noise. Tout = (1 − τ )Tin + τ Toptics where τ is a small absorption probability.
17.
Quantum correction to radio astronomer’s formula Correction factor to linear dependence of occupation number (intensity) on temperature d(ln[N]) x exp[x] = d(ln[T ]) (exp(x) − 1) Correction to intensity ﬂuctuation 2 ¯ ¯ δI N2 + N 1 = ¯ (∆ν)t = 1+ ¯ (∆ν)t I N2 N 2 δT x 2 exp[3x] = (∆ν)t T (exp[x] − 1)2 where x = hν/kB T = (ν/57 GHz).
18.
Characterization of the detector noise (I) Resistor 10 (nV/Hz^0.5) 1 0.01 0.10 1.00 10.00 100.00 1000.00 Frequency (Hz) Yogi bolometer at 110 mK 10-15 (W/Hz^0.5) 10-16 10-17 10-18 0.01 0.10 1.00 10.00 100.00 1000.00 Frequency (Hz) From Giard et al. (astro-ph/9907208 ) To a ﬁrst approximation, the detector noise is (white-noise) + (1/f - noise ). One uses modulation (repeated scanning of the same circle in the sky) to remove 1/f noise.
19.
Characterization of the detector noise (II) As a ﬁrst approximation, one assumes that detector noise in a sky map is uncorrelated between pixels with variance inversely proportional to the pixel integration time. For equal integration time for each pixel N = N0 (i.e., a “white-noise” spectrum, rather than the very “red” approximately scale-invariant CMB spectrum c ∝ −2 .) Non-uniform sky coverage, let t(Ω) be the integration time in the Ω direction and expand t 1/2 (Ω) = t 1/2 m Y m (summation implied). Then ∗ 1/2 (N −1 ) ,m; ,m = t 1/2 mt m (Note that if sky coverage is incomplete N −1 is well-deﬁned, but its inverse N is not.)
20.
Problem of far side-lobes Airy diffraction pattern (from a unapodized circular aperature) Airy diffraction pattern (logarithmic scale) Airy diffraction pattern (linear scale) 1.0 1e−01 0.8 1e−03 0.6 Intensity Intensity 1e−05 0.4 1e−07 0.2 1e−09 0.0 −20 −10 0 10 20 −20 −10 0 10 20 theta/sigma_theta theta/sigma_theta While a Gaussian may be a good approximation near the beam center, diffraction theory tells us that without an inﬁnite aperature exponential fall-off of the point spread function is not possible. 2 2J1 (x) I(x) = I0 × , x = (θ/θbeam ) x
21.
Escaping the Earth, Sun and Moon at L2 L2-Earth distance = 1.5e6 km Earth-Moon distance = 4.0e5 km For Planck standards one needs about 10−9 rejection toward sun ! And a very high rejection toward the Earth as well.
22.
Observations from the ground (I) Atmospheric interference. Calculated optical depth through the atmosphere for a good ground-based site like the South Pole or Dome-C in Winter (black) and at balloon altitude (red). Frequency bands for sub-orbital experiments must be carefully chosen to avoid the emission by molecular lines. Moreover, emission from oxygen lines is circularly polarized and care must be taken to avoid a signiﬁcant polarized signal from the tails of these lines.
23.
Observations from the ground (II) Numerous CMB polarization experiments from the ground and balloons at various stages : QUaD, BICEP ; BRAIN, CLOVER, EBEx, PAPPA, PolarBear, QUIET and Spider Far side lobes Scanning strategy (must scan at constant zenith angle) Polarization from interaction of Zeeman splitting by earth’s magnetic ﬁeld of oxygen lines. Atmospheric backscattering (very polarized) could be a serious problem. [See L. Pietranera et al., “Observing the CMB polarization through ice,” MNRAS 346, 645 (2007)]. Lack of stability and partial sky coverage
24.
