Riassunto• Nelle scorse 3 lezioni abbiamo visto:
– Omogeneità ed isotropia dell’ universo a grande scala (evidenze osservative)
– Espansione dell’ universo (evidenze e interpretazione)– Evoluzione dell’ universo omogeneo (eq. Friedmann e sua
dipendenza dai parametri di densità)– Redshift– La radiazione nell’ Universo – CMB – Spettro del CMB– Osservabili della CMB (Brillanza, Corpo nero, Anisotropie)– Rumore e sua origine fisica
• Moto Browniano• Rumore Johnson
Programma di oggi• Nelle 2 lezioni di oggi vedremo:
– Rumore (continuazione)• Rumore della radiazione
– Rivelatori per il CMB• Termici• Coerenti
– Calibrazione di rivelatori per il CMB– Anisotropia e polarizzazione della CMB– Esperimenti:
• BOOMERanG, WMAP etc.• Planck• B-pol ?
Radiation Noise• 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:
( ) ( ) ( ) WthhWthNhNhE ν
νννν ===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
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
2
45
32
52
/2
//22
/2
1077.24
14
1112
111
111
111
2
1
KHzsrcmW
hck
dfdxe
exTAhckdfW
dfe
WhdfW
eWth
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
Noise and integration time•
• The noise decreases as the square root of the integration time.
• Notice that this applies equally to detector noiseand 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 BLIP
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<ε
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<ε
Rivelatori per il fondo a microonde
• Termici: si integra l’ effetto di moltissimi fotoni per riscaldare un assorbitore. Si misura la variazione di temperatura dell’ assorbitore. Non sono sensibili alla fase dell’ onda elettromagnetica, ma solo all’ energia trasportata. Sono caratterizzati da bassa risoluzione spettrale e altissima sensibilita’. Vanno raffreddati a temperature criogeniche, sub-K.
• Nel caso dei bolometri il termometro e’ una resistenza fortemente dipendente dalla temperatura.
• Coerenti: si converte l’ onda elettromagnetica in una ddp tramite una antenna. Si amplifica direttamente la ddp e poi la si rettifica in un componente non lineare. Sono sensibili ad ampiezza e fase – si possono costruire interferometri. (Radiometri). Sono caratterizzati da alta risoluzione spettrale e buona sensibilita’. Devono essere raffreddati a temperature criogeniche.
Cryogenic Bolometers• The CMB spectrum is continuum and bolometers are wide band
detectors. That’s why they are so sensitive.
filter(frequencyselective)
FeedHorn(angle selective)
IntegratingcavityRadiation
Absorber (ΔT)
Thermometer(Ge thermistor (ΔR)at low T)
IncomingPhotons (ΔB)
• Fundamental noise sources are Johnson noise in the thermistor(<ΔV2> = 4kTRΔf), temperature fluctuations in the thermistor((<ΔW2> = 4kGT2Δf), background radiation noise (Tbkg
5) needto reduce the temperature of the detector and the radiativebackground.
Load resistor
ΔV
• In steady conditions the temperature rise of the sensor isdue to the background radiativepower absorbed Q and to the electrical bias power P:
• The effect of the background power is thus equivalent to an increase of the reference temperature:
Cryogenic Bolometers
PQTTG +=− )( 0
GQTT
TTGGQTTGP
+=
−=⎥⎦⎤
⎢⎣⎡ +−=
00
00
'
)'()(
T0’
Q(pW)0.27K
0.28K
0.26K0 1 2
• In presence of an additional signalΔQ ejωt (from the sky)
• There is a tradeoff between high sensitivity and fast response. The heat capacity C should beminimized to optimize both.
• Using a current biased thermistor toreadout the temperature change:
Cryogenic Bolometers
QTGdt
TdC eff Δ=Δ+Δ
GC
GdQdT
eff
=
+=
τ
ωτ 2211
221
)()(
1
ωταα
αα
+===ℜ
==⇒=
effGRi
dQdTRi
dQdV
RdTiidRdVdT
TdRTR
Small sensorat low
temperature
Responsivity
• A large α isimportant forhigh responsivity.
