# Impact on the tensor-to-scalar ratio of incorrect Galactic foreground modelling

**ABSTRACT** A key goal of many Cosmic Microwave Background experiments is the detection

of gravitational waves, through their B-mode polarization signal at large

scales. To extract such a signal requires modelling contamination from the

Galaxy. Using the Planck experiment as an example, we investigate the impact of

incorrectly modelling foregrounds on estimates of the polarized CMB, quantified

by the bias in tensor-to-scalar ratio r, and optical depth tau. We use a

Bayesian parameter estimation method to estimate the CMB, synchrotron, and

thermal dust components from simulated observations spanning 30-353 GHz,

starting from a model that fits the simulated data, returning r<0.03 at 95%

confidence for an r=0 model, and r=0.09+-0.03 for an r=0.1 model. We then

introduce a set of mismatches between the simulated data and assumed model.

Including a curvature of the synchrotron spectral index with frequency, but

assuming a power-law model, can bias r high by ~1-sigma (delta r ~ 0.03). A

similar bias is seen for thermal dust with a modified black-body frequency

dependence, incorrectly modelled as a power-law. If too much freedom is allowed

in the model, for example fitting for spectral indices in 3 degree pixels over

the sky with physically reasonable priors, we find r can be biased up to

~3-sigma high by effectively setting the indices to the wrong values.

Increasing the signal-to-noise ratio by reducing parameters, or adding

additional foreground data, reduces the bias. We also find that neglecting a 1%

polarized free-free or spinning dust component has a negligible effect on r.

These tests highlight the importance of modelling the foregrounds in a way that

allows for sufficient complexity, while minimizing the number of free

parameters.

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**ABSTRACT:**This clear, straightforward, fundamental introduction to radiative processes in astrophysics is designed to present - from a physicist's viewpoint - radiation processes and their applications to astrophysical phenomena and space science. The book covers such topics as radiative transfer theory, relativistic covariance and kinematics, bremsstrahlung radiation, Compton scattering, some plasma effects, and radiative transitions in atoms. The discussion begins with first principles, physically motivating and deriving all results rather than merely presenting finished formulas. Much of the prerequisite material is provided by brief reviews, making the book a self-contained reference tool. Also included are about 75 problems with solutions, illustrating applications of the material and methods for calculating results01/1979; - SourceAvailable from: Edward J. WollackB. Gold, C. L. Bennett, R. S. Hill, G. Hinshaw, N. Odegard, L. Page, D. N. Spergel, J. L. Weiland, J. Dunkley, M. Halpern, N. Jarosik, A. Kogut, E. Komatsu, D. Larson, S. S. Meyer, M. R. Nolta, E. Wollack, and E. L. Wright[Show abstract] [Hide abstract]

**ABSTRACT:**We present a new estimate of foreground emission in the Wilkinson Microwave Anisotropy Probe (WMAP) data, using a Markov chain Monte Carlo method. The new technique delivers maps of each foreground component for a variety of foreground models with estimates of the uncertainty of each foreground component, and it provides an overall goodness-of-fit estimate. The resulting foreground maps are in broad agreement with those from previous techniques used both within the collaboration and by other authors. We find that for WMAP data, a simple model with power-law synchrotron, free-free, and thermal dust components fits 90% of the sky with a reduced χ2 ν of 1.14. However, the model does not work well inside the Galactic plane. The addition of either synchrotron steepening or a modified spinning dust model improves the fit. This component may account for up to 14% of the total flux at the Ka band (33 GHz). We find no evidence for foreground contamination of the cosmic microwave background temperature map in the 85% of the sky used for cosmological analysis.The Astrophysical Journal Supplement Series 02/2009; 180(2):265. · 16.24 Impact Factor - SourceAvailable from: Mikhail Basko[Show abstract] [Hide abstract]

**ABSTRACT:**A rigorous first order solution with respect to anisotropy is obtained for the equation of polarized radiation transfer in a homogeneous anisotropic universe with a flat co-moving space. The degree of polarization of the background radiation is shown to be very sensitive to the recombination dynamics and to the secondary reheating epoch. Provided that a quadrupole anisotropy of the background radiation is established, the measurements of its polarization degree enable one to set severe limitations on the conditions of secondary ionization.Monthly Notices of the Royal Astronomical Society 03/1980; 191:207-215. · 5.52 Impact Factor

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arXiv:1203.0152v1 [astro-ph.CO] 1 Mar 2012

Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 2 March 2012(MN LATEX style file v2.2)

Impact on the tensor-to-scalar ratio of incorrect Galactic

foreground modelling

Charmaine Armitage-Caplan,1⋆Joanna Dunkley,1Hans Kristian Eriksen,2

Clive Dickinson,3

1Sub-department of Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, UK

2Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo, Norway

3Jodrell Bank Centre for Astrophysics, Alan Turing Building, School of Physics & Astronomy,

The University of Manchester, Oxford Road, Manchester M13 9PL, UK

ABSTRACT

A key goal of many Cosmic Microwave Background experiments is the detection of

gravitational waves, through their B-mode polarization signal at large scales. To ex-

tract such a signal requires modelling contamination from the Galaxy. Using the Planck

experiment as an example, we investigate the impact of incorrectly modelling fore-

grounds on estimates of the polarized CMB, quantified by the bias in tensor-to-scalar

ratio r, and optical depth τ. We use a Bayesian parameter estimation method to

estimate the CMB, synchrotron, and thermal dust components from simulated ob-

servations spanning 30-353 GHz, starting from a model that fits the simulated data,

returning r < 0.03 at 95% confidence for an r = 0 model, and r = 0.09 ± 0.03 for an

r = 0.1 model. We then introduce a set of mismatches between the simulated data

and assumed model. Including a curvature of the synchrotron spectral index with

frequency, but assuming a power-law model, can bias r high by ∼ 1σ (δr ∼ 0.03).

