Misure della CMB
• Da oggi ci concentriamo sul Fondo Cosmico a Microonde (CMB, Cosmic Microwave Background) 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 modern cosmology:An abundant background of photons filling the Universe.
• Generated in the very early universe, less than 4 μs after the Big Bang (109γ for each baryon)
• Thermalized in the primeval fireball (in the first 380000 years after the big bang) by repeated scattering against free electrons
• Redshifted to microwave frequencies and diluted in the subsequent 14 Gyrs of expansion of the Universe
γ
→ 2 + b b
t
10−6s
1013s
1017s
visible NIR MW
visible NIR MW T=3000K
T=3K
em now
em now
r z= =r
+ λ
1 λ
T >1GeV B(ν)
B(ν)
Today 400γ/cm3
• The CMB has been serendipitously
discovered in 1964 by A.
Penzias and R. Wilson, as a thermal noise at a temperature of 3K.
• Robert Dicke and his student David Wilkinson, in Princeton, and Zel’dovich’s group in the Soviet Union were already looking for this radiation when it was discovered.
• Earlier detections have been 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, representing the majority of photons in the universe, escaped detection till 1964.
The CMB brightness is maximum at wavelengths between 10 cm and 0.5 mm: the last range to see the development of sensitive detectors.
The CMB is faint with respect to local emission in the earth environment.
The CMB is isotropic: difficult to separe from instrumental constant emission
The earth atmosphere is not transparent in most of the range.
• But, also, scientists are human beings, are subject to prejudices, and make errors.
observer Interstellar gas cloud Tc=2.3K Why ?
hot star
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 radioreceiver at 33cm of wavelength in 1957. He originally wrote in his thesis that outside the Galactic Plane the temperature was 3K. But his result was then published as an upper limit.
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 with synchronous modulation and demodulation, invented independently by Golay and Dicke around 1946.
• The use of an antenna with very low sidelobes
• The use of an accurate, cold and stable reference of emission to which sky brightness was continuously compared.
• Without any of these improvements, and without the ostination in excluding all the possible spurious effects, they would not detect the CMB.
Importance of a differential detection technique
• Detectors have drifts, due to the fact that their parameters and the environment are not perfectly stable.
• These drifts are very slow low level fluctuations of 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-passed modulated signal (drift removal)
Sky
mirror
chopper
Cold Reference Source
Detector Sky signal
Cold Reference signal
) 2(
) 1 sin(
) (
) ( ) sin(
) ( ) (
ref sky ref sky
T T t
t S
t n t T T t S
−
= +
−
=
∞
→ τ τ
ω
ω
Differential Radiometer
Reference source
• In the case of Penzias and Wilson, the em waves were collected by an off-axis, low sidelobes antenna, and propagated in a waveguide.
• The chopper was a ferrite “Dicke switch”
• The reference source was a “cold load”
cooled at 4.2K with liquid helium.
• The detector was a low-noise maser amplifier.
Importance of low sidelobes
• The power detected is the integral of the brightness times the solid angle, weighted with the angular response of the telescope:
• Typical telescope response RA(θ,φ)
Ω
=A
∫
B RA dW ( , ) ( , )
4
ϕ θ ϕ θ
π RA(θ)
θ main 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 angular response of the telescope:
• Typical telescope response RA(θ,φ)
RA(0)=1 main lobe
side lobes Brightness from direction (θ,φ) Telescope response
in direction (θ,φ) boresight
Ω
=A
∫
B RA dW ( , ) ( , )
4
ϕ θ ϕ θ
π
Importance of low sidelobes
• The power detected is the integral of the brightness times the solid angle, weighted with the angular response of the telescope:
• Typical telescope response RA(θ,φ)
RA(θ)<1 θ main lobe
side lobes Brightness from direction (θ,φ) Telescope response
in direction (θ,φ) boresight
Ω
=A
∫
B RA dW ( , ) ( , )
4
ϕ θ ϕ θ
π
Importance of low sidelobes
• The power detected is the integral of the brightness times the solid angle, weighted with the angular response of the telescope:
• Typical telescope
response RA(θ,φ) θ RA(θ)<<1
main lobe
side lobes Brightness from direction (θ,φ) Telescope response
in direction (θ,φ) boresight
Ω
=A
∫
B RA dW ( , ) ( , )
4
ϕ θ ϕ θ
π
Importance of low sidelobes
• The power detected is the integral of the brightness times the solid angle, weighted with the angular response of the telescope:
• Typical telescope response RA(θ,φ)
RA(θ)<<1 θ
main lobe
side lobes Brightness from direction (θ,φ) Telescope response
in direction (θ,φ) boresight
Ω
=A
∫
B RA dW ( , ) ( , )
4
ϕ θ ϕ θ
π
Importance of low sidelobes
• In the case of CMB observations, the detected brightness is the sum of the brightness from the sky (dominant for the solid angles directed towards the sky, in the main lobe) and the Brightness from ground (dominant for the solid angles directed towards ground, in the sidelobes).
