Calibration of CMB instruments• One of the most difficult steps in the measurement of

CMB spectrum, anisotropy and polarization, is the calibration of the instrument.

• 20% errors (in temperature units) are still normal for these experiments. A 5% measurement is considered high accuracy.

• The problem is due to– The lack of suitable laboratory standards: the best available

source producing known brightness at mm-waves is a cryogenic blackbody – a source diffucilt to operate and to use.

– The lack of well known Galactic sources as celestial standards. Planets are small and can have atmospheric features; AGNsare variable; HII regions are contaminated by surrounding diffuse emission.

Calibration of CMB instruments• Let’s pose the problem in rigorous terms:• We call B(α,δ) the brightness of the sky in W/m2/sr (we will deal with

spectral dependance later; here we consider the signal integrated over the instrument spectral bandwidth)

• A generic photometer observing the direction αο,δο, will detect a signal

• For a point source located in α,δ , the flux F (W/m2) produces a signal

• The responsivity (gain) (V/W) must be calibrated, and the angular response R(θ) as well. This is the response of the system to off-axisradiation as a function of the off-axis angle θ, normalized to the on-axis response to the same source.

• The calibration can be performed observing a source with knownbrightness or known flux.

[ ] Ωℜ= ∫ dRBAV oooo πδαδαϑδαδα

4),,,(),(),(

ℜ

[ ]),,,(),( δαδαϑδα oooo RFAV ℜ=

The CMB dipole as a calibrator• The dipole anisotropy of the CMB is the best responsivity

(gain) calibration source, for several reasons:A. Its amplitude is well known, and derived from astrometric, non-

photometric data.B. Its spectrum is exactly the same as the spectrum of the CMB

anisotropy. No color correction needed. C. It is unpolarized.D. Its signal is only about 10 times larger than the signal of CMB

anisotropy: detector non linearities are avoided.E. The Dipole brightness is present everywhere in the sky

• The disadvantage is that the dipole is a large-scale signal. Significant sky coverage is needed to detect it with an accuracy sufficient for instrument calibration. Moreover, foreground contamination and 1/f noise effects increase as larger sky areas are explored.

Derivation of the CMB dipole• We are moving with a velocity v with respect to the

CMB Last Scattering Surface. • The CMB is isotropic in the reference frame O’ of the

LSS, but is not isotropic in the restframe O of the observer, which is in motion.

• The distribution function f of particles with momentum pis a Lorentz invariant: In fact

where dN is a scalar, so is invariant, and the phase space volume dxidpi can also be shown to be a Lorentz invariant. So

iidpdxdNf =)(p

)()( pp ff =′′

Derivation of the CMB dipole• The Lorentz transformation for the momentum p is

• Applying this eq. to the Planck distribution function for photons we get

• This formula was first derived by Mosengheil (1907), and rederived by Peebles and Wilkinson (1968), Heer and Kohl (1968), Forman (1970).

• For small β :

pccpc rrr

r ′×−

−=

/cnv1/v1 22

TTcTTppT ′

−−

=′×−

−=⇒′

′=

)cos(11

/cnv1/v1 222

θββ

rr

⎥⎦

⎤⎢⎣

⎡++≅ )2cos(

2)cos(1)(

2

θβθβθ oTT

kinematicterm

light aberrationterm

O’

ppn /rr=

vr

Opr

β• The motion of the Earth with respect to the CMB is the

combination of – The motion of the Earth around the Sun (well known)– The motion of the Sun in the Galaxy (well known)– The motion of the Galaxy in the Local Group (known) – The bulk motion of the Local Group (not well known) due to the

gravitational acceleration generated by all other large masses present in the Universe

• The annual revolution of the Earth around the Sun is known extremely well ( v ~ 30 km/s), and produces an annual modulation in the CMB dipole. This is the main signal used in COBE and WMAP for the Dipole calibration, sinceit is known from astrometric measurements much betterthan the total motion of the earth.

• This effect produces a modulation of the order of βTo , i.e. about 300μK, on a total dipole of the order of 3.5 mK.

CMB dipole signal• The CMB temperature fluctuation corresponds to a CMB

brightness fluctuation, which can be found by derivingthe Planck formula with respect to T:

• This conversion from Temperature to Brightness is the same for the dipole and for any smaller scale temperature or polarization anisotropy. For this reason the Dipole spectrum is the same as the spectrum of CMB anisotropy. The maximum of this spectrum is at 271 GHz.