• Ge thermistors:• Superconducting
transition edgethermistors:
Cryogenic Bolometers
221
)()(
1
ωτα
α
+=ℜ
=
effGRidT
TdRTR
110 −−≈ Kα
11000 −≈ KαS.F. Lee et al. Appl.Opt. 37 3391 (1998)
• Johnson noise in the thermistor
• Temperature noise
• Photon noise
• Total NEP (fundamental):
Cryogenic Bolometers
kTRdf
Vd J 42
=Δ
( )22
22
24
fCGGkT
dfWd
eff
effT
π+=
Δ
( )( ) dxeex
hcTk
dfWd
x
xBGPh
∫ −
+−=
Δ2
4
32
552
114 εε
dfWd
dfWd
dfVd
NEP PhTJ222
22 1 Δ
+Δ
+Δ
ℜ=
Again, needof low
temperatureand low
background
Q
Ordini di Grandezza del Rumore• Quindi per un buon bolometro raffreddato a 0.3K :
• In termini di numero di fotoni:
• Siamo quindi lontani dal poter vedere il singolo fontone. • Quello che e’ importante, pero’, e’ che il rumore sia
inferiore alle fluttuazioni intrinseche della radiazione:
HzWNEP /10 17−≈
HzJ
HzWh
NEPN 522
17
1010
/10=≈≈Δ −
−
ν
HzWfwNEP w /105)( 18−×≈<
1900 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 tim
e re
quire
d to
mak
e a
mea
sure
men
t (se
cond
s)
year
Photon noise limit for the CMB
Anni ‘70
Anni ‘80
Spider-Web Bolometers
Absorber
Thermistor
Built by JPL Signal wire
2 mm
•The absorber is micromachined as a web of metallized Si3N4 wires, 2 μm thick, with 0.1 mm pitch.
•This is a good absorber formm-wave photons and features a very low cross section for cosmic rays. Also, the heat capacity isreduced by a large factorwith respect to the solidabsorber.
•NEP ~ 2 10-17 W/Hz0.5 isachieved @0.3K
•150μKCMB in 1 s
•Mauskopf et al. Appl.Opt. 36, 765-771, (1997)
Crill et al., 2003 – BOOMERanG 1998 bolometers, 300 mK
• Constant current bias• Very high impedance voltage follower as close as possible to the
detector (inside dewar)• Very low noise OP amp amplifier (1 nV/sqrt(Hz)
Ge Bolometer Readouthigh impedance (10 MΩ) detector
To A/D converterand data storageB
+
ib
X 1000
Inside dewar Outside dewar
JFET Preamplifier fordifferential AC bias
• AC bias currentis betterbecause the amplifier is usedat a frequencyfar from 1/f noise.
• Sine wave or triangle wavebias
• Demodulatorneeded.
• Differentialbiasing is betterbecausedifferentialamplificationremovescommon interference.
• Low noise, high CNRR amplifierneeded.
BAC
-1+1
10MΩ
5 pF
5 pF
+Vb
-Vb
180Hz +
-VACJFETs,
x100Lock-in Vout
REF
IN
Cryogenic Bolometers• Ge thermistor bolometers have
been used in many CMB experiments:– COBE-FIRAS, ARGO, MAX,
BOOMERanG, MAXIMA, ARCHEOPS
• Ge thermistor bolometers are extremely sensitive, but slow: the typical time constant C/G isof the order of 10 ms @ 300mK
• Transition Edge Superconductor (TES) thermistors can do muchbetter using electro-thermalfeedback (100 μs) – Recentdevelopment
Bolometer Arrays• Once bolometers reach BLIP
conditions (CMB BLIP), the mapping speed can only beincreased by creating largebolometer arrays.
• BOLOCAM and MAMBO are examples of large arrayswith hybrid components (Si wafer + Ge sensors)
• Techniques to build fullylitographed arrays for the CMB are being developed.
• TES offer the naturalsensors. (A. Lee, D. Benford, A. Golding ..hear Richards..)
Bolocam Wafer (CSO)
MAMBO (MPIfR for IRAM)
• Are the future of this field. See recent reviewsfrom Paul Richards, Adrian Lee, Jamie Bock, Harvey Moseley … et al.
• In Proc. of the Far-IR, sub-mm and mm detector technology workshop, Monterey 2002.
TES arrays
• Have very low R, so work better at constant voltage. Lets’write in detail the equations:
Voltage-Biased Superconducting Bolometers
[ ]
PCiG
TP
eTTeCiGT
PP
TeCiTeGeTT
dTdR
RT
RVP
TeCiTeGTedTdR
RdRdVPe
TedtdCTeG
RVPe
dtdQTTeTG
RV
RVPeP
TTGR
VP
ii
iiib
tititib
ti
titibti
otibbti
ob
δωαδδωαδ
ωδδδδ
ωδδδδ
δδδδ
δδδ
ϕϕ
ϕϕϕ
ϕωϕωϕωω
ϕωϕωω
ϕωω
++=⇒⎥⎦
⎤⎢⎣⎡ ++=
+=⎥⎦⎤
⎢⎣⎡−
+=⎥⎦⎤
⎢⎣⎡+
+=⎥⎦
⎤⎢⎣
⎡+⇒
+−+=⎥⎦
⎤⎢⎣
⎡+++
−=+
−
+++
++
+
2
)()()(2
)()(2
)(22
2
1
)(
)(
• The effective thermal conductivity is
• The first part is the Electro-Thermal Feedback (ETF) part.• When δP increases (a signal arrives) T increases; this increases the
resistance which in turns decreases the bias power Pb=Vb2/R. As a
result the total power (P + Pb) does not dcrease as much, and the temperature does not change much.