RA(θ) θ main lobe
side lobes
FWHM=λ/D boresight
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
Ω +
Ω
=A
∫
B RA d∫
B RA dW
lobes side
Ground lobe
main
sky(θ,ϕ) (θ,ϕ) (θ,ϕ) (θ,ϕ)
Importance of low sidelobes
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡
Ω +
Ω
=A
∫
B RA d∫
B RA dW
lobes side
Ground lobe
main
sky(θ,ϕ) (θ,ϕ) (θ,ϕ) (θ,ϕ)
⎥⎦
⎢ ⎤
⎣
⎡ Ω + Ω
≈
lobes side lobes side Ground lobe main lobe main
sky RA B RA
B A
W (θ,ϕ) (θ,ϕ) (θ,ϕ) (θ,ϕ) signal of interest disturbance signal
K
≈3 <<1srad ≈300K ≈2π srad Obtaining : signal of interest >> disturbance signal
requires
) , ( )
,
(θϕ θϕ
lobes side lobe
main RA
RA >>>
≈1
⎥⎦
⎢ ⎤
⎣
⎡ Ω + Ω
≈
lobes side lobes side Ground lobe main lobe main
sky RA B RA
B A
W (θ,ϕ) (θ,ϕ) (θ,ϕ) (θ,ϕ) signal of interest disturbance signal
K
≈3 <<1srad ≈300K ≈2πsrad
600 ) ( ) , (
) , ) (
, ( )
, (
srad B
RA B
RA lobe
main
Ground sky
lobes side lobe main
lobe main lobes
side
Ω
⎥≈
⎦
⎢ ⎤
⎣
⎡
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢
⎣
⎡ Ω Ω
<< θϕ
ϕ ϕ θ
θ ϕ
θ
≈1
<<1x10-10 7x10-8 srad 1’
<<1x10-8 7x10-6srad 10’
<<4x10-7 2x10-4srad 1o
<<4x10-5 2x10-2srad 10o
<RAsidelobes>
Ωmainlobe FWHM
!!!
What is RA( θ,φ ) ?
• Fraunhofer diffraction from a circular aperture (radius a) (at large distances from shield)
• The incident wave is an infinite plane wave (wavevector kinparallel to the z axis).
• The outgoing wave is not infinite, and for this very reason it will have components with wavevectors k in different directions. So it is not a plane wave.
• We want to find out which are the amplitudes of the different components of the outgoing wave.
z
a k θ
kin r dS
What is RA( θ,φ ) ?
• For small anglesθ the vector q=k-kinis in the plane of the aperture and q=kθ.
• The diffracted component with wavevector k is the sum of the contributions from all the elements dS of the aperture, each with its own phase:
z
a k θ kin
∫∫
− ⋅=
S r q i o
q ue dS
u rr
q
r dS
aq aq u J rdr qr J u dr rd e u
u o
a o o a
iqr o q
) ( ) 2
(
2 1
0 0
2
0
cos = =
=
∫ ∫
π − ϕϕ π ∫
Bessel functions q
What is RA( θ,φ ) ?
• The intensity is the square of the field:
• Example: a 2m diameter mirror used at 1 cm and at 1 mm
2
1( )
2 ⎥
⎦
⎢ ⎤
⎣
= ⎡
Ω θ
θ ak
ak I J d dI
o
0.0 0.1 0.2 0.3 0.4 0.5
0.0 0.2 0.4 0.6 0.8 1.0
a=1m λ=1 cm λ=1 mm
angular response
off-axis angle θ (deg)
main lobe
side lobes
What is RA( θ,φ ) ?