Modulation strategies (I) It is not a good idea to measure the polarization as a very small difference between two very large quantities. For example, ∗ ∗ Q = Ex (Ω) Ex (Ω) − Ey (Ω) Ey (Ω) . Signal type Power d(T 2 )/d(log[ ]) CMB Monopole T0 1012.5 µK 2 δT anisotropy 103 µK 2 δE anisotropy 10−1.5 µK 2 δB anisotropy 10−6 µK 2 The “deadly sins” of B polarization measurement can be ranked (from most serious to less serious) as follows : T0 → B leakage. (E.g. polarization from reﬂections, far side lobes, standing waves in instrument) δT → B leakage. (E.g., subtracting two polarized beams having different uncharacterized ellipticity) δE → B leakage. (E.g., poorly calibrated polarization angle, cross polarization)
25.
Toward Measuring the Polarization Directly : PhaseSwitch Modulation Phase switch Mixer Bolometers Ey (Ex ± Ey ) ×(±1) I1 Horn OMT I2 Ex (Ex Ey ) I1 = (Ex ± Ey )2 = Ex 2 + Ey 2 ±Ex Ey I2 = (Ex Ey )2 = Ex 2 + Ey 2 Ex Ey
26.
Rotating Half-Wave Plate Bolometers Ey Mixer (Ex + Ey ) I1 Ex , Ey Horn OMT Ex (Ex − Ey ) I2 Rotating half-wave plate Ex cos(ωt) sin(ωt) 1 0 cos(ωt) − sin(ωt) Ex = Ey − sin(ωt) cos(ωt) 0 −1 sin(ωt) cos(ωt) Ey cos(2ωt) sin(2ωt) Ex = sin(2ωt) − cos(2ωt) Ey 2 I1 = cos(2ωt) + sin(2ωt) Ex + − cos(2ωt) + sin(2ωt) Ey 2 2 2 2 = (Ex + Ey )+(Ex − Ey ) sin(4ωt) + 2Ex Ey cos(4ωt) 2 I2 = cos(2ωt) − sin(2ωt) Ex + + cos(2ωt) − sin(2ωt) Ey 2 2 2 2 = (Ex + Ey )−(Ex − Ey ) sin(4ωt) − 2Ex Ey cos(4ωt)
30.
Foreground datasets WMAP provides an invaluable source of information of synchrotron emission at low-frequencies, including its polarization properties with high signal-to-noise. Haslan map at 405MHz (with higher resolution) Hα emission maps (tracer of free-free emission) Considerably less is known about dust emission extrapolated into the CMB frequencies. DIRBE at 250µ and IRAS dust maps (at 100 µ) can be used but the lever arm of the extrapolation is large. Even less is known about polarized dust. The polarization of starlight at optical frequencies can be used as a tracer of the degree of grain alignment c 100µ ν= = (3000 GHz) · λ λ
31.
Galactic synchrotron emission Hot gas with a non-thermal distribution (high-energy power law tail rather than the exponential fall-off as in a completely thermal distribution) + magnetic ﬁeld produces non-thermal synchrotron emission. Exact spectrum depends on the electron velocity spectrum. Theory says that synchrotron radiation should be smooth because even a monoenergetic distribution of electron velocities becomes smoothed out. Power law is empirical. Observed intensity has a power law ν − 3 approximately relative to a blackbody. Generally highly polarized, because of coherent large-scale magnetic ﬁelds in our galaxy (maybe 30% ) Dominates the emission in WMAP 20Gz maps over most of the sky. Low frequency maps, most notably the 20 GHz and 33 GHz can be used to construct templates.
33.
Galactic free-free emission (Bremsstrahlung) Arises from electron-electron collisions where a photon is also emitted. Proportional to the density squared times a temperature dependent function. Hα emission can be used as a tracer. If all the gas involved were at the same temperature there would be no spread in this correlation. Hα maps can be used as a template for subtracting this component. Is unpolarized because there is no prefered direction associated with this emission mechanism.
34.
Physics of thermal dust emission The following simpliﬁcations lead to a very simple theoretical template : Dust grains are very small compared to the wavelengths of interest All resonances of the dust grain coupled to the electromagnetic ﬁeld lie at frequencies far above those of interest The dust is optically thin 2 ν Idust (ν, Ω) = τ (Ω; ν0 ) B(ν, Tdust ) ν0
35.
Power law of the dust emissivity exponent Assume dust grain polarization is linear and causal ∞ d(t) = dτ K (τ ) E(t − τ ) 0 We deﬁne the grain susceptibility ∞ χ(ω) = dτ K (τ ) exp[+iωτ ) 0 and this function is analytic on the upper-half plane [assuming an exponential decay of the kernel K (τ ).] For free charges there is a pole at the origin. Otherwise, there are poles just below the real line arranged symmetrically about the axis Im(ω) = 0.
