1

Misure della CMB• Da oggi ci concentriamo sul Fondo Cosmico a

Microonde (CMB, Cosmic MicrowaveBackground) e sulla sua misura.

• Il piano delle lezioni e’ il seguente:1. Le osservabili della CMB: brillanza, anisotropie,

polarizzazione e loro dipendenza spettrale. Limiti di detezione intrinseci.

2. Telescopi, fotometri, polarimetri, spettrometri per la misura della CMB. Osservazioni da terra e dallo spazio. Metodi di calibrazione.

3. I programmi di misura della CMB e dei foregrounds, in corso e futuri.

What is the CMB According to moderncosmology:An abundant background of photons filling the Universe.

• Generated in the very earlyuniverse, less than 4 μs after the Big Bang (109γ for each baryon)

• Thermalized in the primevalfireball (in the first 380000 years after the big bang) byrepeated scattering against freeelectrons

• Redshifted to microwavefrequencies and diluted in the subsequent 14 Gyrs of expansion of the Universe

γ2→+ bb

t

10−6s

1013s

1017s

visibleNIRMW

visibleNIRMW

T=3000K

T=3K

em

now

em

now

rrz ==+

λλ1

T >1GeV

B(ν)

B(ν)

Today 400γ/cm3

• The CMB has beenserendipitouslydiscovered in 1964 by A. Penzias and R. Wilson, asa thermal noise at a temperature of 3K.

• Robert Dicke and hisstudent David Wilkinson, in Princeton, and Zel’dovich’s group in the Soviet Union werealready looking for thisradiation when it wasdiscovered.

• Earlier detections havebeen found a-posteirori.

• A 3K blackbody contains 400γ/cm3 ! A lot of photons in the volume of the Universe.

• It is instructive to see why the CMB, representingthe majority of photons in the universe, escapeddetection till 1964.

The CMB brightness is maximum at wavelengthsbetween 10 cm and 0.5 mm: the last range tosee the development of sensitive detectors.The CMB is faint with respect to local emissionin the earth environment.The CMB is isotropic: difficult to separe frominstrumental constant emissionThe earth atmosphere is not transparent in mostof the range.

• But, also, scientists are human beings, are subjectto prejudices, and make errors.

observer

InterstellargascloudTc=2.3KWhy ?

hotstar

G. Herzberg (1950, Spectra of diatomic molecules): “… 2.3K has of course a very restricted meaning”

T=(3+2) K @ λ=33cm

Le Roux surveyed a good fraction of the sky with a sensitive radioreceiverat 33cm of wavelength in 1957. He originally wrote in his thesis thatoutside the Galactic Plane the temperature was 3K. But his result was thenpublished as an upper limit.

2

C.R.Academ.Sci.(1957) 244, 3030

T<3K @ λ=33cm

• Anyway, it was a very poor detection.• The problem was the sensitivity and stability of

radioreceivers.• The breakthroughs of Penzias and Wilson were

• the use of a differential receiver withsynchronous modulation and demodulation, invented independently by Golay and Dickearound 1946.

• The use of an antenna with very low sidelobes• The use of an accurate, cold and stable

reference of emission to which sky brightnesswas continuously compared.

• Without any of these improvements, and withoutthe ostination in excluding all the possible spuriouseffects, they would not detect the CMB.

Importance of a differentialdetection technique

• Detectors have drifts, due to the fact that theirparameters and the environment are notperfectly stable.

• These drifts are very slow low level fluctuationsof the output signal independent of the input signal.

• As a result the spectrum of the detector output has a 1/f component, diverging at f=0.

detector drift(no signal)

detector drift plus modulated signal

High-passedmodulated signal(drift removal)

Sky

mirror

chopper

ColdReferenceSource

Detector

Sky signal

Cold Reference signal

)(21)sin()(

)()sin()()(

refsky

refsky

TTttS

tntTTtS

−=

+−=

∞→ττω

ω

Differential Radiometer

Referencesource

• In the case of Penziasand Wilson, the emwaves were collectedby an off-axis, lowsidelobes antenna, and propagated in a waveguide.

