This document discusses estimating the tensor-to-scalar ratio from cosmic microwave background polarization measurements. It provides background on CMB polarization, including how density perturbations and gravitational waves create E-mode and B-mode polarization patterns. Extracting the CMB polarization signal requires cleaning foreground contamination from galactic synchrotron and dust emission. Component separation methods aim to separate the CMB, synchrotron, and dust signals using multi-frequency data. Estimating the tensor-to-scalar ratio also requires accounting for the covariance from instrument noise and residual foregrounds in power spectrum estimation and parameter constraints.

1. Estimating the
TENSOR-TO-SCALAR RATIO
Yabebal Fantaye (SISSA)
With : Carlo Baccigalupi (SISSA)
Radek Stompor (APC)

2. Outline

CMB Review

CMB polarization

Polarization foregrounds

Extracting CMB Polarization

Parameter estimation

Constraining r
08/23/10 SAAO, Cape Town 2

3. CMB Review I
➢
The cosmic microwave background is the
snapshot of the baby Universe when it was
~400,000 years old.
Simulated PLANCK 1 Year CMB
Intensity Map
08/23/10 SAAO, Cape Town 3

4. CMB Review II
➢ Presence of quadrupolar temperature anisotropy at
decoupling yield a polarized CMB. CMB polarization
carries complementary Information to that of the
CMB temperature.
Q + U
08/23/10 SAAO, Cape Town 4

5. Future and ongoing CMB Experiments
Satellite
PLANCK WMAP
resolution 5 arcmin resolution 14 arcmin
frequency coverage: 9 (LFI and HFI) frequency coverage: 5 (LFI) channels

6. Future and ongoing CMB Polarization
Experiments
Charles Lawrence

7. CMB
POLARIZATION
 Stokes Q and U parameters
 Origin of CMB polarization
 E and B modes in real space
 Polarization power spectrum
08/23/10
08/23/10 SAAO, Cape Town
SISSA student seminar 7

8. Stokes Q and U parameters
I = |Ex| + |Ey|
2 2
Q = |Ex| - |Ey|
2 2
U = 2Re(ExEy ) *
V = 2Im(ExEy ) *
08/23/10 SAAO, Cape Town 8

9. Stokes Q and U parameters
http://www.physics.princeton.edu/cosmology/capmap/polscience.html
Each point on the sphere has a Q or U value
determined by the polarization at that point.
08/23/10 SAAO, Cape Town 9

10. The Origin of CMB Polarization
Red
shifted
 Polarization of the CMB is
inevitable if anisotropies
exist at decoupling.
Blue  Only a quadrupolar
shifted anisotropy (as viewed by the
electron) gives rise to
polarization
Vertically
dσ T polarized
∝∣ε . ε '∣
d
08/23/10 SAAO, Cape Town 10

11. The Origin of CMB Polarization II
Two sources:
●
Density Perturbations
at z=1000 lead to
velocities that create
“E-mode polarization”
(no curl)
●
Gravity waves: create
“B-mode polarization”
(curl)

12. The Origin of CMB Polarization II
Two sources:
●
Density Perturbations
at z=1000 lead to
velocities that create GW
“E-mode polarization”
(no curl)
●
Gravity waves: create
“B-mode polarization”
(curl)

13. E and B mode patterns II
To disentangle the polarization created by the different perturbations we construct the E/
B field which has even and odd polarity. B-mode can only be generated by vector or
tensor per.
Unchanged Q>0 U=0 Q<0 U=0
under parity flip
Sign reverses Q=0 U>0 Q=0 U<0
under parity flip
08/23/10 SAAO, Cape Town 13

14. CMB B-mode polarization
 Large scale Gaussian B-modes from primordial
gravitational waves:
Inflation  GW  local quadrupole around the
last scattering electron  Thomson scattering
 E and B mode polarization
 Non-Gaussian B-modes on small and large scales :
✔
expected signal from lensing of CMB
✔ foregrounds, systematics, etc.
08/23/10 SAAO, Cape Town 14

15. CMB POWER SPECTRUM
• We expand T, E and B CMB modes in
spherical harmonics.
T ∞ l 
n
=∑ ∑a T
 lm Y  lm  n 
T0 l=1 m=−l
• We can then form four possible CMB
power spectra TT, TE, EE and BB.
08/23/10 SAAO, Cape Town 15

16. CMB polarisation spectra
• Have 4 possible spectra: TT, TE, EE, BB.
• TB = EB = 0 by parity.
Gravitational
Lensing
Reionisation
Gravitational
Waves
08/23/10
08/23/10 SAAO, Cape Town
SISSA student seminar 16

17. Tensor-to-scalar ratio
r = AT / A S
AS AT
08/23/10 SAAO, Cape Town 17
http://space.mit.edu/~angelica/polarization.html

18. B-mode power spectrum
 The B-mode power spectrum from
gravitational waves peaks around
ℓ=100. On small and large scales the
contribution from reionization and
lensing dominates .
 The amplitude of the B-mode power
spectrum from gravitational wave is
directly related to the energy scale of
inflation and hence very ideal to probe
the very early Universe.
08/23/10 SAAO, Cape Town 18