CMBCMBx

x

kThxTTB

exeI νν =Δ

−=Δ ;),(

1

0.1 1 10

10-16

10-15

10-14

Brig

htne

ss (W

/cm

2 /sr/c

m-1)

wavenumbers (cm-1)

30 GHz 300 GHz3 GHz

10 cm 1 cm 1 mm

Cosmic Dipoles• If the isotropic source spectrum is not a BlackBody,

the dipole formula is different.• A typical example is the cosmic X-Ray background,

whose specific brightness is basically a power lawwith slope

• In general the dipole anisotropy of the specific brightness induced by our speed β with respect to the cosmic matter emitting the background can be derived as

• This is very sensitive to steep features in the spectrum(α large) which can compensate the smallness of β .

ννα

ddI

Iv

dId

==lnln

...})cos()3(1{)( +−+= θβαθ oII

CMB dipole signal• The signal produced by the CMB dipole temperature

fluctuation is

• Since the dipole signal is almost constant within the beam of the instrument

• So from a scatter plot of the measured signal vs. theexpected CMB Dipole the slope a can be estimated:

[ ]

∫

∫

−=

ΩΔℜ=Δ

ννν

δαδαϑδαδα

dETBexe

TK

dRTAKV

CMBx

x

CMB

ooDIPooDIP

)(),(1

1

),,,(),(),(

[ ] ΩΔℜ≅Δ ∫ dRTAKV ooDIPooDIP ϑδαδα ),(),(

[ ] Ωℜ=

+Δ=Δ

∫ dRAKa

bTaV ooDIPooDIP

ϑ

δαδα ),(),(calibrationconstant (V/K)

[ ] ⇒ΩΔℜ=Δ ∫ dRIAV DIPDIP ϑ

CMB dipole signal• The CMB map obtained from the same instrument in

voltage units (uncalibrated) is

where

• The conversion constant from voltage units to Temperature units is the same we have obtained from the Dipole calibration:

[ ][ ]{ }

),(),(),(

),,,(),(),(

ooTdRAKV

dRTAKV

oo

oooo

δαδαϑδα

δαδαϑδαδα

ΔΩℜ=Δ

ΩΔℜ=Δ

∫∫

aV

T oo

oo

),(),(

),(

δαδα

δα

Δ=Δ

[ ][ ] Ω

ΩΔ=Δ

∫∫

dR

dRTT oo

oo ϑ

δαδαϑδαδα

δα

),,,(),(),(

),(

calibrated T map:

uncalibrated V map:

calibrationconstant (V/K)

CMB dipole signal• Notice that

– since we have defined the calibrated temperature map as the intrinsic CMB map weighted with the angular response, and

– since we have used the CMB dipole as a calibrator …

• the calibrated temperature map does not depend on the detailed angular response, and does not depend on the spectral response of the instrument:

• Where

aV

T oo

oo

),(),(

),(

δαδα

δα

Δ=Δ

[ ]bTaV

indRAKa

ooDIPooDIP +Δ=Δ

Ωℜ= ∫),(),( δαδα

ϑ

Sample CMB dipole signals:• COBE map

Sample CMB dipole signals:• COBE map

Gal. Eq.

apex of motion(WMAP)l=(263.85+0.1)o

b=(48.25+0.04)o

(close to the ecliptic…)

Amplitude(WMAP)ΔT=(3.346+0.017)mK

Dipole signal in the B98 region (filtered in the same way as B98 data)

Detected signal at 150 GHz (detector B150A)

-1,0 -0,5 0,0 0,5 1,0 1,5-0,008

-0,006

-0,004

-0,002

0,000

0,002

0,004 Preliminary CalibrationBOOMERanG LDB1998/99

55' pixels (1610)

Sign

al B

150A

(mV)

COBE dipole (mK)

Slope : a = (4.0+0.4) nV/μK

Point Sources• A point source must be observed anyway to measure the

Angular Response R(θ). This is needed for estimates of the instrinsic power spectrum of the map.

• The point source will inevitably have a spectrum different from the spectrum of the CMB.