• The chopper was a ferrite “Dicke switch”

• The reference sourcewas a “cold load”cooled at 4.2K withliquid helium.

• The detector was a low-noise maser amplifier.

3

Importance of low sidelobes• The power detected is the integral of the brightness

times the solid angle, weighted with the angularresponse of the telescope:

• Typical telescoperesponse RA(θ,φ)

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

RA(θ)θmain lobe

side lobes

Brightness from direction (θ,φ)Telescope response

in direction (θ,φ)

FWHM=λ/D

boresight

Importance of low sidelobes• The power detected is the integral of the brightness

times the solid angle, weighted with the angularresponse of the telescope:

• Typical telescoperesponse RA(θ,φ)

RA(0)=1

main lobe

side lobes

Brightness from direction (θ,φ)Telescope response

in direction (θ,φ) boresight

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

Importance of low sidelobes• The power detected is the integral of the brightness

times the solid angle, weighted with the angularresponse of the telescope:

• Typical telescoperesponse RA(θ,φ)

RA(θ)<1θmain lobe

side lobes

Brightness from direction (θ,φ)Telescope response

in direction (θ,φ) boresight

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

Importance of low sidelobes• The power detected is the integral of the brightness

times the solid angle, weighted with the angularresponse of the telescope:

• Typical telescoperesponse RA(θ,φ)

RA(θ)<<1θ

main lobe

side lobes

Brightness from direction (θ,φ)Telescope response

in direction (θ,φ) boresight

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

Importance of low sidelobes• The power detected is the integral of the brightness

times the solid angle, weighted with the angularresponse of the telescope:

• Typical telescoperesponse RA(θ,φ)

RA(θ)<<1

θ

main lobe

side lobes

Brightness from direction (θ,φ)Telescope response

in direction (θ,φ) boresight

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

Importance of low sidelobes• In the case of CMB

observations, the detectedbrightness is the sum of the brightness from the sky(dominant for the solid anglesdirected towards the sky, in the main lobe) and the Brightness from ground(dominant for the solid anglesdirected towards ground, in the sidelobes).

RA(θ)θmain lobe

side lobes

FWHM=λ/D

boresight

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡Ω+Ω= ∫∫ dRABdRABAW

lobesside

Ground

lobemain

sky ),(),(),(),( ϕθϕθϕθϕθ

4

Importance of low sidelobes

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡Ω+Ω= ∫∫ dRABdRABAW

lobesside

Ground

lobemain

sky ),(),(),(),( ϕθϕθϕθϕθ

⎥⎦

⎤⎢⎣

⎡Ω+Ω≈

lobesside

lobessideGround

lobemain

lobemainsky RABRABAW ),(),(),(),( ϕθϕθϕθϕθ

signal of interest disturbance signal

K3≈ srad1<< sradπ2≈K300≈

Obtaining : signal of interest >> disturbance signalrequires

),(),( ϕθϕθlobesside

lobemain RARA >>>

1≈

⎥⎦

⎤⎢⎣

⎡Ω+Ω≈

lobesside

lobessideGround

lobemain

lobemainsky RABRABAW ),(),(),(),( ϕθϕθϕθϕθ

signal of interest disturbance signal

K3≈ srad1<< sradπ2≈K300≈

600

)(

),(),(

),(),(srad

BB

RARA lobemain

Ground

sky

lobesside

lobemain

lobemain

lobesside

Ω≈⎥

⎦

⎤⎢⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Ω

Ω<<

ϕθϕθ

ϕθϕθ

1≈

<<1x10-107x10-8 srad1’

<<1x10-87x10-6 srad10’

<<4x10-72x10-4 srad1o

<<4x10-52x10-2 srad10o

<RAsidelobes>ΩmainlobeFWHM

!!!

What is RA(θ,φ) ?• Fraunhofer diffraction from a

circular aperture (radius a) (at largedistances from shield)

• The incident wave is an infinite plane wave (wavevector kin parallelto the z axis).

• The outgoing wave is not infinite, and for this very reason it will havecomponents with wavevectors k in different directions. So it is not a plane wave.