36.
How could one explain an emissivity index other thantwo asymptotically as ω → 0 ? (Maybe a series a poles approaching the origin in the ω-plane ? Are very low frequency resonances plausible ? ) Lorentz model (harmonically bound charge) 1 χ(ω) = (e/m) 2 ω0 − ω 2 − iγω σelast = ω 4 χ(ω)2 , σabs = ωIm[χ(ω)], 2 ¨2 1 P= d , Uinc = (E 2 + B 2 ) 3c 2 8π 4 ω 8π e4 8π 2 σRayleigh = σThomson , σThomson = = re ω0 3 m2 c 4 3 where re is the classical electron radius. C. Meny, V. Gromov et al., A& A (astro-ph/0701226)
37.
Current state of our knowledge of dust emission At present the Schlegel, Finkbeiner, and Davis dust templates serve as the best predictor of unpolarized dust emission at CMB frequencies. Nevertheless, they suffer from the drawback of the large amount of extrapolation required because they are based on the 3000 GHz & 1200 GHz, mainly because of uncertainties in the dust temperature and in the dust emissivity index. PLANCK will greatly improve this situation by providing high signal -to-noise full-sky maps at intermediate frequencies (353 GHz, 545 GHz, 857 GHz) with polarization information at 353 GHz. Considerable uncertainty persists as to the polarized dust emission.
38.
Polarized dust emission Potentially big problem for detecting B modes. Thermal dust emission is unpolarized without a mechanism to align dust grains in a coherent way. A preferred polarization direction common that does not average out must be deﬁned. But large-scale coherent magnetic ﬁelds provide such an alignment mechanism The polarization of starlight demonstrates that dust grains are not spherical and that they have been aligned. Unfortunately, the inability to create reliable theoretical models for polarized dust emission and the lack of relevant data renders the extrapolations carried out to date unreliable.
39.
Polarization of starlight by aligned dust grains Source : Compilation of Fosalba et al.
40.
Dust grain alignment mechanism Simple if there is thermal equilibrium and if grain properties and geometries are known. Unfortunately, thermal equilibrium is not a good assumption. Many temperatures enter, and they are not all equal. Dust grains may be modeled as oblate or prolate ellipsoids that are either diamagnetic or paramagnetic. Alignment is much like that of a dielectric in a unform electric ﬁeld. There needlelike structures will want to align in the plane perpendicular to the ﬁeld and ﬂattened structures will want to align parallel to the ﬁeld in order to screen the electric ﬁeld as much as possible. For the magnetic ﬁeld the orientation is opposite. Needlelike structures will want to align perpendicular to the ﬁeld for paramagnetic materials and parallel to the ﬁeld for diamagnetic materials. For oblate structures the alignment is reversed.
41.
Anomalous (“spinning”) dust emission Unexpected correlations between low-frequency (≈ 10–60 GHz) and dust maps (Kogut et al.) suggest the presence of non-thermal dust emission at low frequency where the thermal dust emission is essentially zero. Draine & Lazarian have suggested that spinning grains through electric or magnetic dipole radiation (ie a permanent dipole) could account for this anomaly. For this to work, the rotational degree of freedom must be out of thermal equilibrium or more strongly coupled to the radiation ﬁeld. Credible mechanisms for spinning up the dust have been proposed. Such emission would be expected to have substantial polarization.
42.
Crookes radiometer — a possible analogy for spinningdust 1873, Sir William Crookes
43.
Rotational properties of dust grains For the non-rotational degrees of freedom of dust grains, there is a balance between UV ﬂux from stars that is absorbed and the subsequent re-emission at infrared and microwave frequencies. For small grains, incident photons arrive infrequently, maybe one a day, raising their temperature to mayeb 150 K, and then they decay maybe in 100 sec to a very low temperature slightly above the CMB temperature. These are the grains responsible for the small-wavelength part of the dust emission spectrum. Larger grains, on the other hand, receive photons frequently and maintain a nearly constant temperature, around 17 K. There emission is restricted to long wavelengths by the exponential factor in the Boltzmann distribution. The rotational degrees of freedom, however, are special. A number of effects couple only to the rigid degrees of freedom and not to the other internal degrees of freedom, and can spin up a dust grain. There are also emission mechanisms that are couple only to the rotational mode. For this reason it does make sense to think about “Suprathermal rotational of interstellar dust grains” (a paper title of EM Purcell in 1979 on the subject). Collisions with molecules would tend to spin up the rotational degree of freedom so that its energy is given by the ambient kinetic gas temperature, in the thousands of degrees.