A similar bias is seen for thermal dust with a modified black-body frequency depen-

dence, incorrectly modelled as a power-law. If too much freedom is allowed in the

model, for example fitting for spectral indices in 3 degree pixels over the sky with

physically reasonable priors, we find r can be biased up to ∼ 3σ high by effectively

setting the indices to the wrong values. Increasing the signal-to-noise ratio by reducing

parameters, or adding additional foreground data, reduces the bias. We also find that

neglecting a ∼ 1% polarized free-free or spinning dust component has a negligible ef-

fect on r. These tests highlight the importance of modelling the foregrounds in a way

that allows for sufficient complexity, while minimizing the number of free parameters.

1 INTRODUCTION

Extraction of the polarized Cosmic Microwave Background

signal at large scales is hampered by significant levels of

polarized Galactic emission. The two dominant components

are synchrotron and thermal dust, polarized due to the co-

herent magnetic field in the Galaxy (e.g., Page et al. 2007;

Fraisse et al. 2008). For an experiment observing at mul-

tiple frequencies, one method of separating the signals is

to parameterize the synchrotron and dust, and to fit for

these components, in addition to the CMB, over the re-

gion of sky where Galactic emission is lowest. While demon-

strated to work for E-mode polarization (Page et al. 2007;

Dunkley et al. 2009a; Gold et al. 2009), the signal of inter-

est is the much smaller B-mode signal from inflation (e.g.,

⋆armitage-caplan@physics.ox.ac.uk

Basko & Polnarev 1980; Bond & Efstathiou 1984). A con-

cern with using such methods is that an incorrect model can

lead to bias in the estimated CMB signal.

The

Planck

satellitemission,

2009,ismeasuring the polarization

CMB in seven channels over the frequency range 30-

353 GHz (Planck Collaboration 2006; Tauber et al. 2010;

Planck Collaboration I 2011). While Planck will produce

polarization data, which offer a multitude of opportunities

including possible recovery of inflationary B-modes at large

scales and greater understanding of the polarized nature of

Galactic foregrounds, it also comes with great challenges.

For an all-sky experiment like Planck, component separa-

tion of the polarization signal is more difficult than for the

temperature counterpart, in part because the ratio of the

foreground signal to CMB signal is higher.

In many simulated tests of component separation, the

simulations of the Galactic emission are well matched to

launched

signal

inMay

theof

Page 2

2C. Armitage-Caplan et al.

the model used to describe them. Using a Bayesian compo-

nent separation method which allows us to assume different

models of the Galactic signal, we explore the effect on the

recovery of the CMB in varying scenarios of mismatch be-

tween the model and simulation. We use the recovered CMB

map and its covariance to estimate two cosmological param-

eters: the optical depth to reionization, τ, and the tensor-to-

scalar ratio, r. In this way, we can directly quantify the bias

generated in the parameter estimation as a result of any

particular model-simulation mismatch. Both the Bayesian

component estimation method and the simulated skies used

in this paper were first used and described in a previous pa-

per, Armitage-Caplan et al. (2011). There we examined the

prospects for large-scale polarized map and cosmological pa-

rameter estimation with simulated Planck data for a single

model-simulation combination. This paper is a natural ex-

tension in which we use the same methods to recover maps

and estimate parameters, while varying the simulated data

and separation model.

In §2, we provide a brief overview of the Gibbs sampling

method, the subsequent processing of the sampled distribu-

tion, and the likelihood estimation method. In §3, we explain

how the data are simulated. A detailed account of the mis-

match tests that we examine, and their resulting parameter

estimates, is then presented in §4. We then discuss the re-

sults and methods for mitigating possible biases in §5, and

conclude in S6.

2METHOD

In the Bayesian parameter estimation method of foreground

removal, the emission models of the CMB and foregrounds

are parametrized based on our understanding of their fre-

quency dependence. Focusing on polarization analysis, a

sampling method is then used to estimate the marginal-

ized CMB Q and U Stokes vector maps (and additionally

the marginalized foreground maps) in every pixel over the

sky. In general this extends template-removal methods to al-

low for spatial variation of the foreground spectral indices,

and was first used to clean WMAP polarization data in

Dunkley et al. (2009a). In this analysis, we use HEALPix

(G´ orski et al. 2005) Nside= 16 maps containing Np = 3072

pixels. We use a code called Commander (see Eriksen et al.

(2006) and Eriksen et al. (2008)) to perform the Gibbs sam-

pling. The sampled distribution is then processed into a

mean map and covariance matrix. Finally, we perform a

likelihood estimation for the two cosmological parameters,

τ and r.

2.1 Bayesian Estimation of sky maps

By Bayes’ theorem, the posterior distribution for parame-

ters, s, given a set of maps, d, can be written as

P(s|d) ∝ P(d|s)P(s) (1)

with a prior distribution for the model parameters, P(s).

The Gaussian likelihood of the observed maps is given by

− 2lnP(d|s) =

?

ν

[dν − sν]TN−1

ν [dν − sν] (2)

where dν is the observed sky map at frequency ν, and Nν

is its covariance matrix.

As in Dunkley et al. (2009a); Armitage-Caplan et al.

(2011), we assume that the polarized Galactic emission

is dominated by synchrotron and dust emission, arising

due to the orientation of the Galactic magnetic field (e.g.,

Page et al. 2007). We define a parametric model for the to-

tal sky signal in antenna temperature for a three-component

model (k = 1 for CMB, k = 2 for synchrotron emission, and

k = 3 for thermal dust emission) as

sν = α1(ν)A1+ α2(ν;β2)A2+ α3(ν;β3)A3

(3)

where Akare amplitude vectors of length 2Npand αk(ν;βk)

are diagonal coefficient matrices of side 2Np at each fre-

quency.

Once our model, and priors on the model parameters,

are defined, we estimate the joint CMB-foreground posterior

P(A,β|d) from which we can then obtain the marginalized

distribution for the CMB map vector,

p(A1,d) =

?

p(A,β|d)dA2dA3dβ

(4)

and similarly for the other model parameters.