• We want to find out which are the amplitudes of the differentcomponents of the outgoing wave.

z

a

kθ

kin

r dS

What is RA(θ,φ) ?• For small angles θ the vector

q=k-kin is in the plane of the aperture and q=kθ.

• The diffracted component withwavevector k is the sum of the contributions from all the elements dS of the aperture, each with its own phase:

z

a

kθ

kin

∫∫ ⋅−=S

rqioq dSeuu

rr

q

rdS

aqaqJurdrqrJudrrdeuu o

a

oo

aiqr

oq)(2)(2 1

00

2

0

cos === ∫∫ ∫ − πϕπ

ϕ

Bessel functions

q

What isRA(θ,φ) ?

• The intensity isthe square of the field:

• Example: a 2m diameter mirrorused at 1 cm and at 1 mm

2

1 )(2⎥⎦

⎤⎢⎣

⎡=

Ω θθ

akakJI

ddI

o

0.0 0.1 0.2 0.3 0.4 0.50.0

0.2

0.4

0.6

0.8

1.0

a=1m λ=1 cm λ=1 mm

angu

lar r

espo

nse

off-axis angle θ (deg)

mainlobe

sidelobes

What isRA(θ,φ) ?

• The intensity isthe square of the field:

10-3 10-2 10-1 100 101 10210-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1100

a = 1 m λ=1 cm λ=1 mm

angu

lar r

espo

nse

off-axis angle θ (deg)

mainlobe

sidelobes

2

1 )(2⎥⎦

⎤⎢⎣

⎡=

Ω θθ

akakJI

ddI

o

5

What isRA(θ,φ) ?

• The intensity isthe square of the field:

• The first zero isfor

10-3 10-2 10-1 100 101 10210-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1100

a = 1 m λ=1 cm λ=1 mm

angu

lar r

espo

nse

off-axis angle θ (deg)a2

22.110λθ =

θ10

2

1 )(2⎥⎦

⎤⎢⎣

⎡=

Ω θθ

akakJI

ddI

o

What isRA(θ,φ) ?

• The intensity isthe square of the field:

• The first zero isfor

• The FWHM issimilar

10-3 10-2 10-1 100 101 10210-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1100

a = 1 m λ=1 cm λ=1 mm

angu

lar r

espo

nse

off-axis angle θ (deg)a2

22.110λθ =

0.5 FWHM

θ10

2

1 )(2⎥⎦

⎤⎢⎣

⎡=

Ω θθ

akakJI

ddI

o

What isRA(θ,φ) ?

• The intensity isthe square of the field:

• The envelope of the off-axisresponse scalesas θ-3

approximatelystarting from 0.5 at the FWHM

10-3 10-2 10-1 100 101 10210-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1100

a = 1 m λ=1 cm λ=1 mm

angu

lar r

espo

nse

off-axis angle θ (deg)

3)( −∝θθRA

2

1 )(2⎥⎦

⎤⎢⎣

⎡=

Ω θθ

akakJI

ddI

o

exercise• For the angular response of a circular

aperture, compare the power in the mainlobe to the power in the sidelobes

• Hint:

• But use approximated formulas …

θθθ

θπ

θθθ

θπ

λ

π

λ

dak

akJBA

dak

akJBA

P

P

a

lobemain

a lobesside

lobemain

lobesside

sin)(22

sin)(22

/61.0

0

2

1

/61.0

2

1

∫

∫

⎥⎦

⎤⎢⎣

⎡

⎥⎦

⎤⎢⎣

⎡

=

Low diffraction design• Real world angular responses are worse than the one

studied here.• Sharp edges are in general important sources of

diffraction, and must be avoided in low sidelobesdesign. Use smoothed edges.

• A trumpet has a slow transition to free space at the aperture to avoid diffraction of sound waves.

• The spider supporting the secondary mirror in a Cassegrain telescope is an important source of diffraction.