• For a given incoming power, the ETF reduced the temperature change. • It is the reverse of what happens in a semiconductor bomometer,
where the negative α produces a negative ETF, increasing the temperature change.
• But here we measure the bias current at constant voltage. The currentneeded to keep the bias more stable is increased by the ETF. So wedefine
Voltage-Biased Superconducting Bolometers
PCiG
TP
eTTeCiGT
PPi
i δωαδδωαδ
ϕϕ
++=⇒⎥⎦
⎤⎢⎣⎡ ++=
−
CiGT
PGeff ωα++=
( )oi
L
GCi
GTP
CiGT
P
Lωτω
α
ω
α
ω+
=+
=+
=11
• The Responsivity is
• And using
• We get
• Defining
• We get
Voltage-Biased Superconducting Bolometers
( )PT
TP
VPR
RRV
VPRRV
PRV
Pi b
b
b
b
bbb
δδα
δδ
δδ
δδ
δδ 111/ 2
2 −=−=−===ℜ
)1( o
iii
iLGe
CiGT
Pe
PTP
CiGT
PeT
ωτωαδδδ
ωαδϕϕϕ
++=
++=⇒
++=
−−−
)1(1
)1(11
o
i
bo
ib
b
b
b iLLe
ViLGe
TP
VPT
TP
V ωτωτα
δδα ϕϕ
++−=
++−=−=ℜ
−−
1+=
Loττ
ωτϕ
iLL
Ve
b
i
++−=ℜ
11
)1(1
For large ETF (L>>1): • the time constant is
reduced wrt the standard one by L+1
• For slow signals (ω<<1/τ) and large ETF the responsivity is simply-1/Vb
Such a high value for α(which is >0 for TES) induces a large change in the bias power whenradiation hits the detector (electrothermal feedback)
This results in a largereduction of the time constant and in stabilization of the responsivity.
Bolometri come rivelatori di particelle energetiche
Multiplexing• Ge and Si thermistors are read out using JFETs at
100K. There is no pratical way to multiplex manysensors on a single amplifier. The number of wiresentering the cryostat would be huge for a largeformat array. Practical limit: the JFET boxes of Planck and Hershel…
• TES sensors have very low impedance (about 1 Ω)• They can be readout by a SQUID with no power
dissipation and large noise margin. Time multiplex and frequency multiplex are beingdeveloped (NIST, Berkeley, Helsinki ..)
From: Chervenak et al. 99
Time-domain multiplexing
Vb
Vbf1
f2
NBF
f
columnSQUID
columnSQUID
frequency-domain multiplexing
row i bias
row i+1 bias
j j+1
Ref: Berkeley/NIST design
Coherent Detectors• Here the CMB em waves interact with
an antenna, and a selected mode ispropagated in a waveguide to a probe, where a voltage proportional to the incoming field is generated.
• The voltage is amplified by means of a fast, low noise amplifier (direct receivers)
• Very low noise HEMT amplifiers, cooledat 20K have been developed (NRAO).
• They have been used in many CMB experiments: TOCO, DASI, CBI, WMAP and will be used in Planck-LFI.
