1

Misure della CMB• Nel suo modulo Phil Mauskopf ha trattato in generale

la detezione di radiazione millimetrica.• 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 didetezione intrinseci.

2. I foreground locali ed il loro effetto sulla misura della CMB3. Telescopi, fotometri, polarimetri, spettrometri per la misura

della CMB. Osservazioni da terra e dallo spazio. Metodi dicalibrazione.

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

What is the CMB According to modern cosmology:An abundant background ofphotons 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 ofexpansion of the Universe

γ2→+bb

t

10−6s

1013s

1017s

vis ibleN I RM W

visibleN I RM W

T = 3 0 0 0 K

T = 3 K

em

now

em

now

r

rz ==+

λλ

1

T > 1 G e V

B(ν)

B(ν)

Today 400γ/cm3

Image o f Solar Granulation

8 light minutesHere, now

P l a s m a i n t h e

s o l a r p h o t o s p h e r e

(5500 K)

Image o f Solar Granulation

The BOOMERanG map of the last scattering surface

8 light minutes

14 billion light years

Here, now

Here, now

P l a s m a i n t h e

s o l a r p h o t o s p h e r e

(5500 K)

P l a s m a i n t h e L S S

the c o s m i c

p h o t o s p h e r e

(3000 K)

• The CMB has been serendipitously d iscovered in 1964 by A. Penzias and R. Wilson, as a thermal noise at a temperature of 3K.

• Robert Dicke and hisstudent David Wi lk inson, in Princeton, and Zel ’dovich’s group in the Soviet Union were already looking for this radiat ion when it was d iscovered.

• Earl ier detections have been found a-posteirori.

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

• It is instructive to see why the CMB, represent ing the majority of photons in the universe, escaped detection till 1964.

qThe CMB br ightness is maximum at wavelengths between 10 cm and 0.5 mm: the last range to see the development of sensit ive detectors.

qThe CMB is faint with respect to local emission in the earth env i ronment.

qThe CMB is isotropic: diff icult to separe from instrumental constant emiss ion

qThe earth atmosphere is not transparent in most of the range.

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

2

observer

InterstellargascloudT c=2.3KWhy ?

hotstar

G . H e r z b e r g ( 1 9 5 0 , S p e c t r a o f d i a t o m i c m o l e c u l e s ):

“… 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 sensit ivity and stabil ity of radioreceivers.

• The breakthroughs of Penzias and Wi lson were

• the use of a differential receiver with synchronous modulat ion and demodulat ion, 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 emiss ion to which sky br ightness was cont inuous ly compared.

• Without any of these improvements, and without the ost inat ion in excluding al l 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-passedmodulated signal(drift removal)

3

Sky

mirror

chopper

ColdReferenceSource

Detector

S k y s i g n a l

C o l d R e f e r e n c e s i g n a l

)(2

1)sin()(

)()sin()()(

refsky

refsky

TTttS

tntTTtS

−=

+−=

∞→ττω

ω

Differential Radiometer

Referencesource

• In the case of Penziasand Wilson, the e mwaves were col lectedby an off- axis, l owsidelobes antenna, and propagated in a wavegu ide.

• The chopper was a ferrite “Dicke switch”

• The reference source was a “ cold load” cooled at 4.2K with l iquid helium.

• The detector was a low -noise maser ampli f ier.

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 telescope

response RA(θ,φ)

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

RA(θ)θm a i n l o b e

side lobes

B r i g h t n e s s f r o m d i r e c t i o n (θ,φ)T e l e s c o p e r e s p o n s e

in d i rec t ion (θ,φ)

F W H M =λ/ D

b o r e s i g h t

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 telescope

response RA(θ,φ)

RA(0)=1

m a i n l o b e

side lobes

B r i g h t n e s s f r o m d i r e c t i o n (θ,φ)T e l e s c o p e r e s p o n s e

in d i rec t ion (θ,φ) b o r e s i g h t

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 telescope

response RA(θ,φ)

RA(θ)<1θm a i n l o b e

side lobes

B r i g h t n e s s f r o m d i r e c t i o n (θ,φ)T e l e s c o p e r e s p o n s e

in d i rec t ion (θ,φ) b o r e s i g h t

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 telescope

response RA(θ,φ)RA(θ)<<1θ

m a i n l o b e

side lobes

B r i g h t n e s s f r o m d i r e c t i o n (θ,φ)T e l e s c o p e r e s p o n s e

in d i rec t ion (θ,φ) b o r e s i g h t

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

4

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 telescope

response RA(θ,φ)