• The signal from the source will be:

• where F(ν) is the specific flux of the source (W/m2/Hz).• If the source flux is known, and the instrument makes a

map of the region surrounding the source, the observation can be used to estimate the calibration constant a as follows:

[ ]∫ℜ= νννδαδαϑδα dEFRAV oooo )()(),,,(),(

Point Sources

• So the calibration constant a , needed to convert the uncalibrated map into a calibrated CMB map , can be estimated from:– The uncalibrated map of the source V(α,δ)– The flux of the source F(ν)– The relative spectral response of the instrument E(ν)

[ ]

[ ]

∫∫

∫

∫

∫∫∫

−Ω=

=Ωℜ=

⇒Ωℜ=Ω

ννν

νννδα

ϑ

ϑννν

δα

dEFT

dETBexe

dV

dRKAa

dRAAdEF

dV

CMB

CMBx

x

oo

oo

)()(

)(),(1),(

)()(

),(

Point Sources• CMB anisotropy/polarization experiment have a

typical resolution of a few arcmin.• Known sources much smaller than this typical size

can be considered point-sources and can be usedto measure the angular response and the gain.

• Several kinds can be used:– Planets– Compact HII regions– AGNs

• All kinds have their own peculiarities.

Gaseous Planets :

•The size is in the sub-arcmin range.•Atmospheric features can be important.

Mars• Has a tenuous atmosphere, and no sub-mm features. Its

emitting surface is basically a blackbody at 180 K.• The typical size is 6” (check the ephemeres for the time

of the observation).• The typical signal expected from Mars is equivalent to a

CMB temperature fluctuation. This can be found as follows:

[ ]{ }

[ ]{ }Ω

Ω=Δ⇒

⇒⎪⎩

⎪⎨⎧

ΔΩℜ=Δ

Ωℜ=Δ

∫∫

∫∫

dRK

dETBT

TdRAKV

dETBAV

MarsMarsMars

MarsMars

MarsMarsMars

ϑ

ννν

ϑ

ννν

)(),(

)(),(

),,()(),(

1

)(),(cCMBMars

Beam

MarsCMB

CMBx

xMars

Beam

MarsCMBMars TTfT

dETBexe

dETBTT ν

ννν

ννν

ΩΩ

=

−ΩΩ

=Δ

∫∫

10 100

10

100

1000

ΔT (m

K CM

B)

frequency (GHz)

Signal from Mars (6”) in CMB units in a 5’ FWHM beam : About 1000 times the rms CMB anisotropy in the same beam.More than enough to measure the angular response.But what about linearity ? Is there a saturation risk ?

Beam

MarsMarsT

ΩΩ

Degree-scale anisotropy as a calibrator

• Many experiments focus on a small sky patch, in order to obtain maximum S/N per pixel, to study CMB anisotropy/polarization at intermediate and small scales.

• Large scale signals are not measured and are filtered out to remove the effect of 1/f noise and detector instability.

• The Dipole is not a suitable calibrator for these experiments.

• A possibility is to use the WMAP data in the selectedregion. WMAP has detected CMB anisotropy with S/N~1 for 15’ pixels, and 0.5% calibration accuracy.

• A scatter plot of experiment data vs. WMAP can providethe gain calibration. Point sources (AGN) should be removed first, since their effect is strongly frequencydependent.

b ( d

eg)

b ( d

eg)

b ( d

eg)

b ( d

eg)

l (deg)90GHzl (deg)

b ( d

eg)

220GHz

WM

AP

1st

yr

BO

OM

ER

an

G 9

8

b ( d

eg)

l (deg)150GHz

41GHz l (deg) 60GHz l (deg) 94GHz l (deg)

b ( d

eg)

b ( d

eg)

b ( d

eg)

b ( d

eg)

l (deg)90GHzl (deg)

b ( d

eg)

220GHz

b ( d

eg)

l (deg)150GHz

41GHz l (deg) 60GHz l (deg) 94GHz l (deg)

PKS0537-441

BO

OM

ER

an

G 9

8W

MA

P 1

st y

r

b ( d

eg)

b ( d

eg)

b ( d

eg)

b ( d

eg)

l (deg)90GHzl (deg)

b ( d

eg)