Frequenze troppo alte per amplificazione diretta (<30 GHz)
Oscillatore + Mixer + rivelatore non lineare
Eterodina
L’ ampiezza del segnale a frequenza intermedia e’
)]()([)( IFLIFLIF EEVI νννναν −++=
0 50 100 150 200 2500
50
100
150
200
250
HEMTs Bolometers
NET
CM
B (μK
s1/
2 )
frequency (GHz)
From J.Bock (SPIE 2003)
Space Based, 2010
SINGLE HORN POLARIMETER SENSITIVITY
J. BOCK SPIE 2003
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 forthese experiments. A 5% measurement is consideredhigh 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 surroundingdiffuse 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 angularresponse 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 anaccuracy sufficient for instrument calibration. Moreover, foreground contamination and 1/f noise effects increase aslarger 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 phasespace volume dxidpi can also be shown to be a Lorentzinvariant. 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 photonswe 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 massespresent in the Universe
• The annual revolution of the Earth around the Sun is knownextremely well ( v ~ 30 km/s), and produces an annualmodulation in the CMB dipole. This is the main signalused 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 Dipolespectrum 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 specificbrightness induced by our speed β with respect to the cosmic matter emitting the background can be derivedas
• 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 beamof the instrument
• So from a scatter plot of the measured signal vs. the expected 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 toTemperature 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
),,,(),(),(
),(
calibratedT map:
uncalibratedV 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 differentfrom 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 observationcan be used to estimate the calibration constant a asfollows:
[ ]∫ℜ= νννδαδαϑδα dEFRAV oooo )()(),,,(),(
Point Sources
• So the calibration constant a , needed to convert the uncalibrated map into a calibrated CMB map , can beestimated 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 asfollows:
[ ]{ }
[ ]{ }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
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)
• Le fluttuazioni di densita’ sulla superficie di ultimo scattering producono anisotropia CMB, in piu’ modi.
1) DIRETTAMENTE:• Una sovradensita’ fa perdere energia ai fotoni che provengono da
essa, perche’ devono risalire la buca di potenziale. Per effetto del redshift gravitazionale si ha
• La sovradensita’ causa anche una dilatazione del tempo, per cui noi osserviamo un’ epoca piu’ primordiale (e quindi piu’ calda) laddove ci sono sovradensita’. La dilatazione del tempo e’
• Ma, durante la fase radiativa,
• Quindi in totale
2cTT Φ
==δ
νδνδ
2ctt Φ
=δδ
23/2
32
32/1;
ctt
aa
TTaTta Φ
−=−=−=→∝∝δδδδ
231
cTT Φ
=δδ Effetto Sachs-Wolfe (1967)
• Le fluttuazioni di densita’ sulla superficie di ultimo scattering producono anisotropia CMB in piu’ modi.
2) INDIRETTAMENTE:• Una sovradensita’ attira la materia circostante, e genera quindi un campo di
velocita’ peculiare. I fotoni che subiscono la loro ultima diffusione in zone in movimento con velocita’ peculiare v subiscono un effetto Doppler, quindi
3) ADIABATICAMENTE:• Il mezzo primordiale e’ un plasma di fotoni e materia. • Si dicono perturbazioni adiabatiche quelle in cui le densita’ di radiazione e di
materia fluttuano insieme in modo da mantenere l’ entropia del mezzo costante.• l’ entropia del mezzo e’
• Il numero di particelle di materia e’• Il numero di fotoni e’
• Quindi perturbazioni adiabatiche implica
• La teoria inflazionaria prevede che le fluttuazioni siano di tipo adiabatico.
cTT v
==νδνδ
γγ nnS mm /=
mmn ρ∝4/33
γγγ ρ∝∝ Tn
→−=−==γ
γ
γ
γ
γ
γ
ρδρ
ρδρδδδ
430
m
m
m
m
m
m
nn
nn
SS
m
m
m
m
TT
TT
ρδρδδ
ρδρ
ρδρ
γ
γ
314
43
43
=→==→
Anisotropia CMB• Quindi in totale
• Sperimentalmente si vede che, a parte l’ anisotropia di dipolo, dovuta al moto della Terra (10-3), l’ anisotropia intrinseca ΔT/T e’ molto piccola (dell’ ordine di 10-4-10-5).
• Quindi l’ universo primordiale era estremamente omogeneo. Le strutture presenti oggi nell’ universo si sono formate grazie all’ azione della gravita’, che ha fatto crescere le piccole perturbazioni di densita’ presenti alla ricombinazione, attirando la materia circostante.
Fluttuazioni adiabatiche
Effetto SW Dffusione da elettroni in moto
ccTT
m
m v31
31
2 +ΔΦ
+Δ
=Δ
ρρ
t
UniversoPrimordiale
UniversoStrutturato
Gravita’
vs. Espansio
ne
510−≈Δρρ
510≈Δρρ
Evoluzione delle fluttuazioni• Consideriamo una fluttuazione di densita’ adiabatica, di dimensioni
inizialmente maggiori dell’ orizzonte, prima della ricombinazione. (qui le fluttuazioni isoterme sono congelate).
• Schematizziamo il caso piu’ semplice come una sovradensita’ sferica (densita’ ρ’(t) ) immersa in un universo omogeneo a densita’ ρ(t)critica (universo piatto). Avremo ρ’(t) > ρ(t) =ρc(t) .
• Per la simmetria del problema, la sovradensita’ evolve come un mini-universo a densita’ piu’ alta di quella critica (la dinamica dipende solo dalla massa contenuta all’ interno della sfera, teorema di Birkhoff).