• Penzias and Wilson used an under-illuminated off-axis paraboloid, to get low sidelobes

10dB = a factor 10 in power

6

Other example of low sidelobes design: Planck

STRAY LIGHT

Main Spillover

Main Beam

Sub Spillover

Main Spillover

F. Villa, LFI

Main Beam

Near Sidelobes

Angle from boresight

Resp

onse

Far Sidelobes

F. Villa, LFI

107

Anotherlow sidelobes design:

BOOMERanG

Even more stringentnequirements due

near earth operation

R

TS

P

1.3 m

Ground Shield

Sun Shield

Cold Lyot Stop+ baffles

7

Angle ofsunshield

• A single point (T=3K @ λ=4.3 cm) is notenough to demonstrate that the CMB isthermal radiation.

• Coverage of the interesting frequencyrange to establish the black-body nature of the CMB has been a difficult challange, involving for decades researchers in Princeton (Wilkinson), Berkeley (Smoot, Lubin), Firenze (Melchiorri), Milano (Sironi), Bologna (Mandolesi) …

• At wavelengths > 20 cm our Galaxybecomes brighter than the CMB.

• At wavelengths < 10 cm the correction foratmospheric emission is significant even at high altitude sites. It is necessary to movethe experiment outside the earthatmosphere.

• Balloon experiments (Paul Richards) and finally the COBE satellite (FIRAS, JohnMather) proved the Planckian nature of the CMB.

CMB and cosmology• 1992: COBE-FIRAS

measures the spectrum of the CMB with incredibleprecision (1/10000)

• The thermal spectrum at 2.735K and the high photons to barions ratio together with the measuredprimordial abundances of light elements is evidencefor a hot initial phase of the Universe.

J. Mather et al. 1992

COBE-FIRAS• COBE-FIRAS was a Martin-Puplett

Fourier-Transform Spectrometer withcomposite bolometers. It was placed in a 400 km orbit.

• A zero instrument comparing the specific sky brightness to the brightnessof a cryogenic Blackbody

• The output was nulled (within detector noise) for Tref=2.725 K

• The brightness of empty sky is a blackbody at the same temperature !

• The early universe was in thermalequilibrium at high Temperature.

σ (cm-1) wavenumber

8

• The spectrum

KTec

hTB

CMB

x

725.21

2),(3

2

=−

=νν

mmTBTB 06.1),(),( max =⇒= λλνλν

GHzkThx

CMBCMB 56

νν≅=

)31.5(159

82.23

1

1maxmax

maxmaxmax

−

−

==

⇒=⇒=−

cmGHz

xxe x

σν

WienRJ

• Techniques ?

???160bolometers160

detectorscoherent 160

max

max

max

⇒=≈⇒=>>⇒=<<

GHzGHzGHz

νννννν

WienRJ

A microwave receiver• This antenna for satellite

broadcast is a microwavereceiver at a frequency of 10 GHz.

• If we point this receiver awayfrom the geostationary satellite, it will receive the CMB !

• Unfortunately, the noise of thiskind of receiver, due to thermalmotions of the electrons in the input circuit, is too high for thispurpose.

• The only way to reduce the noise is to cool down the receiver at cryogenictemperature.

• The CMB is only slightly anisotropic.

• The brightness (temperature) fluctuationsare due to small density fluctuationspresent in the primeval fireball, and totheir motions:

ppmTT

CMB

CMB 1010 5 =≈Δ −

PhotonDensityfluctuations

Gravitationalredshift

Scatteringagainstmoving e-

ccTT v

31

41

2 +Δ

+Δ

=Δ ϕ

ρρ

γ

γ

Image of Solar Granulation

8 light minutesHere, now

Plasma in the solar photosphere

(5500 K)

Image of Solar Granulation

The BOOMERanG map of the last scattering surface

8 light minutes

14 billion light years

Here, now

Here, now

Plasma in the solar photosphere

(5500 K)

Plasma in the LSS the cosmicphotosphere

(3000 K)

9

CMB observables

• The spectrum

• The angulardistribution

kThx

echTB x

ννν =−

=1

2),(3

2

GHzxTTTB

exeTB

TTBdTdTB

x

x

21783.3

),(1

),(

),(),(

maxmax =⇒=

Δ−

=Δ

Δ=Δ

ν

νν

νν

1010 1011 1012

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

W m-2 sr-1 Hz-1

W (m2 sr Hz)-1/2 Hz-1/2

W m-2 sr-1 Hz-1

W m-2 sr-1 Hz-1

T=2.725 K

average brightness anisotropy (rms) polarization (E rms) photon noise (rms) polarization (B rms)