220GHz

b ( d

eg)

l (deg)150GHz

41GHz l (deg) 60GHz l (deg) 94GHz l (deg)

PMNJ0519-4546

BO

OM

ER

an

G 9

8W

MA

P 1

st y

r

b ( d

eg)

b ( d

eg)

b ( d

eg)

b ( d

eg)

l (deg)90GHzl (deg)

b ( d

eg)

220GHz

b ( d

eg)

l (deg)150GHz

41GHz l (deg) 60GHz l (deg) 94GHz l (deg)

PKS0454-46

WM

AP

1st

yr

BO

OM

ER

an

G 9

8

10010

100

1000

10000

20030

CMB rms

PKS0537-441 PMNJ0519-4546 PKS0454-46

μKC

MB

in a

20'

bea

m

frequency (GHz)

55.2'20

100430/ −

⎟⎠⎞

⎜⎝⎛=⎟⎟

⎠

⎞⎜⎜⎝

⎛ ΩGHzK

F

CMB

νμ

• There are additional AGNs lost in the confusion of the CMB fluctuations.

• The WOMBAT catalogue and tools predict quite well the flux observed for the 3 detected AGN, and can be used to estimate the contamination due to unresolved AGNs.

• In the 3% of the sky mapped by B98 the contamination of the PS at 150 GHz is less than 0.3% at l=200, and less than 8% at l=600.

• This is reduced by 50% if the resolved sources (at 150 GHz) are removed, and by 80% if are removed those resolved at 41 GHz.

0.0 0.5 1.0 1.5 2.0 2.5 3.01

10

100

Flux (Jy) @ 150 GHzco

unts

WOMBAT catalog

http://astron.berkeley.edu/wombat/foregrounds/radio.html

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

5000

6000 CMB all sources resolved @150GHz removed resolved @40GHz removed

l(l+1

)cl/2

π (μ

K2 )

multipole l

150GHz

b ( d

eg)

b ( d

eg)

b ( d

eg)

b ( d

eg)

l (deg)90GHzl (deg)

b ( d

eg)

220GHz

WM

AP

1st

yr

BO

OM

ER

an

G 9

8

b ( d

eg)

l (deg)150GHz

41GHz l (deg) 60GHz l (deg) 94GHz l (deg)

Scatter Plot• Once point sources have been removed, one can

scatter-plot the experiment data (in V) vs. the WMAP data (in KCMB), and obtain the calibration constant (in V/K) from the slope of the best fit line.

• Problems of this approach:a) The experiment beam is different from the WMAP beam

(see next slide). b) The experiment response to large scales can be different

from the WMAP response c) The noise level of the two experiments is different: this

biases the best fit slope. • A solution for a) is to re-bin the maps in pixels larger

than the beams.• Problem b)&c) can be corrected carrying out detailed

simulations to estimate the bias.

B98-150GHz

WMAP 94GHz

B98150GHz

WMAP 94GHz11’G

aussian13’ G

aussian

13’ Gaussian

11’ Gaussian

a)

b)

b)

b)• Experiment data = yi ; WMAP data = xi

• In the case of BOOMERanG 98 and WMAP, in 7’pixels, σ(yi) ~30μK, σ(xi) ~80μK .

• Best slope estimate : 1) Remove averages from data sets yi and xi , so that y=ax2) Find the value of a which minimizes χ2 :

3) Make simulations of best fit lines for correlated data with different levels of noise for xi and yi. To understand if - for the noises of the experiment and of WMAP - there is a bias.