• Scrivendo l’ equazione di Friedmann nei due casi si ottiene:
22
22
22
2
83'
03
8
3'8
⎟⎠⎞
⎜⎝⎛=
−=⇒
⎪⎪
⎩
⎪⎪
⎨
⎧
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦⎤
⎢⎣⎡ −
−=⎟⎟⎠
⎞⎜⎜⎝
⎛⎥⎦⎤
⎢⎣⎡ −
aa
Gkc
aaGH
kcaaGH
o
o
o
ρπρρρ
ρδρ
ρπ
ρπ
Evoluzione delle fluttuazioni
• Nell’ epoca dominata dalla radiazione quindi
• Nell’ epoca dominata dalla materia quindi
• La perturbazione quindi cresce fino al momento in cui l’ orizzonte supera le sue dimensioni. Si dice che la perturbazione “entra nell’orizzonte”. E diventa connessa causalmente. Da qui in poi la sua evoluzione dipende dalla sua massa e da come questa si confronta con la massa di Jeans.
22
83
⎟⎠⎞
⎜⎝⎛=
aa
Gkc o
ρπρδρ
42/1 ; −∝=∝ ata radρρ
inin ttt
aa ρδρ
ρδρ
ρδρ
=→∝∝ − 2411
33/2 ; −∝=∝ ata matρρ3/2
3/223
11⎟⎟⎠
⎞⎜⎜⎝
⎛=→∝∝ −
inin ttt
aa ρδρ
ρδρ
ρδρ
Spettro di potenza• Ci permette di studiare i contributi delle fluttuazioni con diverse
dimensioni. • Siccome siamo in un regime di piccole perturbazioni (regime
lineare), la perturbazione totale e’ semplicemente la somma dei contributi delle diverse scale.
• Si usa espandere le perturbazioni di densita’ in serie di onde piane:
dove k e’ il numero d’ onde (inverso della lunghzza d’ onda) della k-ma onda piana che contribuisce alla fluttuazione totale di densita’.
• Il bello del regime lineare e’ che tutte le onde piane evolvono indipendentemente. Se si trova l’ evoluzione dell’ onda piana generica, basta poi sommare tutte le soluzioni.
∫∫ ⋅⋅ == kdekdetktx xkik
xki 33),(),(rrrrr δδ
ρδρ
Spettro di potenza• La quantita’
e’ lo spettro di potenza delle perturbazioni di densita’.• Questa e’ una quantita’ importante perche’
– La sua evoluzione porta alla formazione di strutture piu’ o meno grandi (galassie, ammassi di galassie, superammassi) nell’ universo presente.
– Tramite i suoi effetti sul potenziale e sui campi di velocita’ alla ricombinazione, porta alla formazione di anisotropie del fondo a microonde di diverse dimensioni (grandi scale, piccole scale).
• Abbiamo quindi diverse osservabili che permettono di porre vincoli sulla forma dello spettro di potenza delle fluttuazioni di densita’.
• Gran parte della cosmologia moderna si occupa di studiare l’evoluzione dello spettro di potenza P(k) ed i suoi effetti.
2)( kkP δ=
Spettro di potenza• Se non si vogliono introdurre scale privilegiate lo spettro
di potenza delle perturbazioni di densita’ deve essere ad es. una legge di potenza:
• In questo caso la perturbazione di densita’ in box di dimensione R e’
• La teoria inflazionaria produce nell’ universo primoridialeuno spettro di potenza iniziale delle perturbazioni di densita’ detto invariante di scala, con . In tal caso
• Questi spettri poi evolvono sotto l’ azione della gravita’ e dell’ evoluzione (espansione) dell’ universo
nin AkkP =)(
22 −∝Δ RR
ρ1≈n
)3(/1
0
2/1
0
22max
4)( +−+=
∫∫ ∝∝=Δ nR
nRk
RRdkkdkkkP πρ
Grandi scale• Se supponiamo che lo spettro di potenza sia invariante di scala, la
parte di anisotropia CMB proporzionale a Δφ e’ indipendente dalla scala angolare.
• Infatti, considerando una fluttuazione sferica di raggio R:
dove nell’ ultimo passaggio si e’ usata l’ equazione di Friedmann.• Abbiamo visto prima che Δρ/ρ va come 1/R2, quindi ΔT/T e’
indipendente dalla scala.• Quindi, dove domina l’ effetto SW (grandi scale), le fluttuazioni di
temperatura sono le stesse a tutte le scale.