CM

B (M

KS

uni

ts)

Frequency (Hz)

• The angular distribution of the CMB isthe map

( ) ( )bTTdecRA

TT

TT

CMB

CMB

CMB

CMB

CMB

CMB ,or, lΔΔ

=Δ

Red=slightly hotterGreen=average (2.725 K)Blue=slightly colder

WMAP, 94 GHz

• The map gives punctual information: it is a picture of the last scattering surface at redshift 1000. It is useful to– test the purity of the detected CMB signal, i.e.

the absence of foreground radiation– test the absence of instrumental noise or

systematics– test the gaussianity of the detected CMB signal

• However, we have only a statisticaltheory of the fluctuations in the earlyuniverse.

• From theory we can then predict only the general, statistical properties of the CMB anisotropy map, not its detailed pattern.

IF ΔTCMB(l,b) IS A GAUSSIAN RANDOM FIELD,

THEN ALL INFORMATION IS ENCODED IN ITS

ANGULAR POWER SPECTRUM

CMB observables• The rms anisotropy

has contributionsfrom many angularscales

• The angular power spectrum cl of the anisotropy definesthe contribution tothe rms from the different multipoles:

( ) ( )

∑

∑

+=Δ

=

=Δ

ll

ll

lll

l cT

ac

YaT

m

m

mm

)12(41

,,

2

2

,

π

ϕθϕθ

1010 1011 1012

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

W m-2 sr-1 Hz-1

W (m2 sr Hz)-1/2 Hz-1/2

W m-2 sr-1 Hz-1

W m-2 sr-1 Hz-1

T=2.725 K

average brightness anisotropy (rms) polarization (E rms) photon noise (rms) polarization (B rms)

CM

B (M

KS

uni

ts)

Frequency (Hz)

CMB anisotropy observables

• The angular power spectrum cl of the anisotropy definesthe contribution tothe rms from the different multipoles:

( ) ( )

∑

∑

+=Δ

=

=Δ

ll

ll

lll

l cT

ac

YaT

m

m

mm

)12(41

,,

2

2

,

π

ϕθϕθ∑ +=Δl

lll cwTmeas

)12(412

π

2)1( σ+−= lll ew LP

• A real experiment will notbe sensitive to all the multipoles of the CMB.

• The window function wldefines the sensitivity of the instrument to differentmultipoles.

• The detected signal will be:

• For example, if the angularresolution is a gaussianbeam with s.d. σ, the corresponding window function is

10

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

5000

6000

l(l+1

)cl/2

π (μ

K2 )

multipole l

( ) ( )

∑

∑

+=Δ

=

=Δ

ll

ll

lll

l cT

ac

YaT

m

m

mm

)12(41

,,

2

2

,

π

ϕθϕθ

Expected power spectrum:

Why this way ?

cl

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

5000

6000

l(l+1

)cl/2

π (μ

K2 )

multipole l

( ) ( )

∑

∑

+=Δ

=

=Δ

ll

ll

lll

l cT

ac

YaT

m

m

mm

)12(41

,,

2

2

,

π

ϕθϕθ

Expected power spectrum:

Because the early universeis a plasma of photons and baryons, and because of horizon effects.

cl

Density perturbations (Δρ/ρ) were oscillating in the primeval fireball (as a result of the opposite effects of gravity and photon pressure). After recombination, density perturbation can grow and create the hierarchy of structures we see in the nearbyUniverse.

Before recombination

After recombination T < 3000 KT > 3000 K

overdensity

Due to gravity, Δρ/ρ increases, and so does T

Pressure of photonsincreases, and the perturbation bounces back

T is reduced enoughthat gravity wins again

Here photons are not tightlycoupled to matter, and theirpressure is not effective. Perturbations can grow.

tthe Universe is a plasma

the Universe is neutral

Size of sound horizon

timeBig-bang recombination Power Spectrum

mul

tipo l

e22

045

0

1st peak

2nd peak

LSS

300000 ly

In the primeval plasma, photons/baryons density perturbations start to oscillate only when the sound horizonbecomes larger than their linear size . Small wavelength perturbations oscillate faster than large ones.