• The intensity is the square of the field:
10-3 10-2 10-1 100 101 102 10-14
10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
a = 1 m λ=1 cm λ=1 mm
angular response
off-axis angle
θ
(deg)main lobe
side lobes 2
1( )
2 ⎥
⎦
⎢ ⎤
⎣
= ⎡
Ω θ
θ ak
ak I J d
dI
o
What is RA( θ,φ ) ?
• The intensity is the square of the field:
• The first zero is for
10-3 10-2 10-1 100 101 102 10-14
10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
a = 1 m λ=1 cm λ=1 mm
angular response
off-axis angle
θ
(deg) a222 .
10 1 θ = λ
θ10
2
1( )
2 ⎥
⎦
⎢ ⎤
⎣
= ⎡
Ω θ
θ ak
ak I J d dI
o
What is RA( θ,φ ) ?
• The intensity is the square of the field:
• The first zero is for
• The FWHM is similar
10-3 10-2 10-1 100 101 102 10-14
10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
a = 1 m λ=1 cm λ=1 mm
angular response
off-axis angle
θ
(deg) a222 .
10 1 θ = λ
0.5 FWHM θ10
2
1( )
2 ⎥
⎦
⎢ ⎤
⎣
= ⎡
Ω θ
θ ak
ak I J d
dI
o
What is RA( θ,φ ) ?
• The intensity is the square of the field:
• The envelope of the off-axis response scales asθ-3 approximately starting from 0.5 at the FWHM
10-3 10-2 10-1 100 101 102 10-14
10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100
a = 1 m λ=1 cm λ=1 mm
angular response
off-axis angle
θ
(deg) ) 3(θ ∝θ− RA
2
1( )
2 ⎥
⎦
⎢ ⎤
⎣
= ⎡
Ω θ
θ ak
ak I J d dI
o
exercise
• For the angular response of a circular aperture, compare the power in the main lobe to the power in the sidelobes
• Hint:
• But use approximated formulas …
θ θ θπ θ
θ θ θ
π θ
λ π
λ
ak d ak B J
A
ak d ak B J
A P P
a
lobe main a lobes side
lobe main lobes side
) sin ( 2 2
) sin ( 2 2
/ 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 sidelobes design. 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
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
Response
Far Sidelobes
F. Villa, LFI 107
Another low sidelobes design:
BOOMERanG Even more stringent
nequirements due near earth operation
R T
S
P 1.3 m
Ground Shield Sun Shield
Cold Lyot Stop + baffles
Angle of sunshield
• A single point (T=3K @ λ=4.3 cm) is not enough to demonstrate that the CMB is thermal radiation.
• Coverage of the interesting frequency range to establish the black-body nature of the CMB has been a difficult challange, involvingfor decadesresearchers in Princeton (Wilkinson), Berkeley (Smoot, Lubin), Firenze (Melchiorri), Milano (Sironi), Bologna (Mandolesi) …
• At wavelengths > 20 cm our Galaxy becomes brighter than the CMB.
• At wavelengths < 10 cm the correction for atmospheric emission is significant even at high altitude sites. It is necessary to move the experiment outside the earth atmosphere.
• Balloon experiments (Paul Richards) and finally the COBE satellite (FIRAS, John Mather) proved the Planckian nature of the CMB.
CMB and cosmology
• 1992: COBE-FIRAS measures the spectrum of the CMB with incredible precision (1/10000)
• The thermal spectrum at 2.735K and the high photons to barions ratio together with the measured primordial abundances of light elements isevidence for a hot initial phase of the Universe.
J. Mather et al. 1992
COBE-FIRAS
• COBE-FIRAS was a Martin-Puplett Fourier-Transform Spectrometer with composite bolometers. It was placed in a 400 km orbit.
• A zero instrument comparing the specific sky brightness to the brightness of 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 thermal equilibrium at high Temperature.
σ (cm-1) wavenumber
• The spectrum
K T
e c T h B
CMB
x
725 . 2
1 ) 2
, (
3
2
= = ν −
ν
mm T
B T
B
( , ) = ( λ , ) ⇒ λ
max= 1 . 06 ν
ν λ
GHz kT
x h
CMB
CMB 56
ν ν ≅
=
) 31 . 5 ( 159
82 . 3 2
1
1 max
max
max max max
−
−
=
=
⇒
=
⇒
=
−
cm GHz
x x ex
σ ν
Wien RJ
• Techniques ?