2
22
2
3
22 234
cHR
RcRG
RcMG
cTT
RR
RR⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ≈⎟⎟
⎠
⎞⎜⎜⎝
⎛ Δ=
Δ=
ΔΦ≈
Δρρπ
ρρρ
Grandi scale e piccole scale• Noi non osserviamo le fluttuazioni nel volume, ma il loro effetto
proiettato sulla superficie di ultimo scattering. Grandi scale Rcorrisponderanno a grandi separazioni angolari nel cielo. In unaespansione in multipoli l’ andamento invariante di scala corrispondera’ a un andamento costante a bassi multipoli.
• Questo andamento e’ una delle previsioni forti del modello inflazionario. Le scale piu’ grandi sono anche maggiori dell’orizzonte, quindi lo spettro di fluttuazioni generato dall’ inflationresta congelato prima della ricombinazione, e direttamente osservabile.
• A scale piu’ piccole (alti multipoli) sono importanti gli effetti delle oscillazioni del plasma.
1
Spettro di potenza• La mappa di anisotropie che possiamo misurare contiene contributi
da diverse scale angolari :
( ) ( )bTTdecRA
TT
TT
CMB
CMB
CMB
CMB
CMB
CMB ,or, lΔΔ
=Δ
Perturbazione “fredda”di grandi dimensioni
Piano Galattico
Perturbazione “calda”di piccole dimensioni
• The map gives punctual information: it is a pictureof the last scattering surface at redshift 1000. It isuseful 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 anisotropymap, 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
Spettro di Potenza
( ) ( )
∑
∑
+=Δ
=
=Δ
ll
ll
lll
l cT
ac
YaT
m
m
mm
)12(41
,,
2
2
,
π
ϕθϕθlc
l
?• Che forma deve avere lo spettro di potenza ?• Cioe’ quanta anisotropia abbiamo a ciascuna scala
angolare ?• Abbiamo visto che a grandi scale (maggiori dell’
orizzonte, che alla ricombinazione sottende un angolo di circa 1 grado) domina l’ effetto SW.
• Se le perturbazioni sono invarianti di scala, lo spettro di potenza va come 1/l/(l+1).
• A scale piu’ piccole dell’ orizzonte si devono vedere gli effetti delle oscillazioni del plasma:
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
tipol
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
Th e
an g
le su
bten
ded
depe
n ds o
n th
e ge
ome t
ryof
spa c
e
size of perturbation(wavelength/2)
300000 y0 y
v vv
v v
v v
v
Processed bycausal effects like
Acoustic oscillations
Unperturbed
Quantum fluctuationsin the earlyUniverse IN
FLA
TIO
NP(
k)=A
kn
k
horizon horizon
l
l ( l +
1) c
l
horizon
Scal
essm
alle
rtha
nho
rizon
Scal
esla
r ger
tha n
horiz
o n
tBig-Bang10-36s 300000 yrs0
plasma neutral
Power spectrum of perturbations
Power spectrumof CMB temperaturefluctuations
Paradigm of CMB anisotropies
Radiation pressurefrom photonsresists gravitationalcompression
Inflation
(ΔT/T) = (Δρ/ρ) /3 + (Δφ/c2)/3– (v/c)•n
Gaussian,adiabatic(density)
Nucleosynthesis3 min
Recombination
τ>1
Grandi scale e piccole scale• A scale ancora piu’ piccole lo spessore finito della superficie di
ultimo scattering e la tendenza dei fotoni a diffondere fuori dalle perturbazioni di densita’ diminuiscono velocemente l’ ampiezza delle anisotropie. plasma
ricombinazione
neutro ++-
-
τ<<1
Le perturbazioni di piccole dimensioni si mediano a 0 lungo la linea di vista nello spessore della ricombinazione
Linea di v
ista
Linea di vista+
3
• Qualitativamente ci aspettiamo il seguente spettro di potenza delle anisotropie CMB:
Am
piez
za d
elle
flut
tuaz
ioni
Log[Multipolo] (inverso della scala angolare)
Sachs-Wolfe
Oscillazioni acustiche
Dampingtail
)gradi(200
θθπ
≈≈l
2
13
lll c)1( +
200=l
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 FWHMw
l
multipole
2)1( σ+−= lll ew LP
cl
Parametri Cosmologici e Spettro di Potenza
• La forma dello spettro di potenza delle anisotropie CMB dipende soprattutto da tre parametri cosmologici:
Ωο, n, Ωb
• Vediamo perche’.
• Il parametro di curvatura Ωο descrive la geometria dell’ universo. In un universo curvo la luce si propaga su geodetiche curve. Anche se la curvatura e’ molto piccola, il suo effetto diventa importante sull’ immagine della CMB, perche’ i fotoni percorrono una distanza enorme prima di arrivare fino a noi.