R

R

C

C

C

C

1st dip

2nd dip

The

an g

le su

bten

ded

depe

n dso

n th

e ge

ome t

ryof

spac

e

size of perturbation(wavelength/2)

300000 y0 y

v vv

v v

v v

v

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

5000

6000

l(l+1

)cl/2

π (μ

K2 )

multipole l

( ) ( )

∑

∑

+=Δ

=

=Δ

ll

ll

lll

l cT

ac

YaT

m

m

mm

)12(41

,,

2

2

,

π

ϕθϕθ

An instrumentwith finite angularresolution is not sensitive to the smallest scales(highest multipoles). For a gaussian beam with s.d. σ:

Expected power spectrum:

0 200 400 600 800 1000 1200 14000.0

0.2

0.4

0.6

0.8

1.0 20' FWHM 10' FWHM 5' FWHM

7o FWHM

wl

multipole

2)1( σ+−= lll ew LP

cl

0 200 400 600 800 1000 1200 14000

1000

2000

3000

4000

5000

6000

l(l+1

)cl/2

π (μ

K2 )

multipole l

( ) ( )

∑

∑

+=Δ

=

=Δ

ll

ll

lll

l cT

ac

YaT

m

m

mm

)12(41

,,

2

2

,

π

ϕθϕθ

∑ +=Δl

lll cwT LP

meas)12(

412

π2)1( σ+−= ll

l ew LP

rms signal in an instrumentwith gaussian beam σ :

Expected power spectrum:

1000 100 100

20

40

60

80

100

120

FWHM - gaussian beam (arcmin)

ΔTrms (μK)

CO

BE

WM

AP

BO

OM

ER

anG

5102.4 −×=Δ

rmsTT

cl

11

CMB observables• The polarization

state

• Extremely weak !

7

6

102

104

),(1

),(

−

−

×≈Δ

×≈Δ

Δ−

=Δ

rmsB

P

rmsE

P

Px

x

P

TTTT

TTTB

exeTB νν

1010 1011 1012

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

W m-2 sr-1 Hz-1

W (m2 sr Hz)-1/2 Hz-1/2

W m-2 sr-1 Hz-1

W m-2 sr-1 Hz-1

T=2.725 K

average brightness anisotropy (rms) polarization (E rms) photon noise (rms) polarization (B rms)

CM

B (M

KS

uni

ts)

Frequency (Hz)

• CMB photons are Thomson scattered at recombination.• If the local distribution of incoming radiation in the rest

frame of the electron has a quadrupole moment, the scattered radiation acquires some degree of linearpolarization.

Last scattering surface

CMB polarization

-

-

+

-

+x

y

--

+

-

+

x

y

-x

y

-10ppm +10ppm

= e- at last scattering

+ +-

-+

+-- + +

-

-+

+--

Overdensity Underdensity Overdensity Underdensity

v v v v v v v v

EE

EE

E-modes in the polarization pattern

Convergingflux

Same flux asseen in the

electronreference frameDiverging

flux

Quadrupole anisotropydue to Doppler effect

redshift

blueshift

blueshift

redshift

+ +

+

+

- -

-

-

resultingCMB polarizationfield (E-modes)

Velocity fieldsat recombination • This component of the CMB polarization field is called E

component, or gradient component. This is the only kind of polarization produced at recombination.

• It is related to velocity fields. For acoustic oscillations, it will bemaximum for perturbations with maximum velocity and zero density contrast.

• So we expect peaks in this polarization power spectrum where wehave minima in the temperature anistropy power spectrum.

• The amplitude of the polarization signal depends on the length of the recombination process (it is not produced before, nor later).

12

• Tensor perturbations (gravity waves) also produce quadrupoleanisotropy. The generation of a faint stochastic background of gravity waves is a generic feature of all inflationary processes.

• The resulting polarization pattern is shear-like. • The amplitude of the effect is very small.• This component of the CMB polarization field is called B-modes

component, or curl component.