44.
Rotational properties of dust grains (II) Catalysis of the formation of molecular hydrogen on the dust grain, perhaps at a particular non-random site releases 4.2eV, a huge amount of recoil applying a torque to the rotational degree of freedom, likely in a coherent way. This is more like an engine extracting energy from an out-of-equilibrium situation. Non-uniform radiation pressure will also exert a torque. In general the absorption properties across the grain will be non-uniform, causing the incident radiation to exert a torque in a coherent. Again, thermodynamically this is much like an engine. The hot temperature is the UV radiation expelled as at a lower temperature (the thermal IR radiation). The net torque would vanish if the two temperatures were equal. Dust grains in general have a permanent electric dipole moment, and like non-symmetric diatomic molecules (eg CO). If they are charged, their center of charge is unlikely to coincide precisely with their center of mass. This means that rotating dust grains act as dipole radiators, with the radiated power proportional to the fourth power of the angular velocity. This provides a mechanism to limit the angular velocity of a grain. This is all very complicated. The necessary information for reliable modelling is lacking. (Papers by Draine, Lazarian, and especially older papers by Purcell offer an invaluable source of information.)
45.
Sunyaev-Zeldovich Effect (thermal) Scattering of CMB photons by very hot gas (≈ 107 − 108 K . Contrary to expectation, the effect cools in the Rayleigh-Jeans part of the spectrum and heats in the Wien part, by moving photons from the red to the blue on the average. Electron scattering does not change the number of electrons but only changes their distribution in frequency. At microwave frequencies only scattering with electrons from the ionized gas is relevant. kB Te y= dl ne σT mc 2 ∆T x(ex + 1) Z = −4 y T S ex − 1
46.
ACT cluster maps (148 GHz channel only) FWMH σbeam ≈ 1.37arcmin Hinks et al., “The Atacama Cosmology Telescope (ACT) : Beam Proﬁles and First SZ Cluster Maps” (astro-ph/0907.0461)
47.
High- mono-frequency power spectrum Fowler et al. (ACT collaboration), “The Atacama Cosmology Telescope : A Measurement of the 600 < < 8000 Cosmic Microwave Background Power Spectrum at 148 GHz,” astro-ph/1001.2934 (2010).
48.
SPT (South Pole Telescope) IR point sources Hall et al. (SPT collaboration), “Angular Power Spectra of the Millimeter Wavelength Background Light from Dusty Star-forming Galaxies with the South Pole Telescope (astro-ph/0912.4315)
49.
Kinetic Sunyaev-Zeldovich & Ostriker-Vishniac effect (much smaller than the thermal SZ effect and easily confused with cosmological perturbations because of its blackbody spectral form) η0 ∆T (Ω) = dηa(η)ne (η)σT exp[−τ ] Ω · v(Ω(η0 − η), η) T Ostriker −Vishniac 0 ∞ = dτ σT exp[−τ ] Ω · v(Ω(η0 − η(tau)), η) 0 ∆T vpec = τcluster T kSZ c Three related effects are incorporated within a single formula : (1) kinetic Sunyaev-Zeldovich (fully collapsed and virialized objects) (2) Ostriker-Vishniac (in the ﬁeld, higher-order perturbation theory, later times, (3) patchy reionization (emphasizes the effect of sharp edges of the electron density, due to Strömgen spheres....). These should be though of as aspect of a single effect because the distinctions between them cannot be cleanly differentiated.
50.