RA(θ)<<1

θ

m a i n l o b e

side lobes

B r i g h t n e s s f r o m d i r e c t i o n (θ,φ)T e l e s c o p e r e s p o n s e

in d i rec t ion (θ,φ) b o r e s i g h t

Ω= ∫ dRABAW ),(),(4

ϕθϕθπ

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)θm a i n l o b e

side lobes

F W H M =λ/ D

b o r e s i g h t

Ω+Ω= ∫∫ dRABdRABAW

lobesside

Ground

lobemain

sky ),(),(),(),( ϕθϕθϕθϕθ

Importance of low sidelobes

Ω+Ω= ∫∫ dRABdRABAW

lobesside

Ground

lobemain

sky ),(),(),(),( ϕθϕθϕθϕθ

Ω+Ω≈

lobesside

lobessideGround

lobemain

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

s ignal of in te res t d i s t u rbance s i gna l

K3≈ srad1<< sradπ2≈K300≈

O b t a i n i n g : s ignal o f in te res t >> d i s tu rbance s i gna l

r equ i r e s

),(),( ϕθϕθlobesside

lobemain RARA >>>

1≈

Ω+Ω≈

lobesside

lobessideGround

lobemain

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

s ignal of in te res t d i s t u rbance s i gna l

K3≈ srad1<< sradπ2≈K300≈

600

)(

),(

),(),(),(

srad

B

BRARA lobe

main

Ground

sky

lobesside

lobemain

lobemain

lobesside

Ω≈

Ω

Ω<<

ϕθ

ϕθϕθϕθ

1≈

< < 1 x 1 0 -107 x 1 0 -8 s r a d1 ’

< < 1 x 1 0 -87 x 1 0 -6 s r a d1 0 ’

< < 4 x 1 0 -72 x 1 0 -4 s r a d1o

< < 4 x 1 0 -52 x 1 0 -2 s r a d1 0o

< R A sidelobes >Ωmain lobeF W H M

!!!

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 k in parallel 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

k in

rdS

What is RA(θ,φ) ?

• For small angles θ the vectorq=k-k in is 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

k in

∫∫ ⋅−=S

rqi

oqdSeuu

rr

q

rdS

aq

aqJurdrqrJudrrdeuu o

a

oo

a

iqr

oq

)(2)(2 1

00

2

0

cos === ∫∫ ∫ − πϕπ

ϕ

B e s s e l f u n c t i o n s

q

5

What is RA(θ,φ) ?

• The intensity isthe square of the

field:

• Example: a 2m

diameter mirror used at 1 cm and at 1 mm

2

1)(2

=

Ω θθ

ak

akJI

d

dIo

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)

m a i n

l o b e

side

lobes

What is RA(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)

m a i n

l o b e

side

lobes

2

1)(2

=

Ω θθ

ak

akJI

d

dIo

What is RA(θ,φ) ?

• The intensity isthe square of the

field:

• The first zero is for

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 re

spon

se

off-axis angle θ (deg)

a222.110

λθ =

θ10

2

1)(2

=

Ω θθ

ak

akJI

d

dIo

What is RA(θ,φ) ?

• The intensity isthe square of the

field:

• The first zero is for

• The FWHM is similar

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 re

spon

se

off-axis angle θ (deg)

a222.110

λθ =

0 . 5 F W H M

θ10

2

1)(2

=

Ω θθ

ak

akJI

d

dIo

What is RA(θ,φ) ?

• The intensity isthe square of the field:

• The envelope of the off -axis response scales as θ -3

approximately starting 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 re

spon

se

off-axis angle θ (deg)

3)( −∝θθRA

2

1)(2

=

Ω θθ

ak

akJI

d

dIo

exercise

• For the angular response of a circularaperture, compare the power in the main lobe 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)(2

2

sin)(2

2

/61.0

0

2

1

/61.0

2

1

∫

∫

=

6

Low diffraction design

• Real world angular responses are worse than the one studied here.

• Sharp edges are in general important sources ofdiffraction, 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

• 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 establ ish the black-body nature of the CMB has been a diff icult challange, involving for decades researchers in Princeton ( Wilkinson), Berkeley (Smoot, Lubin), Firenze (Melchiorri), Milano (Sironi), Bologna ( Mandolesi) …

• At wavelengths > 20 cm our Galaxy becomes br ighter than the CMB.

• At wavelengths < 10 cm the correction for atmospheric emission is signif icant even at high altitude sites. It is necessary to move the experiment outside the earth atmosphere.

• Bal loon experiments (Paul Richards) and finally the COBE satel l ite (FIRAS, JohnMather) 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 is evidence for a hot initial phase of the Universe.