??
? 160
bolometers 160
detectors coherent 160
max max max
⇒
=
≈
⇒
=
>>
⇒
=
<<
GHz GHz GHz ν
ν ν ν
ν ν
Wien RJ
A microwave receiver
• This antenna for satellite broadcast is a microwave receiver at a frequency of 10 GHz.
• If we point this receiver away from the geostationary satellite, it will receive the CMB !
• Unfortunately, the noise of this kind of receiver, due to thermal motions of the electrons in the input circuit, is too high for this purpose.
• The only way to reduce the noise is to cool down the receiver at cryogenic temperature.
• The CMB is only slightly anisotropic.
• The brightness (temperature) fluctuations are due to small density fluctuations present in the primeval fireball, and to their motions:
T ppm T
CMB
CMB
≈ 10
5= 10
Δ
−Photon Density fluctuations
Gravitational redshift
Scattering against moving e-
c c T
T
v
3 1 4 1
2
+
+ Δ
= Δ
Δ ϕ
ρ ρ
γ γ
Image of Solar Granulation
8 light minutes Here, 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 cosmic photosphere (3000 K)
CMB observables
• The spectrum
• The angular distribution
kT x h e c T h
B
ν ν
x =ν
= − 1 ) 2 , (
3 2
GHz x
T T T e B T xe B
T T dTB T d B
x x
217 83
. 3
) , 1 ( ) , (
) , ( ) , (
max
max= ⇒ =
Δ
= − Δ
Δ
= Δ
ν ν ν
ν ν
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)
CMB (MKS units)
Frequency (Hz)
• The angular distribution of the CMB is the map
( ) ( ) b
T dec T
T RA T T
T
CMB CMB CMB
CMB CMB
CMB
Δ , or Δ l ,
Δ =
Red=slightly hotter Green=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 statistical theory of the fluctuations in the early universe.
• From theory we can then predict only the general, statistical properties of the CMB anisotropy map, not its detailed pattern.
IF ΔT
CMB(l,b) IS
A GAUSSIAN RANDOM FIELD, THEN ALL INFORMATION
IS ENCODED IN ITS ANGULAR POWER SPECTRUM
CMB observables
• The rms anisotropy has contributions from many angular scales
• The angular power spectrum clof the anisotropy defines the contribution to the rms from the different multipoles:
( ) ( )
∑
∑
+
= Δ
=
= Δ
l l
l l
l l l
l c
T a c
Y a T
m m
m m
) 1 2 4 (
1
, ,
2 2
,
π
ϕ θ ϕ
θ
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)
CMB (MKS units)
Frequency (Hz)
CMB anisotropy observables
• The angular power spectrum clof the anisotropy defines the contribution to the rms from the different multipoles:
( ) ( )
∑
∑
+
= Δ
=
= Δ
l l l
l
l l l
l c
T a c
Y a T
m m
m m
) 1 2 4 (
1
, ,
2 2
,
π
ϕ θ ϕ
θ Δ =
∑
+l l wlcl
T meas (2 1) 4
2 1 π
) 2 1 (+ σ
= −ll
l e
wLP
• A real experiment will not be sensitive to all the multipoles of the CMB.
• The window function wl defines the sensitivity of the instrument to different multipoles.
• The detected signal will be:
• For example, if the angular resolution is a gaussian beam with s.d. σ, the corresponding window function is
0 200 400 600 800 1000 1200 1400 0
1000 2000 3000 4000 5000 6000
l(l+1)cl/2π (μK2)
multipole l
( ) ( )
∑
∑
+
= Δ
=
= Δ
l l
l l
l l l
l c
T a c
Y a T
m m
m m
) 1 2 4 (
1
, ,
2 2
,
π
ϕ θ ϕ
θ
Expected power spectrum:
Why this way ?
c l
0 200 400 600 800 1000 1200 1400 0
1000 2000 3000 4000 5000 6000
l(l+1)cl/2π (μK2)
multipole l
( ) ( )
∑
∑
+
= Δ
=
= Δ
l l
l l
l l l
l c
T a c
Y a T
m m
m m
) 1 2 4 (
1
, ,
2 2
,
π
ϕ θ ϕ
θ
Expected power spectrum:
Because the early universe is a plasma of photons and baryons, and because of horizon effects.
c l
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 growand create the hierarchy of structures we see in the nearby Universe.