( ) ( )( ) ( )∑ +

−=

i ii

ii

xayaxya 222

22

σσχ

230.0 240.0 -20.0 -10.0 1380. 0.116 7.700

240.0 250.0 -20.0 -10.0 7332. 0.296 2.900

250.0 260.0 -20.0 -10.0 7353. 0.381 2.300

260.0 270.0 -20.0 -10.0 7289. 0.246 2.400

270.0 280.0 -20.0 -10.0 885. 0.221 2.900

230.0 240.0 -30.0 -20.0 3940. 0.305 3.200

240.0 250.0 -30.0 -20.0 6954. 0.305 2.500

250.0 260.0 -30.0 -20.0 6954. 0.285 2.700

260.0 270.0 -30.0 -20.0 6869. 0.279 2.500

270.0 280.0 -30.0 -20.0 143. 0.154 7.100

230.0 240.0 -40.0 -30.0 5518. 0.178 4.600

240.0 250.0 -40.0 -30.0 6213. 0.270 2.800

250.0 260.0 -40.0 -30.0 6213. 0.337 3.000

260.0 270.0 -40.0 -30.0 6158. 0.288 2.200

270.0 280.0 -40.0 -30.0 520. 0.227 2.600

230.0 240.0 -50.0 -40.0 5117. 0.188 4.200

240.0 250.0 -50.0 -40.0 5416. 0.285 3.100

250.0 260.0 -50.0 -40.0 5417. 0.291 2.900

260.0 270.0 -50.0 -40.0 5200. 0.245 3.000

270.0 280.0 -50.0 -40.0 1196. 0.243 2.500

230.0 240.0 -60.0 -50.0 230. 0.262 3.200

240.0 250.0 -60.0 -50.0 3415. 0.171 3.600

250.0 260.0 -60.0 -50.0 2583. 0.164 3.800

l(o) b(o) N R amin• Results for several regions:• First 4 columns define the

region, in Galactic coordinates; 5th column is the number of pixels observed by both experiments; 6th column is Pearson’s correlation coefficient; 7th column is the best fit calibration constant.

• Resulting average calibration: (3.5+0.3)ADU/μK

A variant of this correlation method is based on the cross-power spectrum:

– Compute the Angular Power Spectrum of the uncalibrated experiment, XX(l), and the Cross Power Spectrum between the uncalibrated experiment and WMAP, XW(l):

1/a(l)[V/K]= XW(l)/XX(l)– Using the same region, cosmic variance is not

effective– The method is computationally more costly– Beam and low multipoles response differences can

be taken into account easily: – 1/a(l)[V/K]= [XW(l)/(BX(l)BW(l))]/[XX(l)/BX

2 (l)] where B2 are the spherical harmonic transforms of the beams/responses (Hivon E. et al. 2003, Polentaet al. 2004).

100 200 300 400 500 6000.7

0.8

0.9

1.0

1.1

1.2

1.3B98 - 150 GHz A+A1+A2+B1 raw

c l[WM

APx

B98

]/cl[B

98*B

98]

multipole

No obvious trend vs multipole: beam calibration OKGain calibration: to be multiplied by 0.95 + 0.01

Hivon E. et al. 2003

All clcorrectedfor beam andfinite sky coverage

WMAP/B98 (gain recalibration)

0.95 +/- 0.010.95 +/-0.03Sum

0.97 +/- 0.030.96 +/- 0.03B150B2

0.95 +/- 0.020.97 +/- 0.03B150A2

0.85 +/- 0.020.89 +/- 0.03B150A1

0.96 +/- 0.020.95 +/- 0.03B150A

C(l) basedPixel basedChannel

Raw maps on 1.8% of the sky(Netterfield et al. cut)

0.95 +/- 0.010.95 +/-0.03Sum

0.95 +/- 0.020.95 +/- 0.03B150B2

0.98 +/- 0.020.98 +/- 0.03B150A2

0.92 +/- 0.030.91 +/- 0.03B150A1

0.96 +/- 0.020.95 +/- 0.03B150A

C(l) basedPixel basedChannel

Destriped maps on 1.8% of the sky(Netterfield et al. cut)

Hivon E. et al. 2003

• The nominal calibration of the 150 GHz map was off by 5% (well within the published 10% error)

• The new calibration is accurate to 1%, which is very good news for the calibration of B2K

Beam pattern calibration• We have seen before why beam calibration is so

important. • For example it affects directly the estimates of the

angular power spectrum at high multipoles:

• Where B is the spherical harmonics transform of the beam, a steeply decreasing function at high multipoles !

2,

l

ll B

cc measured=

B98-150GHz

WMAP 94GHz

B98150GHz

WMAP 94GHz11’G

aussian13’ G

aussian

13’ Gaussian

11’ Gaussian

R(θ)Bl2 SHT

BOOM98: 150 GHz window function

Combination of:

Pixelization (14’ healpix)Effective beam(including estimated 2’ rmspointing jitter)

Freq. FWHM90GHz 18’+2’

150GHz 10’+1’240GHz 14’+1’410GHz 13’+1’

Pointing jitter• As an example of how important can be the estimate of the instrument

beam and of systematic errors, let’s consider what happened for the first release of the BOOMERanG data (B98 Nature paper).