For the multivariate problem that we are considering,

Gibbs sampling draws from the joint distribution by sam-

pling each parameter conditionally as follows

Ai+1← P(A|β,d)

βi+1← P(β|A,d).

(5)

(6)

We use Commander to implement the sampling of the

amplitude-type and spectral index parameters. Comman-

der is a flexible code for joint component separation and

CMB power spectrum estimation; the reader is directed to

Armitage-Caplan et al. (2011) for a full description of its use

for sampling only the sky signal.

2.2 Likelihood estimation of cosmological

parameters

The product of a Bayesian parametric map estimation

method is both a CMB map (which is taken to be the

mean map calculated from the Gibbs chain after some burn-

in) and a covariance matrix (which can be estimated from

the marginalized posterior distribution) and together these

products can be used to place constraints on cosmological

parameters. We compute the likelihood of the estimated

maps, given a theoretical angular power spectrum, using

the exact pixel-likelihood method described in Page et al.

(2007); Armitage-Caplan et al. (2011).

The two cosmological parameters constrained by the

large scale CMB polarization signal are the optical depth to

reionization, τ, and the tensor-to-scalar ratio, r. The signa-

ture of reionization is at ℓ<

∼20 in CEE

of the reionization signal is proportional to τ2. The tensor-

to-scalar ratio r directly scales the CBB

and is best probed at two angular scales: at the low ℓ<

‘reionization bump’ before CBB

ℓ

or at the smaller scale ℓ ∼ 100 ‘recombination bump’ where

foregrounds are expected to be lower but lensing is a con-

taminant. In this study we are considering constraints from

the large-scale reionization bump, using ∼75% of the sky.

ℓ

where the amplitude

ℓ

power spectrum

∼20

due to lensing dominates,

Page 3

3

By varying only the optical depth to reionization, and

fixing the temperature anisotropy power at the first acous-

tic peak (ℓ = 220), we calculate the likelihood for each value

of τ. Separately, we vary only the tensor-to-scalar ratio, and

calculate the likelihood at each value of r. The resulting one-

dimensional distributions for r and τ then include marginal-

ization over foreground uncertainty. To account for imper-

fect foreground cleaning in the Galactic plane, we apply a

Galactic mask when calculating the likelihoods. In this anal-

ysis we use the standard WMAP ‘P06’ mask (Page et al.

2007), which masks 26% of the sky.

3 SIMULATED MAPS

We generate simulated maps at the seven polarized nom-

inal frequency channels for Planck (30, 44, 70, 100, 143,

217, and 353 GHz). In our analysis, we do not apply

beams or smoothing to the data; these would be included

in a more realistic analysis but are not expected to signif-

icantly affect results. Realizations of the CMB are gener-

ated from a power spectrum computed using ΛCDM cos-

mological parameters (Komatsu et al.

r = 0 or r = 0.1. Diagonal white noise realizations are

generated based on the noise levels taken from the Planck

Bluebook (Planck Collaboration 2006), and we scale the

given noise levels at beam-sized pixels to the correspond-

ing noise level at Nside = 16 sized pixels, with side 3.6◦.

This noise model is over-simplified as it contains no 1/f-

noise or other spatial correlations that are reported in the

‘early’ Planck papers, which would increase effective noise

levels (Planck HFI Core Team 2011; Zacchei et al. 2011).

For the foreground components, we use two baseline

tests to benchmark the level of bias in the mismatch tests.

In Test 1 (baseline with uniform βs), spectral indices

given by simple power-laws are used to simulate the syn-

chrotron and dust foregrounds, and as a model in the compo-

nent estimation. The simulated synchrotron Q and U emis-

sion maps are modelled as power-law and given as an ex-

trapolation in frequency of the polarized 23 GHz WMAP

map:

2011), with either

Qν(p) = Q23(p)

?ν

23

?βs(p)

(7)

Uν(p) = U23(p)

?ν

23

?βs(p)

(8)

We set the synchrotron spectral index to βs = −3 uni-

formly over the whole sky, consistent with observations

by WMAP (Page et al. 2007; Gold et al. 2009). The sim-

ulated thermal dust Q and U emission maps are also mod-

elled as power-law emission and generated by extrapolat-

ing the predicted 94 GHz map in Finkbeiner at al. (1999):

Sν(p) = S94(p)?ν

angles we use a software package called the Planck Sky

Model (PSM, version 1.6.6) developed by the Planck Work-

ing Group 2. They closely match the sychrotron angles. The

dust polarization fraction is set at 12%, which is scaled

by a geometric depolarization factor due to the expected

magnetic field configuration, resulting in an observed po-

larization fraction of ∼ 4%. We set the dust spectral index

94

?βd. To generate the dust polarization

to βd = 1.5 uniformly over the whole sky. This is consis-

tent with polarization observations by WMAP at frequen-

cies below 100 GHz, although at higher frequencies ther-

mal emission is observed to deviate from power-law (e.g.,

Planck Collaboration XXIV 2011).

For the parametric model, we assume that the spectral

index of the Galactic components do not vary over the fre-

quency range considered, so the coefficients are given by

α2(ν,β2) = diag[(ν/ν30)β2] (9)

α3(ν,β3) = diag[(ν/ν353)β3]. (10)

Here we have defined the two spectral index vectors β2 and

β3 for synchrotron and dust, respectively. We set the pivot

frequencies to 30 GHz and 353 GHz. We impose Gaussian

priors on the spectral index parameters of β2 = −3.0±0.3 for

synchrotron and β3 = 1.5±0.5 for dust. The priors we have

chosen have central value and standard deviation at approx-

imately the average and range of values typically observed

and predicted theoretically (see, for example, Fraisse et al.

(2008); Dunkley et al. (2009b) for further discussion).