Which is the geometry of our Universe ?• According to General Relativity, the presence of mass and energy curves the
space (see e.g. gravitational lensing effects). Also the large scale geometry of the Universe is affected by the average mass and energy: their presencecurves the background metric of the universe.
Flat space in 2-D
Flat space in 3-D
Curved space in 2-D(positive curvature)
Curved space in 3-D(positive curvature)
Curved space in 2-D(negative curvature)
Curved space in 3-D(negative curvature)
Ω > 1 Ω < 1Ω = 1
Il righello• Nell’ immagine della CMB e’ impressa una scala
caratteristica: e’ la dimensione dell’ orizzonte all’epoca della ricombinazione.
• Le fluttuazioni piu’ grandi dell’ orizzonte sono rimaste “congelate”, mentre quelle piu’ piccole hanno avuto il tempo di oscillare.
• Siccome la ricombinazione avviene 380000 anni dopo il Big-Bang, abbiamo a disposizione un “righello” di 380000 anni luce a redshift 1100.
• In geometrie diverse, il righello sottende angoli diversi.
Critical density Universe
Ω>1
Ω<1
High density Universe
Low density Universe
1o
2o
0.5o
hori
zon
Ω=1
14 Gly
LSS
hori
zon
hor i
zon
Ω>1 Ω=1 Ω<1
2o 1o
0.5o
High density Universe Critical density Universe Low density Universel
PS
l
PS
l
PS
200 200 2000 0 0
Distanza di Diametro Angolare• Quando oggi (t0) guardiamo una sorgente lontana, la vediamo
come era all’ epoca t1 quando la luce e’ partita.• Se la sua dimensione fisica all’ epoca t1 e’ ΔS , e la sua
coordinata comobile e’ χ1 , la sua distanza e’ χ1 (a1/ao), e la sua dimensione angolare apparente e’ Δθ = ΔS/[χ1(a1/ao)]
• Definiamo distanza di diametro angolare DA = χ1 (a1/ao) , cosi’Δθ = ΔS / DA
• Questa e’ esprimibile in funzione della distanza di luminosita’che gia’ conosciamo: DL = (1+z1)χ1
• Quindi DA = χ1a1/ao = [DL /(1+z1)](a1/ao) = DL /(1+z1)2. • Avevamo ricavato esplicitamente DL in funzione di z e dei
parametri cosmologici:
qui,oraΔθ ΔS
χ1(a1/ao)
• Dall’ eq. di Friedmann risultava
• E quindi si puo’ esplicitare
• Casi particolari:
• I dati piu’ recenti di CMB e SN1a indicano Ωo=1, ma ΩΛ=0.7, quindi si deve usare la formula con l’ integrale.
∫+
−Λ
−− Ω−+Ω+Ω+Ω+=+=
1
)1(11 2/12342 ]ˆ)1(ˆˆ[ˆ
ˆ)1()1(
z aaaaad
HczzD
oMoRooL χ
∫+
−Λ
−− Ω−+Ω+Ω+Ω+=
⇒+
+=
+=
1
)1(1
1
2/12342
22
]ˆ)1(ˆˆ[ˆˆ
)1(1
)1()1(
)1(
z aaaaad
Hc
zD
zz
zDD
oMoRooA
LA
χ
Distanza di Diametro Angolare
( )22 )1(
11)2(20z
zzH
cDo
ooo
oARo +Ω
−Ω+Ω−−Ω=⇒=Ω=ΩΛ
2/3)1(1121;0
zz
HcDo
AoRo +−+
=⇒=Ω=Ω=ΩΛ
∫+
−Λ
−− Ω−+Ω+Ω+Ω+=
1
)1(1 2/12342 ]ˆ)1(ˆˆ[ˆ
ˆ)1(
1z aaaa
adHc
zD
oMoRooA
Distanza di Diametro Angolare
• L’ andamento del diametro angolare di una sorgente di dimensioni lineari date in funzione del redshift mostra una caratteristica molto interessante: allontanandosi sempre di piu’, le dimensioni diminuiscono fino ad un valore minimo. Poi riaumentano !
• Il motivo e’ che la materia interposta tra noi e la sorgente agisce da lente gravitazionale, ingrandendola.
• Per un ammasso di galassie di diametro proprio pari a 1 Mpc, la minima dimensione angolare e’ dell’ ordine di 230”. Una galassia di 20 kpc ha una minima dimensione di 4”.
