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 sky
coverage)
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
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 the
in 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 : Phase
Switch 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 than
two 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.
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 spinning
dust
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 external
templates
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.
63. 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.