In Test 2 (baseline with non-uniform βs), simple power-

laws are again used to both simulate the foreground com-

ponents and also as a model in the separation estimation,

but the synchrotron index varies spatially over the sky. Dust

emission is simulated as in baseline Test 1 but synchrotron

emission is modelled as power-law with a spatially vary-

ing βs. The degree of spatial variation in the polarization

spectral index has not yet been well-measured, but a realis-

tic model is taken to be model 4 of Miville-Deschenes et al.

(2008), given by

βs =log(P23/gfsS408)

log(23/0.408)

(11)

where P23 is the WMAP polarization map at 23 GHz, g

is a geometrical reduction factor (reflecting depolarization

due to magnetic field structure), fs is the intrinsic polar-

ization fraction from the cosmic ray energy spectrum, and

S408 is the 408 MHz map of Haslam et al. (1982). The val-

ues of βs range from −3.3 to −2.8. The parametric model

is as described in baseline Test 1, where we fit to power-law

synchrotron and dust components.

4 MISMATCH TESTS

The set of tests described below are given a label identi-

fier (A through I) and a short descriptive name to help

the reader understand the results. In each test, we describe

the model used to simulate the Galactic foreground compo-

nent maps (known as the simulation) and then we describe

the model used for the parametric component separation

(known as the model). The mismatch tests are summarized

in Table 1. We categorize the mismatch tests into the follow-

ing three categories: incorrect model (§4.1); extra simulated

components (§4.2); incorrect priors (§4.3).

In every case, we define the parametric model for

the sky signal using equation 3. Given that the CMB ra-

diation is blackbody, the coefficient for α1 is given by

α1(ν,β1) = α1(ν) = f(ν)I, where the function f(ν) con-

verts the CMB signal I from thermodynamic to antenna

temperature. Though the spectral indices for Q and U in

a given pixel are expected to be similar (following from the

Page 4

4C. Armitage-Caplan et al.

Label Name Simulation Model

Baseline Tests

1 Baseline uniform βs

sync power-law βs = -3

dust power-law βd= 1.5

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.5 ± 0.5

2 Baseline non-uniform βs

sync power-law βs= −3.3 to −2.8

dust power-law βd= 1.5

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.5 ± 0.5

Incorrect Model

A Dust 2-component-a

sync power-law βs = -3

2-component dust

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.5 ± 0.5

B Dust 2-component-b

sync power-law βs = -3

2-component dust

sync power-law βs= −3 ± 0.3

1-component dust

C Synchrotron curvature

sync curvature

dust power-law βd= 1.5

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.5 ± 0.5

Extra Components

D 1% Free-free

sync power-law βs = -3

dust power-law βd= 1.5

1% polarized free-free

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.5 ± 0.5

no free-free

E1% Spinning dust

sync power-law βs = -3

dust power-law βd= 1.5

1% polarized spinning dust

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.5 ± 0.5

no spinning dust

Incorrect Priors

F Strong βs prior mismatch

sync power-law βs= −3.3 to −2.8

dust power-law βd= 1.5

sync power-law βs= −2.5 ± 0.5

dust power-law βd= 1.5 ± 0.5

G Weak βs prior mismatch

sync power-law βs= −3.3 to −2.8

dust power-law βd= 1.5

sync power-law βs= −2.8 ± 0.5

dust power-law βd= 1.5 ± 0.5

H Strong βdprior mismatch

sync power-law βs= −3

dust power-law βd= 1.5

sync power-law βs= −3 ± 0.3

dust power-law βd= 2.0 ± 0.5

I Weak βdprior mismatch

sync power-law βs= −3

dust power-law βd= 1.5

sync power-law βs= −3 ± 0.3

dust power-law βd= 1.7 ± 0.5

Table 1. Summary of mismatch tests.

assumption that the polarization angle does not change with

frequency), unless otherwise stated, we allow the option for

the indices to be sampled independently for Q and for U.

Thus, our model is completely described by 6Np amplitude

parameters A = (A1,A2,A3) and 4Np spectral index pa-

rameters β = (βQ

amplitude-type parameters and Gaussian priors on the spec-

tral index parameters. The model is estimated from 14Np

data points (seven frequencies with two Stokes parameters).

2,βQ

3,βU

2,βU

3). We impose a flat prior on

We plot the likelihood curves for the estimated parame-

ters, r and τ, for each mismatch case and show the compari-

son likelihood curve from its corresponding baseline test. By

holding all parameters constant, except for the mismatch be-

ing tested, we are able to quantify the level of bias induced

by each type of mismatch. In this section we describe each

test and present the numerical results; in Section 5 we dis-

cuss their implications.

4.1Incorrect model

Here we consider a subset of cases where the frequency de-

pendence of the synchrotron and dust emission are modelled

incorrectly.

4.1.1Thermal dust frequency dependence

Thermal dust emission is well-approximated by a modi-

fied black-body, with intensity scaling as νβBν(T), where

Bν(T) is a black-body spectrum with temperature T. In the

Rayleigh-Jeans limit, this approximates to the power-law as-

sumed in our baseline simulations. Over a broader frequency

range, the power-law approximation breaks down, and mod-

elling the curvature becomes important. In the simplest ex-

tension to a power law, it is common to fit for one or two

parameters to describe the integrated dust emission from

any line of sight: either the emissivity index β, or emissivity

plus temperature T. More realistically, the integrated dust

emission arises from dust grains at various temperatures,

so could best be represented by the sum of modified black-

bodies. In Finkbeiner at al. (1999), a model with just two

Page 5

5

Figure 1. Recovered distributions for the tensor-to-scalar ratio, r, for mismatched simulation and models, comparing the baseline (test

1) with three mismatched cases for r = 0 (left), and r = 0.1 (right). Modelling a two-component thermal dust simulation (modified

black-body emission with dust at mean temperatures 16 K and 10 K) with a power-law dust spectral index (test A) biases r high by

about 1σ, as does neglecting a curvature in the synchrotron spectral index (test C). Modeling a two-component dust simulation with a

one-component modified black-body model (Test B) has only a minor effect.

components at mean temperatures 9.6 K and 16.4 K was

found to be a good fit to the IRAS data.