Log θ
Log z
∫+
−Λ
−− Ω−+Ω+Ω+Ω+=
1
)1(1 2/12342 ]ˆ)1(ˆˆ[ˆ
ˆ)1(
1z aaaa
adHc
zD
oMoRooA
Distanza di Diametro Angolare
J.C.Jackson, J. Cosmol. Astropart. Phys. JCAP11 (2004) 007“Ultra-compact radio sources”
Ωm=0.24ΩΛ=0.76
Ωm=0ΩΛ=0
• Sorgenti radio ultracompatte (diametro apparente pochi millesimi di secondo d’arco, risolte solo con l’ uso di interferometri molto grandi, VLBI)
• a
Big
Ban
g(z
=∞)
Ric
ombi
nazi
one
(z=1
100)
qui, ora
Dimensionidell’ orizzonte alla ricombinazione:Rad. : 2ct (solo primi 50000 anni)Poi mat. (50000-380000 anni)In generale:
∫=rect
orech tacdttatd
0 )()()(
Misura degli orizzonti• L’ orizzonte alla ricombinazione e’ un righello osservabile e
lontanissimo (z=1100). E’ osservabile per il suo effetto sulla CMB: regioni che sono state causalmente connesse devono apparire diverse da quelle disconnesse.
• Nell’ immagine della CMB si deve quindi riconoscere una scala caratteristica: quella dell’ orizzonte alla ricombinazione.
• L’ orizzonte ha dimensioni fisiche dΗ(trec), e sottende un angolo θΗ che possiamo calcolare come θΗ =dΗ(trec)/DA
• In generale l’ angolo θΗ e’ una funzione delle due variabili ΩΜο e ΩΛ, ma sempre dell’ordine di un grado. Se Ω = ΩΜο + ΩΛ =1 (linea tratteggiata), θΗ e’indipendente dai valori relativi di ΩΜο e ΩΛ e vale circa 0.85 gradi.
r = 0r1
Δθ ΔS
∫+
−ΛΛ
− Ω−Ω−+Ω+Ω+=
1
)1(1 2/1232 ]ˆ)1(ˆ[ˆ
ˆ)1(
1z aaa
adHc
zD
MoMooA
ΛΩ−Ω
=2Mo
oq
∫=rect
orech tacdttatd
0 )()()(
Ω=1Ω=1.2
Ω=0.7
• L’ angolo di 0.015 radianti e’una specie di spartiacque: angoli maggiori corrispondono a Ω>1; angoli minori corrispondono a Ω<1.
• Misurando Δθ si puo’ quindi ricavare la densita’ media di massa-energia nell’ universo.
• In Y2000, the breakthrough: wide, detailed images of the CMB
de Bernardis et al. 2000Hanany et al. 2000 0.5o
• Dalle misure di spettro di potenza e’ evidente un picco ad un multipolo di circa 200, corrispondente ad un angolo sotteso di circa 0.9 gradi.
• Questo significa che la maggior parte delle strutture ha dimensioni di circa 0.9 gradi.
• Questa misura ha un certo errore, per cui definisce non una singola linea, ma una area elongata nel diagramma ΩMo, ΩΛ.
• L’ area consistente con le misure (punti neri) comprende l’ universo critico (Ω=1, linea rossa), mentre non e’consistente con Ω=0.8 o con Ω=1.2 (linee blu tratteggiate)
• Possiamo quindi concludere che l’ universo ha parametro di densita’ (1.0+0.2).
P. de Bernardis, et al., Nature, 404, 955-959, 2000.
WMAP & BOOM/98: Power Spectra
Ωo=1.02+0.02
Parametri Cosmologici e Spettro di Potenza
• La forma dello spettro di potenza delle anisotropie CMB dipende soprattutto da tre parametri cosmologici:
Ωο, n, Ωb• Il parametro n descrive la pendenza dello spettro di
potenza iniziale delle perturbazioni di densita’.
• Se queste vengono generate dal processo inflazionario, ci si aspetta n=1 (in realta’ leggermente meno di 1).
• Se n e’ >1 c’e’ un eccesso di fluttuazioni a scale piccole (k grandi), cioe’ uno spettro “blu”
• Se n e’ <1 c’e’ un eccesso di fluttuazioni a scale grandi (k piccoli), cioe’ uno spettro “rosso”
• Lo spettro di potenza delle anisotropie “ricalca” quello delle fluttuazioni di densita’ che le generano.
ninkdef
AkkPkP == )()( 2δ
• Infine, la densita’ di barioni influisce sulla simmetria delle oscillazioni acustiche. Aumentandola, si sopprimono i picchi di ordine pari rispetto a quelli di ordine dispari.
• 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 Ecomponent, 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).
• 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