Point sources Two basic types : radio point sources [arising from synchrotron emission in compact objects (e.g., radio-loud AGNs, "ﬂat spectrum" radion galaxies, BL-Lacs,....), and IR points sources (dusty galaxies with hot thermal dust emission) Point sources dominate at high- . May be approximated as a Poissonian distribution of pointlike objects on the celestial sphere. But these two simplifying approximations have their limitations, especially as increasingly smaller scales are being probed. Clustering complicates constructing a template to account for unsubtracted point sources. Unfortunately, their spectral indices vary considerably ; therefore, to identify either in high frequency or low frequency maps and mask in the intermediate "CMB" frequencies is a good strategy for the most luminous objects. Nevertheless, a background of unresolved point sources is predicted to subsist.
51.
Foreground removal techniques There is no silver bullet for this problem. There are many approaches and since the problem is hard and no unambiguous solution suggests itself, one want to try every reasonable approach and compare results. Approaches can be based on : (1) Understanding and modelling the detailed physics of the foreground components and other contaminants. (2) Data analysis. Study of correlations. Template subtraction. (3) Blind analyses (e.g. independent component analysis, internal linear combination). Best for looking for unexpected new components in the data. (4) Bayesian modelling. Expressing what is known as best as possible in terms of priors which compete with eacg other and are rationally resolved in accordance with Bayes theorem.
52.
Linearized multi-component model βsync (ν) βfree (ν) βdust (ν) ν ν ν δTR−J (ν) = Tsync +Tfree +δTCMB a(ν)+Tdust νK νK νK Comments : When the intensity as a function of wavelength is expressed as a function as a Rayleight-Jeans (“brightness”) temperature, the low-frequency part of the Planck blackbody spectrum (I ∝ T ) is extrapolated into the Wien part of the spectrum, where there should be an x/(exp(x) − 1) correction factor. This is done so that brightness temperatures add linearly when ﬂuxes are combined. β ≈ −3(−2.5 – −3.1). βCMB = 0 at low frequencies where before the Wien regime correction factor kicks in. β ≈ 2 for dust at large wavelengths, at frequencies well before the dust temperature. βfree ≈ −2.14.
53.
A simplest model of component separation Step I : Formulate a model for the log-likelihood (For simplicity we assume each pixel can be analyzed independently.) χ2 = yobs − x T M T N −1 (yobs − Mx) + x T Cprior x T −1 yobs ≡ frequency channel vector (e.g., 30 Ghz, 100 GHz, 217 GHz, 350 GHz, . . .) x ≡ underlying components (e.g., primordial CMB, dust, synchrotron,.....) yobs = Mx + n N = nnT (i.e., detector noise, instrumental error) Cprior = xx T (limits on reasonable values or even non-informative (ﬂat) prior)
54.
A simplest model of component separation (II) Step II : Complete the square in the variable of interest discarding constant terms. χ2 = yobs − x T M T N −1 (yobs − Mx) + x T Cprior x T −1 = x T M T N −1 M + Cprior x + yobs N −1 M x + x T M T N −1 yobs −1 T +(irrelevant constant)1 T = x − xML M T N −1 M + Cprior −1 x − xML + (irrelevant constant)2 where −1 xML = M T N −1 M + Cprior −1 yobs and T −1 x − xML x − xML = M T N −1 M + Cprior −1 −1 Note that (1) in general M is rectangular and not square and (2) Cprior can have zero eigenvalues of be identically zero.
55.
A simplest model of component separation (III) :Interpretation Note that the inverse covariance relation for x −1 Cposterior = M T N −1 M + Cprior −1 −1 , which is effectively a special case of Bayes’ theorem for Gaussian distributions, expresses the additivity of information. Some comments : The number of unknowns can be greater than, equal to, or less than the number of data points. For (1), N −1 indicates with what weight to −1 reconcile inconsistent equations. For case (3), Cprior provides the missing information.
56.
Marginalization (over “nuisance” variables) x1 Suppose x = where in general both x1 and x2 can have x2 more than one component. Suppose that we only care about x1 and could care less about x2 . We show that the resulting marginalized inverse covariance matrix is (Cmarg ) = (C −1 )11 − (C −1 )12 C22 (C −1 )21 −1 where −1 −1 C11 C12 C −1 = −1 −1 . C21 C22
57.