Here we consider two mismatches between model and

simulation. In Test A (two-component-dust-a), the dust

emission is simulated with two temperature components,

while the parametric model fits to a dust power-law. We

use model 7 of Finkbeiner at al. (1999), with [Q,U](ν) ∝

A1νβ1Bν(T1)+A2νβ2Bν(T2). In this model the first compo-

nent is sub-dominant, with A2/A1 = 24.6. The dust emis-

sivity indices are β1 = 1.5, β2 = 2.6 over the whole sky.

Synchrotron emission is simulated as power-law with a spa-

tially uniform βs. The parametric model fits to power-law

dust and synchrotron, neglecting the curvature of the dust

spectrum. In Test B (two-component-dust-b), dust emission

is again simulated with two temperature components (as in

Test A), while the parametric model fits to a one-component

dust model, [Q,U](ν) ∝ νβBν(T). We fix the temperature

T over the sky to the values of T2 from the simulation, and

estimate a single index βd in every pixel.

Using these test cases, we perform component separa-

tion and use the resulting CMB maps to compute the like-

lihoods for parameters τ and r for the r = 0 and r = 0.1

simulations. The distributions are shown in Fig. 1, and re-

covered mean values for r, and τ, for these and all other

tests are summarized in Table 2. For r = 0 we quote 95%

upper limits; for r = 0.1 and τ we give 68% confidence lev-

els. For r = 0.1 we find a non-negligible bias on r of 1σ

high for Test A, fitting a two-component dust model with a

power-law, and a similar bias high for the optical depth, τ.

Using a one-temperature component model to fit the two-

component simulation (Test B), recovers r with only ∼ 0.2σ

bias. We see a similar effect for the r = 0 case, where for

Test A the recovered r value is greater than zero at 1σ, but

Test B is consistent with the baseline case.

4.1.2 Synchrotron frequency dependence

Synchrotron emission is expected to be roughly power-law in

frequency (see e.g., Rybicki & Lightman 1979), the result of

relativistic cosmic-ray electrons accelerated in the Galactic

magnetic field (Strong et al. 2007). However, a steepening of

the index with frequency is also expected, due to increased

energy loss of the electrons (e.g., Banday & Wolfendale

1991; Strong et al. 2007). The WMAP data are consistent

with power-law emission, but a modest steepening would fit

the data, and can be parameterized by a curvature of the

spectral index. In a pessismistic scenario, the degree of steep-

ening could vary significantly over the sky, or the frequency

dependence could be ill-fit by a single curvature parameter.

In Test C (synchrotron curvature), the simulated Galac-

tic foreground includes a steepening of the synchrotron index

with frequency while the parametric model retains power-

law synchrotron emission. The synchrotron emission has

spectral curvature such that the index decreases by 0.3 above

23 GHz. Figure 1 and Table 2 show the results from this

third test case. The effect on the recovered CMB is non-

negligible. We find that a synchrotron curvature simulation

generates a bias of about 1σ high in r, or δr ∼ 0.03, roughly

the same level as the two-component dust simulation with

power-law model. This mismatch also results in a 1.5σ pref-

erence for r > 0 for the r = 0 model.

4.2Additional polarized components

Our model and simulations contain only synchrotron and

thermal dust emission components. Other emission com-

ponents are not expected to be significantly polarized (see

e.g., Fraisse et al. (2008), and Section 5 for further discus-

sion). However, both free-free and spinning dust emission

are detected in intensity, and they may be minimally po-

larized at the few-percent level. Macellari et al. (2011) find

an upper limit on spinning dust of 5% and an upper limit

on free-free polarization of < 3%. Dickinson et al. (2011);

L´ opez-Caraballo et al. (2011) reduce the upper limits on

spinning dust polarization to ∼ 1 − 2%.

Test D (free-free) simulates a Galactic foreground that

includes a 1% polarized free-free emission in addition to

the synchrotron and dust emission. Free-free Q and U emis-

Page 6

6C. Armitage-Caplan et al.

TestRecovered r

r = 0

Recovered r

r = 0.1

Bias (σ) Recovered τ

τ = 0.1

Bias (σ)

Baseline Tests

(1) Baseline (uniform βs)

(2) Baseline (non-uniform βs)

< 0.03†

< 0.03†

0.092 ± 0.033

0.092 ± 0.033

—

—

0.094 ± 0.005

0.094 ± 0.005

–

–

Incorrect Model

(A) Dust 2-component-a

(B) Dust 2-component-b

(C) Synchrotron curvature

0.02 ± 0.016

< 0.04†

0.03 ± 0.020

0.125 ± 0.037

0.096 ± 0.036

0.125 ± 0.039

+0.9

+0.2

+0.9

0.097 ± 0.005

0.094 ± 0.005

0.097 ± 0.005

+0.6

< +0.1

+0.6

Extra Components

(D) 1% free free

(E) 1% spinning dust

< 0.03†

< 0.04†

0.091 ± 0.032

0.094 ± 0.033

< −0.03

< +0.03

0.094 ± 0.005

0.094 ± 0.005

< +0.1

< −0.1

Incorrect Priors

(F) Strong βs prior mismatch

(G) Weak βs prior mismatch

(H) Strong βdprior mismatch

(I) Weak βdprior mismatch

0.168 ± 0.047

0.029 ± 0.021

0.133 ± 0.044

< 0.04†

0.197 ± 0.047

0.117 ± 0.039

0.224 ± 0.040

0.111 ± 0.034

+2.1

+0.6

+3.3

+0.6

0.104 ± 0.006

0.096 ± 0.005

0.107 ± 0.005

0.096 ± 0.005

+1.7

+0.4

+2.6

+0.4

Table 2. Marginalized estimates and corresponding biases for r for simulations with r = 0 and r = 0.1, and for τ for simulations with

τ = 0.1.†These values are the upper 95% confidence levels for r = 0.