J. Mather et a l . 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.

7

• How did FIRAS work ?

• This is aga in a differential ins t rument

• FIRAS is a cryogenic FTS ( Fourier Transform Spectrometer).

Movable

Mirror

Mirror

Detector

Source

BS BS

x/2

FTS ( Michelson Interferometer)

Chopper

Zero Path Difference:T h e t w o w a v e s are c o m b i n e d in

p h a s e a t the de tec to r

Mirror

Detector

Source

BS BS

x/2

FTS ( Michelson Interferometer)

Chopper

x Optical Path Difference:T h e t w o w a v e s are c o m b i n e d w i t h a

p h a s e d i f f e r e n c e w h i c h d e p e n d s on λ

FTS• If the optical path difference is x, the electrical

field of the em wave at the detector will be

• Here σ=1/λ is the wavenumber. The power detected will be

• If the source is not monochromatic, we have to add all the frequency components

)](2cos[)()2cos()()( xctEctEtE oo ++= πσσπσσ

( ) ( )[ ]xEtExI o πσσ 4cos1)()( 22 +=∝

( ) ( ) ( ) ( ) σπσσσπσσ dxSdxEIxI otot 4cos4cos)(00

2 ∫∫∞∞

∝∝−

Interferogram (measuredchanging x)

Spectrum of theradiation under analysis

FTS

• If we could measure the interferogram up to infinite distances x, it would be possible to derive the spectrum of the radiation under analysis by Fourier transformingthe interferogram.

• In practice we can only measure

• xm a x depends on the instrument; its effect is to degrade the resolution of the spectral measurement.

( ) ( ) σπσσ dxSIxI tot 4cos)(0

∫∞

∝−

( ) ( ) ( )dxxIxIS tot∫∞

∞−

−∝ πσσ 4cos)(

( ) ( ) ( )dxxIxIS

x

x

tot∫−

−∝max

max

4cos)(' πσσ

FTS• Consider a rectangular spectrum

• In the limit

• This means that instead of obtaining a delta function in σo, we obtain a sinc function, with a width inversely proportional to x m a x .

• This limits the resolution of the instrument, because two closeby lines will produce two sinc functions slightly offset. If the offset corresponds to more than the first zero of the sinc function, the two lines will be distinguished. This requires that . In the case of the CMB, 1 cm-1

is enough, so x>1cm: a compact interferometer.

( ) ( ) ( ) ( )( )]4[

]4sin[4cos4cos'

max

max

max

maxσσπ

σσππσπσσ

−−

∝∝ ∫− o

o

x

x

ox

xdxxxS

S(σ)

σσo

u

S o/u

0→u

max/1 x>∆σ

8

The Martin-Puplett FTS

Roof Mi r ro r

BS Pol

.

S 1

S 2

D1

D2In

put P

ol.

Outpu

t Pol.

• FIRAS was a Martin-Puplettinterferometer .

• The MPI has two input ports (Source1 and Source2) and two output ports (Det.1 and Det.2)

• Beamsplitting is acheived by means of a polarizer. This requires two additional polarizers: input and output.

The Martin-Puplett FTS

Roof Mi r ro r

BS Pol

.

S 1

S 2

D1

D2

Inpu

t Pol

.

Outpu

t Pol.

• At mm wavelengths, wire grids are almostideal polarizers.

• The input polarizer has wires parallel tothe plane of the drawing. So it transmits only the polarized component of S1 orthogonal to the drawing, and reflects only the polarized component of S2 parallel to the drawing.

The Martin-Puplett FTS

Roof Mi r ro r

BS Pol

.

S 1

S 2

D1

D2

Inpu

t Pol

.

Outpu

t Pol.

• The Beam Splitter polarizer has wires tilted by 32o26’ wrt the plane of the drawing.

• In this way its wires are seen at an angle of 45o by the input beam 3.

• So reflected (4) ad transmitted (4’) beams will contain radiation from both sources in both polarizations.

3

4

4’

The Martin-Puplett FTS

Roof Mi r ro r

BS Pol

.

S 1

S 2

D1

D2

Inpu

t Pol

.

Outpu

t Pol.

• Roof mirrors are used as retroreflectors, so that the polarization direction is rotated by 90o.

• So the beam which had been transmitted by the BS polarizer (6’) is now reflected towards the output polarizer, and the beam which had been reflected (6) is now transmitted, alsotowards the output polarizer.

3

4

4’6’

6

The Martin-Puplett FTS

Roof Mi r ro r

BS Pol

.

S 1

S 2

D1

D2

Inpu

t Pol

.

Outpu

t Pol.

• So we have recombined the two beams, after having introduced a delay in one of the two.