Before recombination
After recombination T < 3000 K T > 3000 K
overdensity
Due to gravity, Δρ/ρ increases, and so does T
Pressure of photons increases, and the perturbation bounces back
T is reduced enough that gravity wins again
Here photons are not tightly coupled to matter, and their pressure is not effective.
Perturbations can grow.
t
the Universe is a plasma the Universe is neutral
Size of sound horizon
time
Big-bang recombination Power Spectrum
multipole220450
1st peak 2nd peak
LSS
300000 ly
In the primeval plasma, photons/baryons density perturbations start to oscillate only when the sound horizon becomes larger than their linear size . Small wavelength perturbations oscillate faster than large ones.
R
R C
C
C
C
1st dip 2nd dip
The angle subtendeddependson the geometryof space
size of perturbation (wavelength/2)
300000 y 0 y
v v
v
v v
v v
v
0 200 400 600 800 1000 1200 1400 0
1000 2000 3000 4000 5000 6000
l(l+1)cl/2π (μK2)
multipole l
( ) ( )
∑
∑
+
= Δ
=
= Δ
l l
l l
l l l
l c
T a c
Y a T
m m
m m
) 1 2 4 (
1
, ,
2 2
,
π
ϕ θ ϕ
θ
An instrument with finite angular resolution 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 1400 0.0
0.2 0.4 0.6 0.8 1.0
20' FWHM 10' FWHM 5' FWHM 7o FWHM
wl
multipole ) 2
1 (+ σ
=
−lll e
wLP
c l
0 200 400 600 800 1000 1200 1400 0
1000 2000 3000 4000 5000 6000
l(l+1)cl/2π (μK2)
multipole l
( ) ( )
∑
∑
+
= Δ
=
= Δ
l l
l l
l l l
l c
T a c
Y a T
m m
m m
) 1 2 4 (
1
, ,
2 2
,
π
ϕ θ ϕ
θ
∑
+= Δ
l
l
l wlc
T LP
meas (2 1)
4
2 1 π
) 2 1 (+ σ
= −ll
l e
wLP
rms signal in an instrument with gaussian beamσ : Expected power spectrum:
10000 100 10
20 40 60 80 100 120
FWHM - gaussian beam (arcmin)
ΔTrms (μK) COBE WMAP BOOMERanG
105
2 . 4× − Δ = Trms
T
c l
CMB observables
• The polarization state
• Extremely weak !
7 6
10 2
10 4
) , 1 ( ) , (
−
−
× Δ ≈
× Δ ≈
Δ
= − Δ
rmsB P
rmsE P
P x
x P
T T T T
T T T e B T xe
B ν ν
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)
CMB (MKS units)
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 linear polarization.
Last scattering surface
CMB polarization
-
-
+
-
+
xy
-
-+
-
+
x y
- x
y
-10ppm +10ppm
= e-at last scattering
+ +
-
- +
+ -
- + +
-
- +
+ - - Overdensity Underdensity Overdensity Underdensity
v v v v v v v v
E
E
E
E
E-modes in the polarization pattern
Converging flux
Same flux as seen in the electron reference frame Diverging
flux
Quadrupole anisotropy due to Doppler effect redshift
blueshift
blueshift
redshift
+ +
+
+
- -
-
-
resulting CMB polarization
field (E-modes) Velocity fields
at recombination • This component of the CMB polarization field is calledE
component, or gradient component. This is the only kind of polarization produced at recombination.
• It is related to velocity fields. Foracoustic oscillations, it will be maximum for perturbations with maximum velocity and zero density contrast.
• So we expectpeaks in this polarizationpower spectrum where we haveminima in the temperature anistropypower spectrum.
• The amplitude of the polarization signal depends on the length of the recombination process (it is not produced before, nor later).
• Tensor perturbations (gravity waves)also produce quadrupole anisotropy. The generation of a faint stochastic background of gravity waves is a generic feature of allinflationary processes.
• The resulting polarization pattern is shear-like.
• The amplitude of the effect is very small.