• Velocity fields cannot produce B modes.• Weak lensing can, but is subdominant at scales larger than 1 deg.• Mathematical alghoritms exist to separate B modes and E modes.

• The amplitude of this effect is very small, but depends on the Energy scale of inflation. In fact the amplitude of tensormodes normalized to the scalar ones is:

• and

• There are theoretical arguments to expectthat the energy scale of inflation is close tothe scale of GUT i.e. around 1016 GeV.

• The current upper limit on anisotropy at large scales gives T/S<0.5 (at 2σ)

GeV107.3 16

4/14/1

2

24/1

×≅⎟⎟

⎠

⎞⎜⎜⎝

⎛≡⎟

⎠⎞

⎜⎝⎛ V

CC

ST

Scalar

GW Inflation potential

The background of Gravitational Waves

⎥⎥⎦

⎤

⎢⎢⎣

⎡

×≅

+GeV102

1.02

)1(16

4/1

maxVKcB μ

π l

ll

T/S=0.28

E ?

rms B-modespolarization signal

>2 ordersof magnitudesmaller than

rms T anisotropy !

Summary of CMB observables:

• CMB spectrum• CMB anisotropy angular

power spectrum• CMB polarization angular

power spectra:–E-modes–B-modes

IncreasingDifficulty

KT 725.2=

KTrms μ80=Δ

KT rmsE μ2, =Δ

KT rmsB μ1.0, <Δ

Intrinsic CMB measurements limits

• Now that we know what the CMB is, and how it was measured, we can study its noiseproperties, which limit the ultimate sensitivity of its measurement.

Radiation Noise• Is the fundamental limit of any measurement.• Photon noise reflects the particle-wave duality of

photons.• It is the sum of Poisson noise (particles) PLUS

interference noise (waves)• Poisson noise:

( ) ( ) ( ) WthhWhNhNhE νν

ννν ===Δ=Δ 22222

This is a typical random-walk process (variance prop.to time).Using Einstein’s generalizationwe get the power spectrum and the varianceof radiative power fluctuations:

kBTdfdfkBTt f 42 22 =⇒=Δ θθ &

WdfhdfW f ν22 =Δ

13

Radiation Noise• Orders of magnitude example: A He-Ne 1 mW

laser beam has a perfect poisson statistics, so

• Notice the power spectrum units (remember thatthe integral of the PS over frequency is the variance).

• In this case the intrinsic fluctuations per unitbandwidth are >7 orders of magnitude smaller thanthe signal.

• It is useless to build a complex detector with a noise of for this measurement: the precision of the measurement will be limited at a level of

HzWWhW f

112 105.22 −×==Δ ν

HzW /10 15−

HzW /105.2 11−×

Radiation Noise• Thermal radiation (like the CMB) has also

wave interference noise: the correct statisticsis Bose-Einstein.

gN

NN

dNd

TNe

gNVT

kTE

22

,

2/)( ;

1

+=Δ⇒

=Δ−

= − μμ

Poisson noiseWave interference noise

Radiation Noise• For a blackbody

⎥⎦⎤

⎢⎣⎡

−+=Δ⇒

−=

=−

= −

111

118

42;1

/2

/3

2

3

2

/)(

kTh

kTh

kTE

eNN

Vdec

N

Vdc

ge

gN

ν

ν

μ

νπν

νπν

Poisson noise, important at shortwavelengths

Wave interference noise,Important at low frequencies

Radiation Noise

( )

52

1832

5

45

32

52

/2

//22

/2

1077.24

14

1112

111

111

111

2

1

KHzsrcmW

hck

dfdxe

exTAhckdfW

dfe

WhdfW

eWh

eNhE

eNN

x

xx

x

kTh

kThkTh

kTh

−×=

−Ω=Δ

⎥⎦⎤

⎢⎣⎡

−+=Δ

⎥⎦⎤

⎢⎣⎡

−+=⎥⎦

⎤⎢⎣⎡

−+=Δ

⎥⎦⎤

⎢⎣⎡

−+=Δ

∫

ν

νν

ν

ν

νν

CMB observables• The spectrum

• The angulardistribution

• The polarizationstate

• The noise

kThx

echTB x

ννν =−

=1

2),(3

2

TTTB

exeTB x

x Δ−

=Δ ),(1

),( νν

TTTB

exeTB Px

x

PΔ

−=Δ ),(

1),( νν

1010 1011 1012

10-24

10-23

10-22

10-21

10-20

10-19

10-18

10-17

W m-2 sr-1 Hz-1

W (m2 sr Hz)-1/2 Hz-1/2

W m-2 sr-1 Hz-1

W m-2 sr-1 Hz-1

T=2.725 K

average brightness anisotropy (rms) polarization (E rms) photon noise (rms) polarization (B rms)