Figure 2. Recovered distributions for the tensor-to-scalar ratio, r, for simulations containing polarized components that are neglected

in the models. The baseline results (test 1) are compared to those with a 1% polarized free-free component (test D), and a 1% polarized

spinning dust component (test E), for r = 0 (left), and r = 0.1 (right). At this polarization level, these components are sufficiently

sub-dominant that they do not bias the recovered parameters.

sion are given by Qff(ν) = 0.01Iff(ν)cos(2γ) and Uff(ν) =

0.01Iff(ν)sin(2γ), where Iff(ν) is a free-free intensity map at

frequency ν and γ are the thermal dust angles. This assumes

that the free-free polarization angles match the thermal dust

angles, which is unrealistic but should not significantly af-

fect conclusions. The free-free intensity is generated from

the PSM, which is consistent with WMAP data. The para-

metric model fits for power-law synchrotron and dust but

omits the free-free component.

Test E (spinning dust) includes a 1% polarized spinning

dust emission in addition to synchrotron and thermal dust.

Spinning dust Q and U emission are given by Qsd(ν) =

0.01Isd(ν)cos(2γ) and Usd(ν) = 0.01Isd(ν)sin(2γ), where

Isd(ν) is a spinning dust intensity map at frequency ν es-

timated from the PSM, and the angles γ are the same as

the thermal dust angles. The parametric model omits the

spinning dust component.

The resulting likelihoods are shown in Fig. 2, and pa-

rameters given in Table 2. We find that these small unmod-

elled components have a negligible effect on the estimated

parameters; the induced biases are within 0.04σ of the base-

line measurement in each case.

4.3Incorrect priors

In our baseline model estimation we imposed Gaussian pri-

ors of βs = −3±0.3 for the synchrotron spectral index, and

βd = 1.5 ± 0.5 for the thermal dust emissivity index. This

Page 7

7

Figure 3. Recovered distributions for r, if prior distributions are imposed on spectral indices that do not exactly match the simulation

inputs. The baseline (test 2) has a dust index βd= 1.5, and a synchrotron index with mean βs= −3 over the sky. Indices for Stokes Q

and U are fit in 3-degree pixels over the sky, with Gaussian priors βs= −3 ± 0.3 and βd= 1.5 ± 0.5. Offsetting the synchotron prior by

1σ to −2.5 ± 0.5 (test F), significantly biases the recovered r high (top panels, for r = 0, left, and r = 0.1, right). A ∼ 0.5σ offset (test

G) results in a smaller but non-negligible bias. Similar biases are found for offsets in the dust prior (bottom), for βd= 2.0 ±0.5 (test H)

and βd= 1.7 ± 0.5 (test I). These biases arise from over-parameterizing the model in low signal-to-noise regions.

allowed an estimate of the CMB in areas of the sky with a

low signal-to-noise ratio. Even with seven frequencies, if the

signal-to-noise ratio is low, the synchrotron and dust com-

ponent can become degenerate with the CMB unless priors

are imposed.

The priors are astrophysically motivated; synchrotron

emission is expected to have an index in the typical range

−3.5<

∼βs

∼−2.5, depending on the injection spectrum and

nature of diffusion and cooling (Rybicki & Lightman 1979;

Fraisse et al. 2008). Thermal dust emission is expected to

have emissivity index in the range 1<

Fraisse et al. 2008). The 2σ range of the prior therefore cap-

tures physically reasonable beheaviour. However, our sim-

ulations are perfectly matched to these priors: the simu-

lated synchrotron indices are either exactly −3.0 in Test

1, or have a mean over the sky of −3 in Test 2, and the

dust was simulated to have an index of 1.5. The real sky

will likely not match so well: we expect the emission to lie

in the prior range, but will not precisely match the mean.

Dickinson et al. (2009) conducted a similar study to quan-

tify the effect of priors using real data. Though they found

that the priors had a small impact on the CMB spectra, they

considered unpolarized emission, where foregrounds are rel-

atively smaller.

<

∼β<

∼2.5 (see e.g.,

We test the effects of these prior choices by fixing the

simulation spectral behavior, but choosing alternative Gaus-

sian priors with means that are offset from the simulation

inputs.

Test F (‘strong’ βs prior mismatch) examines a reason-

ably strong case of mismatch between the model prior and

simulation for synchrotron. Using Test 2 as the baseline, it

simulates synchrotron emission with values of βs that range

between −3.3 and −2.8, but the parametric model assumes

power-law synchrotron with a prior on βs of −2.5±0.5. Test

G (‘weak’ βs prior mismatch) assumes a prior of −2.8±0.5.

Test H (strong βd prior mismatch) has a mismatch between

the model prior and simulation for dust. Using the base-

line simulations, the dust emission has βd = 1.5 while the

parametric model assumes a prior on βd of 2.0 ± 0.5. Test I

(‘weak’ βd prior mismatch) assumes a prior of 1.7 ± 0.5.

The likelihoods for these cases are plotted in Fig. 3,

with parameters reported in Table 2. These mismatches

result in the most significant biases. For synchrotron, the

strong mismatch case results in a 3.5σ spurious detection of

r (0.17 ± 0.05), for a model with no tensor component. The

recovered value for r is also biased about 2σ high for the

r = 0.1 case, and the optical depth τ is high by almost 2σ.

The weak mismatch case, with prior −2.8±0.5, is biased by

Page 8

8 C. Armitage-Caplan et al.

∼ 0.6σ in r, with a spurious signal at the 1σ level. Similar

results are seen for the dust emission. For the strong mis-

match a signal is significantly detected at 3σ when r = 0,

and biased more than 3σ for r = 0.1 (returning 0.22±0.04).

The weak mismatch case suffers from a bias of 0.6σ in r,

and 0.4σ in τ.

5 DISCUSSION

We have found that modelling polarized Galactic fore-

grounds incorrectly can lead to significant biases in the re-

covered CMB signal. In this section we discuss the reasons

these biases are observed, and how they might be mitigated.