• This component of the CMB polarization field is calledB-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 tensor modes normalized to the scalar ones is:
• and
• There are theoretical arguments to expect that the energy scale of inflation is close to the scale of GUT i.e. around 1016GeV.
• The current upper limit on anisotropy at large scales gives T/S<0.5 (at 2σ)
GeV 10 7 .
3 16
4 / 4 1 / 1
2 2 4 / 1
≅ ×
⎟⎟⎠
⎜⎜ ⎞
⎝
≡⎛
⎟⎠
⎜ ⎞
⎝
⎛ V
C C S T
Scalar
GW Inflation potential
The background of Gravitational Waves
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡
≅ × +
GeV 10 1 2 . 2 0
) 1 (
16 4 / 1
max
K V
cB μ
π l
l l
T/S=0.28 E ?
rms B-modes polarization signal
>2 orders of magnitude smaller than rms T anisotropy !
Summary of CMB observables:
• CMB spectrum
• CMB anisotropy angular power spectrum
• CMB polarization angular power spectra:
– E-modes – B-modes
Increasing Difficulty
K T=2.725
K Trms=80μ Δ
K TE,rms=2μ Δ
K TB,rms<0.1μ Δ
Intrinsic CMB measurements limits
• Now that we know what the CMB is, and how it was measured, we can study its noise properties, 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:
( ) ( ) ( )
h Wth h W N h N h
E ν
ν ν ν
ν Δ = = =
=
Δ 2 2 2 2 2
This is a typical random-walk process (variance prop.to time).
Using Einstein’s generalization we get the power spectrum and the variance of radiative power fluctuations:
kBTdf df
kBTt f 4
2 2
2 = ⇒ =
Δθ θ&
Wdf h df Wf2 =2 ν Δ
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 that the integral of the PS over frequency is the variance).
• In this case the intrinsic fluctuations per unit bandwidth are >7 orders of magnitude smaller than the 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
Hz W W
h Wf
11
2 = 2 =2.5×10−
Δ ν
Hz W / 10−15
Hz W / 10 5 . 2 × −11
Radiation Noise
• Thermal radiation (like the CMB) has also wave interference noise: the correct statistics is Bose-Einstein.
g N N N
d N Td e N
N g
V T kT
E
2 2
, 2
/ )
( ;
1 +
= Δ
⇒
=
− Δ
= −
μ
μ
Poisson noise
Wave interference noise
Radiation Noise
• For a blackbody
⎥⎦⎤
⎢⎣⎡ + −
= Δ
⇒
= −
− =
= −
1 1 1
1 1 8
4 2 1 ;
/ 2
/ 3
2
3 2
/ ) (
kT h kT h kT E
N e N
e Vd N c
c Vd e g
N g
ν ν μ
πν ν
ν ν π
Poisson noise, important at short wavelengths
Wave interference noise, Important at low frequencies
Radiation Noise
( )
5 2 18 3
2 5
4 5 3 2
5 2
/ 2
/ /
2 2
/ 2
10 77 . 4 2
1 4
1 1 1
2
1 1 1
1 1 1
1 1 1
2
1
K Hz sr cm
W h
c k
df e dx
e T x h A c df k W
e df W h df W
W e e h
N h E
N e N
x
x x
x kT h
kT h kT
h kT h
× −
=
Ω −
= Δ
⎥⎦⎤
⎢⎣⎡ + −
= Δ
⎥⎦⎤
⎢⎣⎡ + −
⎥⎦=
⎢⎣ ⎤
⎡ + −
= Δ
⎥⎦⎤
⎢⎣⎡ + −
= Δ
∫
ν
ν ν
ν
ν
ν ν
CMB observables
• The spectrum
• The angular distribution
• The polarization state
• The noise kT x h e c T h
Bν νx = ν
= − 1 ) 2 , (
3 2
T T T e B T xe
B x
x Δ
= −
Δ (, )
) 1 ,
(ν ν
T T T e B T xe
B x P
x P
Δ
= −
Δ (, )
) 1 ,
(ν ν
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)
CMB (MKS units)
Frequency (Hz)
(
e)
dxe x h c
T dx k T
W x
x 2 4 3 2
5 5 2
1 ) 4
,
( = −
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 thanτ will not be independent.
• 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 df W
N
f
f W Wt
∫
Δ=
=
max
min 2
σ σ
W t