Marginalization formula derivation We show that T −1 exp − 1 x1 Cmarg x1 2 T −1 −1 x1 C11 C12 x1 = (constant) · d k x2 exp − 2 1 −1 −1 x2 C21 C22 x2 In other words, we project an ellipsoid onto a hyperplane. T −1 T −1 T −1 T −1 RHS = exp − 1 x1 C11 x1 2 d k x2 exp − 2 x2 C22 x2 + x2 C21 x1 + x1 C12 x2 1 T −1 −1 −1 = exp − 1 x1 (C11 − C12 C22 C21 x1 2 T −1 −1 −1 × 2 T d k x2 exp − 1 x2 + x1 C12 C22 C22 x2 + C22 C21 x1 1 T −1 −1 −1 = (constant) × exp − 2 x1 (C11 − C12 C22 C21 x1
58.
How is the previous analysis too simplistic ? We assume a small, ﬁnite number of components, each with a spatially uniform and known frequence dependence. We know for example that the synchrotron spectral index varies between different parts of the sky, and a common dust temperature and emissivity index at low frequency is likely just a ﬁrst approximation. The positivity (and hence non-Gaussianity) of the foreground emissions is not taken into account. Spatial information is not utilized for the cleaning. This would be OK if the foregrounds were close to white-noise in spectrum, but there observed spectrum is very red, closer to that of the primordial ﬂuctuations. Some of the defects can be remedied within a linear framework, but characterizing and then taking into account the non-linearity is not easy.
59.
Template ﬁtting and correlations with externaltemplates In each frequency channel one has an Ansatz of the form Tcorrected = Traw + αTtemplate α can be determined by maximum likelihood. One minizes the variance (in a suitably weighted way) of Tcorrected by varying the coefﬁcient α. If there are many degrees of freedom involved and the template is good, then the corrected map would have the contaminant entirely removed and one degree of freedom of the real signal as well (due to fortuitous overlap). The success of the method depends on the quality of the template. Noise in the template and inadequacy of the model (i.e., spatial variation is the spectral index between the frequencies over which the template is constructed and the frequecy at which the template removal is applied will lead to errors.
60.
Internal Linear Combination (ILC) Methods We are given maps at different frequencies (labelled by i). The maps are all normalized so that the CMB signal contributes with coefﬁcient one but the maps also contain noise and contamination from foregrounds : yi (p) = s(p) + fi (p) + ni (p), We seek a set of weights wi such that i = 1 and s(p) = wi yi (p) i has minimum variance. When there are no foregrounds and just noise, this prescription yields inverse variance weighting. When there are independent foregrounds, a linear combination is chosen to mask the foregrounds, and when both are present, an optimal compromise is found.
61.
Analysis of ILC The variance is given by χ2 = w T Cw We can neglect the variance from the CMB because the constraint 1T w = 1 prevents it from affecting the minimization. We ﬁnd that N −1 1 wopt = T −1 1 N 1 Foregrounds may be considered as just another type of noise.
62.
Weaknesses of the ILC Given that the statistical properties of the foregrounds are not uniform over the sky, the variance of the overall ILC map as a ﬁgure of merit is not appropriate. This prescription favors linear combinations that work well in the galactic plane where the foregrounds are largest, but the linear combination obtained is likely not to be obtained in the low-noise regions of the map, which carry the most information. The ﬁneness of the pixelization also enters into determining the optimal combination. One may want to use different linear combinations in different regions of harmonic space. Prior information is not exploited. In practice these problems have been alleviated by separating the sky into several zones suitably matched, with ILC applied independently to the several regions.
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Independent component analysis (ICA) This ‘blind’ method seeks to extract a number of components from the data without any prior information. Let x be the component vector and d the data vector. One seekis a mixing matrix M and a component vector x such that χ2 = (d − Mx)T N −1 (d − Mx) is minimized where d is the data vector. One seeks that the components be “independent” or orthogonal in an appropriately deﬁned way. One of the problems is that is all the distributions are supposed to be Gaussian, the distinction between the components disppears. Apart from multiplicative (rescalings of the components), one can mix the components among themselves using an orthogonal matrix O, so that x → 0x, and M → MO T while maintaining their independence. Several solutions have been proposed to this problem, one of which is to maximize the degree of non-Gaussianity of the components. By the central limit theorem, mixtures of the non-Gaussian components (what one seeks to avoid) increases the Gaussianity. In practice ICA-based methods work quite well, despite their lack of a rigorous foundation.
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