5.1Effect of priors

When marginalizing over foreground uncertainty using a pa-

rameterized method, components are distinguished by their

frequency dependence. This provides a way of separating the

black-body CMB signal from the foreground components. In

the low signal-to-noise regime a prior on this spectral behav-

ior breaks the degeneracy between CMB and foregrounds.

However, we find that choosing an incorrect, yet phys-

ically reasonable, prior for the frequency dependence can

have a significant impact on the estimated cosmological sig-

nal. With a simulated synchrotron spectral index between

−3.3 and −2.8, and a Gaussian prior of −2.5±0.5 on the in-

dex in each pixel, the tensor-to-scalar ratio is overestimated

by ∼ 3σ for an r = 0.1 model, or a spurious detection made

when r = 0. The effect is less extreme when the mean of the

Gaussian prior is closer to the input, −2.8, but a bias of 1σ

is still observed. In the limit of a low signal-to-noise ratio,

this can be understood as equivalent to setting the spec-

tral index to the wrong value over the whole sky. A prior

of βs = −2.5 ± 0.5 results in an index that is everywhere

∼ −2.5, instead of the mean simulated value βs ∼ −3. Simi-

larly, a prior on the dust index, or emissivity, of βd= 2.0±0.5

results in an index of ∼ 2.0 instead of the simulated 1.5.

This incorrect recovery in regions having a low signal-

to-noise ratio is demonstrated in the left panels of Fig. 4

for the synchrotron Q-Stokes component. Away from the

Galactic plane, the index is estimated to be roughly −2.5±

0.5. We also show in Fig. 5 the frequency dependence of

the components, rms averaged over the masked sky in 3.6◦

pixels, and compared to the CMB signal in both E-modes

and B-modes for r = 0.1. Assuming that the synchrotron

pivot is fixed at 30 GHz, an index that is too shallow by

βs ∼ 0.5 overestimates the synchrotron power by of order

0.1 µK in antenna temperature at the foreground minimum

of 100 GHz. This is significant compared to the r = 0.1 B-

mode signal, so a bias is expected. Similarly for dust, with

a pivot at 353 GHz, a dust emissivity index too steep by

βd ∼ 0.5 would underestimate the dust at 100 GHz by up

to ∼ 0.1 µK in antenna temperature; significant compared

to the r = 0.1 signal.

This specific case where the prior is systematically dif-

ferent to the input by up to 1σ everywhere on the sky is

a pessimistic scenario, but not implausible. To avoid the

risk of bias, one must therefore take care in how the fore-

ground model is parameterized. In the Bayesian framework,

our chosen model has too many free parameters, given the

low signal-to-noise ratio, so the result is being driven by the

prior. To mitigate this, there are several ways of increas-

ing the signal-to-noise ratio in the indices: including an-

cillary data from complementary experiments like WMAP

and C-BASS (King et al. 2010), assuming common temper-

ature and polarization spectral indices, using larger pixels to

define the indices, or defining spectral indices in harmonic

space to allow spatial coherence.

We consider two of these possible improvements. Each

three-degree pixel can have a distinct spectral index for I, Q,

and U. The first natural improvement is to fix the Q and U

spectral indices to be common in each pixel, βs

ically this is reasonable; the polarized signal comes from the

same region of the Galaxy for both Q and U-type, and can

be expected to have the same frequency dependence, con-

sistent with observations (Kogut et al. 2007; Dunkley et al.

2009a; Gold et al. 2009). We repeat Tests F and G with this

condition (Tests F2 and G2), and show the recovered in-

dex map in Fig. 4, with the likelihoods for r in Fig. 6. The

index map now has a higher signal-to-noise ratio, and the

bias on r reduced from more than 2σ to 1σ (for a prior of

βs = −2.5 ± 0.5). Fixing the temperature and polarization

indices to be common is less physically motivated so we do

not consider this here; depolarization effects could lead to

different regions of the Galaxy contributing to the integrated

polarization signal.

The signal-to-noise ratio can also be improved by

adding ancillary data that better traces the foregrounds.

Since the synchrotron signal dominates at lower frequencies,

additional data at the low frequency range will increase the

synchrotron signal-to-noise ratio. We repeat Test F again

(F3), adding simulated data from the WMAP 23 GHz K-

Band channel, and projected C-BASS data at 5 GHz, to the

simulated Planck data from 30-353 GHz. Figure 4 shows the

significantly improved estimate of the synchrotron index in

this case, which translates into a reduction in bias on r from

2σ to 1σ for a prior of βs = −2.5 ± 0.5. With the low fre-

quency data, the indices are better constrained by the data.

A final obvious way to reduce the model freedom is to al-

low less spatial variation in the indices. In the limit of no spa-

tial variation, this reduces to template cleaning (Page et al.

2007; Kogut et al. 2007; Efstathiou et al. 2009), with one

spectral index over the whole sky. However, a concern with

these methods is that they may not capture realistic spatial

variation. The optimal balance is likely in between, requir-

ing fewer than ∼3000 parameters to describe the spatially

varying frequency dependence. Such an approach has been

considered for polarization analysis in e.g., Dunkley et al.

(2009a), where 48 synchrotron spectral index parameters

were used for WMAP component separation. In making this

choice with real data, it will be important to test that results

do not depend on the prior placed on frequency dependence.

If so, the number of parameters should be reduced, or ex-

ternal data included where available.

Q= βs

U. Phys-

5.2Effect of over-simplified model

In Section 4.1 we found that over-simplifying the frequency

dependence of the two components can also lead to a bias in

recovered parameters. Modeling the synchrotron as a power

law everywhere on the sky, when it actually has a spectral

curvature of C = −0.3, results in a ∼ 0.03 bias high in r.