CM

B (M

KS

uni

ts)

Frequency (Hz)( ) dxe

exhcTkdxTW

x

x

2

4

32

552

14),(

−=Δ ν

Noise and integration time• Any detector has a response time τ which limits its

sensitivity at high post-detection frequencies. Data taken at intervals shorter than τ will not beindependent.

• The error on the estimate of , the average power in the observation time t, is

•

• where N is the number of independent measurements. In the integration time t, it will be N=t/τ.

N

dfW

N

f

fWW t

∫ Δ

==

max

min

2

σσ

tW

14

Noise and integration time•

• The noise decreases as the square root of the integration time.

• Notice that this is applies equally to detector noise and to intrinsic radiation noise.

tW

tt

W

t

dfW

t

dfW

N

dfW

N

T

f

f

f

fWW T

22

/1

/1

2

22

/

11

/

/

max

min

max

min

Δ≅

⎥⎦⎤

⎢⎣⎡ −Δ

=

Δ

≅

=

Δ

=

Δ

==

∫

∫∫

ττ

τ

τ

σσ

τ

Noise and integration time• Numerical example: CMB anisotropy (or

polarization) measurement limited only byradiation noise:

( )

( )t

dxe

exAhcTk

dxe

exAhcTk

TT

dxTxBe

xeB

TT

dxTxBe

xeTTTB

x

xx

x

x

xx

x

x

xx

x

x

xx

x

1

)1(2

14

),(1

),(1

),(

2

1

2

1

2

1

2

1

2

4

32

44

2

4

32

55

∫

∫

∫

∫

−Ω

−Ω

=⎟⎠⎞

⎜⎝⎛ Δ

−

Δ=⎟

⎠⎞

⎜⎝⎛ Δ

−Δ

=Δ

σ

σσ

ν

1 10 100 1000 100000.01

0.1

1

10

150 GHz,10% BW, λ2

150 GHz, 10% BW, 1 cm2sr

30 GHz, 10% BW, λ2

erro

r per

pix

el (μ

K)

integration time (s)

The ultimate sensitivity plot !!

CMB BLIP1900 1920 1940 1960 1980 2000 2020 2040 2060

102

107

1012

1017

Langley's bolometerGolay Cell

Golay Cell

Boyle and Rodgers bolometer

F.J.Low's cryogenic bolometer

Composite bolometer

Composite bolometer at 0.3K

Spider web bolometer at 0.3KSpider web bolometer at 0.1K

1year

1day

1 hour

1 second

Development of thermal detectors for far IR and mm-waves

time

requ

ired

to m

ake

a m

easu

rem

ent (

seco

nds)

year

Radiation Noise

∫ −+−

Ω=Δ

⎥⎦⎤

⎢⎣⎡

−+=Δ

=

2

11

)1(4

11''

'

45

32

52

/2

x

xx

x

kTh

dfdxeexTA

hckdfW

eNN

NN

εε

εε

ε

ν

• For a grey-body with emissivity• relevant cases:

– Radiation emitted by a mirror)– Radiation emitted by the atmosphere in the

atmospheric windows

1<εRadiation Noise

∫ −+−

Ω=Δ

⎥⎦⎤

⎢⎣⎡

−+=Δ

=

2

11

)1(4

11''

'

45

32

52

/2

x

xx

x

kTh

dfdxeexTA

hckdfW

eNN

NN

εε

εε

ε

ν

• For a grey-body with emissivity• relevant cases:

– Radiation emitted by a mirror– Radiation emitted by the atmosphere in the

atmospheric windows

1<ε