• The two beams directed to the output polarizer have orthogonal polarizations.

• The wires of the output polarizer are seen at an angle of 45o from the two beams, so both the final beams (9 and 9’) contain components from both the delayed and undelayed beams.

3

4

4’6’

6

9

9’

1

2

The Martin-Puplett FTS

Roof Mi r ro r

BS Pol

.

S 1

S 2

D1

D2

Inpu

t Pol

.

Outpu

t Pol.

• Two independent interferograms are recorded by the two detectors D1 and D2.

• In FIRAS the two detectors were used for high and low frequencies.

• Each interferogram contains the DIFFERENCE of the spectra of the two sources S1 and S2.

• In FIRAS S1 was from the sky, and S2 was from an internal cryogenic blackbody.

3

4

4’6’

6

9

9’

1

2

9

FIRAS• The FIRAS guys were able to change the temperature of

the internal blackbody until the interferograms were 0.

• This is a null measurement, which is much more sensitive than an absolute one: (one can boost the gain of

the instrument without saturating it !).

• This means that the difference between the spectrum of

the sky and the spectrum of a blackbody is 0, i.e. the spectrum of the sky is a blackbody with that temperature.

• This also means that the internal blackbody is a real blackbody: it is unlikely that the sky can have the same deviation from the Planck law characteristic of the source built in the lab. σ (cm- 1) wavenumber

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.

• Phil has already spoken at length about photon noise, so this will be just a quick application.

CMB observab les

• The spectrum

1

2),(

3

2 −=

xec

hTB

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)1

2),(

2

2 −=

xecTn

νν

GHzkT

hx

CMB

CMB69

νν==

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:

( ) ( ) ( ) Wthh

WhNhNhE ν

νννν ===∆=∆ 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 4222

=⇒=∆ θθ &

WdfhdfW f ν22 =∆

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

WWhW f

112105.22

−×==∆ ν

HzW /10 15−

HzW /105.211−

×

10

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

wave interference noise: the correct statisticsis Bose-Einstein.

g

NNN

d

NdTN

e

gN

VT

kTE

2

2

,

2

/)(;

1

+=∆⇒

=∆−

= − µµ

Poisson noiseWave interference noise

Radiation Noise• For a blackbody

−+=∆⇒

−=

=−

= −

1

11

1

18

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

18

32

5

4

5

32

5

2

/

2

//

22

/

2

1077.24

1

4

1

112

1

11

1

11

1

11

2

1

KHzsrcm

W

hc

k

dfdxe

exTA

hc

kdfW

dfe

WhdfW

eWh

eNhE

eNN

x

x

x

x

kTh

kThkTh

kTh

−×=

−Ω=∆

−+=∆

−+=

−+=∆

−+=∆

∫

ν

νν

ν

ν

νν

CMB observab les

• The spectrum

• The angular distribution

• The polarization state

• The noise

kT

hx

ec

hTB

x

ννν =

−=

1

2),(

3

2

T

TTB

e

xeTB

x

x ∆−

=∆ ),(1

),( νν

T

TTB

e

xeTB P

x

x

P

∆−

=∆ ),(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 H z-1

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

W m -2 sr -1 H z-1

W m -2 sr -1 H z-1

T=2.725 K

average brightness

anisotropy (rms) polarizat ion (E rms)

photon noise (rms)

polarizat ion (B rms)

CM

B (M

KS

uni

ts)

Frequency (Hz)( )dx

e

ex

hc

TkdxTW

x

x

2

4

32

552

1

4),(

−=∆ 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 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

dfW

N

f

fW

Wt

∫ ∆

==

max

min

2

σσ

tW

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.

t

W

t

tW

t

dfW

t

dfW

N

dfW

N

T

f

f

f

fW

WT

22

/1

/1

2

22

/

11

/

/

max

min

max

min

∆≅

−∆

=

∆

≅

=

∆

=

∆

==

∫

∫∫

ττ

τ

τ

σσ

τ

11

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

polarization) measurement limited only by radiation noise:

( )

( )t

dxe

exA

hc

Tk

dxe

exA

hc

Tk

T

T

dxTxBe

xe

B

T

T

dxTxBe

xe

T

TTB

x

x

x

x

x

xx

x

x

x

x

x

x

x

x

x

1

)1(

2

1

4

),(1

),(1

),(

2

1

2

1

2

1

2

1

2

4

32

44

2

4

32

55

∫

∫

∫

∫

−Ω

−Ω

=

∆

−

∆=

∆

−∆

=∆

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 (

µK)

integration time (s)

The ultimate sensitivity plot !!

CMB BLIP