• The effective beam is the convolution of the telescope beam[(9.2+0.5)’FWHM @ 150 GHz] with the telescope pointing jitter.

• The results in Nature were based on a jitter estimate of (2+1)’rmsfrom a few scans of RCW38 done in CMB mode. This is, however, on the edge of the area surveyed for CMB measurements. We understand now that this result is not representative of all the data in CMB mode.

• With the improved pointing solution it is possible to infer the effective beam (and the jitter) from many more measurements of 3 AGN in the center of the CMB area. We see that the old pointing solution had a jitter of (4+2)’ rms -> Nature results should be corrected: the effective beam was (12.7+1.4)’FWHM instead of the assumed (10+1)’FWHM.

• The new pointing solution has a jitter of (2.5+2.0)’ rms. The effective beam for the new data with new pointing solution is (10.9+1.4)’FWHM.

Corresponding Effect on the PS

• Original data, as publishedin Nature, with published random and systematic errors

0 100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

original Nature data Nature +1σ gain Nature -1σ gain Nature +1σ gain +1σ beam Nature -1σ gain -1σ beam

l(l+1

)cl/2

π (μ

K2 )

multipole

Corresponding Effect on the PS

Original dataand data corrected forjitter underestimate

Correction substantial at l=600 (+35%, butstill within published

errors)

0 100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

original Nature data jitter underestimate corrected Nature +1σ gain Nature -1σ gain Nature +1σ gain +1σ beam Nature -1σ gain -1σ beam

l(l+1

)cl/2

π (μ

K2 )

multipole

Also Calibration Correction

• We also found a better treatment of the effect of high pass filters in the Dipole calibration

• 10% (1σ) decrease of gain i.e. additional20% coherent increase of the PS values

0 100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

original Nature data jitter underestimate

and gain corrected Nature +1σ gain Nature -1σ gain Nature +1σ gain +1σ beam Nature -1σ gain -1σ beam

l(l+1

)cl/2

π (μ

K2 )

multipole

0 100 200 300 400 500 6000

1000

2000

3000

4000

5000

6000

7000

B98 data corrected

l(l+1

)cl/2

π (μ

K2 )

multipole

Corrected data : the 2nd peak is not evident yet

• After the correction, there isa hint of a 2nd peak, but it is notstatisticallysignificant.

• the data (from a single bolometer) are still not sensitive enough.

192

196

200

204

208lo

catio

n of

pea

k

lp (m

ultip

ole)

450046004700480049005000

peak

am

plitu

de

lp(lp

+1)c

lp/2

π (μ

K2 )

6 8 10 12 14

0.260.280.300.32

beam FWHM (arcmin)

(l>32

0 av

erag

e)/

(pea

k am

plitu

de)

Effect of jitter underestimate in preliminary results: 10’ –> 12.7’

1%

4%

4%

Old beam New beam

Effects of jitter underestimate and calibration correction on science

• Cosmological parameters extraction from the corrected B98 together with the COBE-DMR PS data:

• Ωo remains the same – (1% effect is negligible)• Ωbh2 changes from 0.036+0.006 to 0.027+0.006

(same weak priors, l<625): • (cfr. BBN: Ωbh2 = 0.020+0.002) • We are comparing the density of baryons 3 minutes after the

big-bang (assuming it is the same as at z=3) to the density ofbaryons 300000 yrs after the Big Bang.

• Different physics (nuclear reactions vs acoustic waves in a plasma), different experimental methods and systematic effects!

ArtificialPlanetDiam = 20, 40 cmDist. 2 km(4, 8 arcmin)

Telescope BEAM CalibrationAt ground calibration with artificial planet (tethered blackbody + CCD monitor)

Telescope Beam Calibration : scans on RCW38For BOOMERanG this is a point source, very

useful to get our beam size. We have hundreds of scans for each detector, so we can obtain

both the telescope beam and the pointing jitter

2.5’

P.de Bernardis Oct.2000

Compact HII region in an areafree from Galactic confusionAcbar data at 1.4 mm = 2.5’ diam.