19. POLARIZATION
FOREGROUNDS
 Foreground levels
 CMB vs. Foregrounds
08/23/10 SAAO, Cape Town 19

20. Polarization foregrounds
• The two dominant sources of polarization
foregrounds are : Synchrotron, produced by
cosmic-ray electrons orbiting in the total
Galactic magnetic field, and Dust, absorption
of starlight by aligned non-spherical dust
grains.
• Free-free emission is unpolarized and spinning
dust grains are expected to have polarization
fractions of 1-2%.
• The signal from polarized radio sources is
negligible.
08/23/10 SAAO, Cape Town 20

21. Polarization foregrounds
Page et al. 2007
r = 0.3, τ = 0.09
08/23/10 SAAO, Cape Town 21

22. Foregrounds at ℓ=100
Ebex science proposal 2007
Synch
Dust
B-mode
Planck-Noise
Ebex-Noise
CMB-B, Dust, Synch outside P06 and in EBEX region, and Planck + EBEX
sensitivity
08/23/10 SAAO, Cape Town
08/23/10 SISSA student seminar 22

23. EXTRACTING CMB
POLARIZATION
 Foreground cleaning a.k.a.
component separation
 Power spectrum estimation
08/23/10 SAAO, Cape Town 23

24. CMB ANALYSIS PIPELINE FLOW
 Data Collection
 Data Selection (cutting bad data)
 Data Filtering
 Data Calibration (relative calibration)
 Noise Estimation
 Map Estimation
 Foreground Correction
 Power Spectrum Estimation
 Cosmological Parameter Estimation
08/23/10 SAAO, Cape Town 24

25. Simulation setup
 Stivoli et al. 2010 -
Suborbital experiments
1) Balloon-borne (EBEX like)
fsky – 1%
FWHM – 8‘
F channels ( GHz ) - 150, 250, 410
Noise levels (μK/3.5' pix) - 1.5, 4, 40
2) Ground-based (POLARBeaR like)
fsky – 2.5%
FWHM – 8‘
F channels ( GHz ) – 90, 150, 220
Noise levels (μK/3.5' pix) – 3, 3, 9
High
Noise levels (μK/3.5' pix) – 1, 1, 1
low
08/23/10 SAAO, Cape Town 25

26. Component separation I
 Parameteric component separation
(Stompor et al. 2009 ): solves the data
model,
=A( )+n
where A is the mixing matrix (contains the frequency scaling of
the components) and n is noise
08/23/10 SAAO, Cape Town 26

27. Component separation II

For experiments with N channels, at most
N parameters can be solved. In our case
we have 3 channels and 4 parameters,
CMB, dust and synchrotron amplitudes
plus dust frequency scaling.

Assumptions :
Basic – Self-contained FG separation.
Synchrotron is assumed negligible (true for
balloon experiment)
No Sync – Synchrotron component is not
added in the simulation
08/23/10 SAAO, Cape Town 27

28. Stivoli et al. 2010
Ground and
Balloon input spectra for the
150GHz channel
08/23/10 SAAO, Cape Town 28

29. Component separation III
 Using ML parametric component separation
Stivoli et al. concluded that
1) basic foreground + balloon, detection of
r=0.04 is possible at 2-sigma
Stivoli et al. 2010
08/23/10 SAAO, Cape Town 29

30. Component separation III
2) Ground observation requires external
information to reach similar precision to that
of the balloon experiment.
08/23/10 Stivoli et al. 2010 SAAO, Cape Town 30

31. Component separation III
 Given the estimate of the residual
foreground, the total covariance matrix
for the estimated power is written as
' ' ' '
Where b(b') denotes the multipole bin
number and Δ the residual foreground
map
08/23/10 SAAO, Cape Town 31

32. PARAMETER
ESTIMATION
 component separation
 Power spectrum estimation
08/23/10 SAAO, Cape Town 32

33. Open problems
 No one has yet done proper propagation of errors from
foreground separation, power spectrum estimation to
error on r.
 Stivoli et al. result assumes that all parameters except r
are perfectly known, which is not the case in reality.
 Biases on the power spectrum by the presence of
residual foregrounds needs to be properly accounted.
 Covariances from instrument and foregrounds needs to
be accurately propagated
08/23/10 SAAO, Cape Town 33

34. This work
 Degeneracies of r with other parameters: we use the six
WMAP parameters plus r, (Ωb,Ωm,θ,τ,ns,As, r), in our
MCMC analysis. We choose CosmoMC to do this.
 Residual foregrounds : we introduce a two parameter
model to study the bias caused by the presence of the
residual foreground
 Covariance matrix : For now we assume uncorrelated
instrumental noise + Cov from 100% correlated residual
components.
 We use a 9-dimensional Gaussian likelihood.
08/23/10 SAAO, Cape Town 34