Page 9

9

Figure 4. Estimated synchrotron spectral index for the Q-Stokes parameter (showing mean, top, and uncertainty, bottom), for a

simulation with mean βs = −3 and prior −2.5 ± 0.5 (Test F). Allowing free Q and U spectral indices, and using just 30-353 GHz

data (left), the prior of −2.5 is returned in low signal-to-noise regions. If Q and U signals are assigned a common index (centre), the

signal-to-noise is increased. If low-frequency simulated data from WMAP (23 GHz) and C-BASS (5 GHZ) is added (right), the spectral

index map is recovered with high signal-to-noise.

Figure 5. Frequency scaling of the foreground components in the baseline simulation (test 1), rms averaged over the unmasked sky in

3.7◦pixels (ℓ ∼ 50 scales), and compared to the CMB E-mode signal for τ = 0.1 (solid blue curve) and B-mode signal for r = 0.1 (dashed

blue curve). If an incorrect spectral index in synchrotron or thermal dust is assumed (e.g., by imposing a prior: tests F, G, H, and I), or

a synchrotron curvature neglected (test C), the over- or under-subtraction of foregrounds at ∼ 100 −150 GHz is significant compared to

an r = 0.1 signal.

Page 10

10C. Armitage-Caplan et al.

Figure 6. Recovered distributions for input r = 0.1 for the base-

line simulation with mean synchrotron index input βs= −3, and

Gaussian priors −2.5 ± 0.5 or −2.8 ± 0.5 (test F and G). The

prior-dependent biases are reduced when the signal-to-noise is in-

creased by assigning Q and U common indices (top, F2), or adding

low-frequency data from WMAP or C-BASS (bottom, F3).

As in Sec 5.1, this can be understood as an overestimation

of synchrotron at the 100 GHz range by up to ∼ 0.05µK

in antenna temperature, illustrated in Fig. 5. Since some

steepening is expected from synchrotron cooling, a strategy

to prevent this bias would be to additionally marginalize

over a curvature parameter. If the estimated CMB power

does not change significantly with its inclusion, and the cur-

vature is consistent with zero, this would justify neglecting

the additional complexity.

While we have examined only the case for synchrotron

having a negative spectral curvature, there is some evidence

to suggest that the spectral curvature could be positive

(e.g., Dickinson et al. 2009; de Oliveira-Costa et al. 2008;

Kogut et al. 2007). This is not unexpected since multiple

spectral components can give a flattening of the effective

synchrotron index. With real data, a positive curvature as

large as 0.3 could be realistically considered.

At the high frequency end, thermal dust emission is

typically modelled as a modified black-body, characterized

by an emissivity and temperature, with I(ν) ∝ νβBν(T),

and similarly for Q and U. This corresponds to our ‘one-

component’ dust model. A more complicated model has a

sum of two or more components with different temperatures.

In Sec 4.1 we found that modelling a two-component dust

model as a one-component dust model has only a small effect

on the estimated CMB signal. This reflects that the sum of

two modified black-bodies, one sub-dominant, scales with

frequency similarly to a single black-body.

A larger bias was found for a modified black-body mod-

elled as a power law. In this case we find a 1σ shift in recov-

ered r, with the power-law model typically over-subtracting

dust. The effect is similar to neglecting synchrotron curva-

ture. While it is unlikely in practice that the dust would

be modelled as a pure power-law, it is possible that one

could make the wrong choice for the dust temperature. In

these tests we fixed the temperature to the input value that

was common over the whole sky, and varied just the emis-

sivity in each pixel. To check for a possible bias with real

data, one would ideally additionally fit for the dust tem-

perature. Another approach to determine the dust temper-

ature would be to use the temperature data, including the

higher-frequency unpolarized channels of Planck (545 and

857 GHz), and IRAS/DIRBE data up to ∼ 3000 GHz. The

dust temperature could then be assumed to be common for

the polarization data.

5.3Effect of neglected components

We find that neglecting sub-dominant polarized free-free

and spinning dust components has a negligible effect on

the results. This can be understood from Fig. 7. The sim-

ulations include a 1% polarized signal, with the rms sig-

nal of each component, averaged outside the Galactic mask,

shown to be sub-dominant to an r = 0.1 signal in the

range ν > 100 GHz. The true polarization of these com-

ponents is unknown, but is not expected to exceed this

level. Observations of the Ophiuchi and Perseus cloud limit

the polarization of spinning dust to be less than 2% at

20-30 GHz (Planck Collaboration XX 2011), and WMAP

observations limit it to less than 1% over the whole sky.

These levels are consistent with the spinning dust model

by Draine & Lazarian (1999). For this mask, spinning dust

polarization has a slightly larger effect on r than free-free

polarization. The spinning dust component is currently the

most uncertain, so will be worth re-visiting with real data.

There are fewer observational constraints on the po-

larization of free-free emission. However, it should be in-

trinsically unpolarized because the scattering directions are

random. Secondary polarization can be generated at the

edges of bright free-free features from Thomson scattering

(Rybicki & Lightman 1979; Keating et al. 1998), but lead-

ing to less than 1% polarization at high Galactic latitudes.

We have not therefore considered larger polarization levels.

We have also not considered more exotic components, such

as a polarized ‘Haze’ (Dobler & Finkbeiner 2007), or mag-

netic dust models (Draine & Lazarian 1999).

6 CONCLUSIONS

Extracting robust estimates for the tensor-to-scalar ratio

rely on modelling and subtracting polarized foregrounds.

Since the polarized CMB signal is many times smaller than

Page 11

11

Figure 7. Frequency scaling of the 1% polarized free-free and

spinning dust foregrounds included in Tests D and E, rms aver-

aged over the unmasked sky in 3.7◦pixels. At ν>

are over an order of magnitude lower than the CMB E-mode sig-

nal for τ = 0.1 (solid blue curve), and below the B-mode signal

for r = 0.1 (dashed blue curve).

∼100 GHz they

the foreground emission, the need to get this right is par-

ticularly acute. Many methods have been considered and

implemented for foreground removal, but given the lack of

data, the simulations are usually simple in form.

In this paper we have begun to quantify the impact on

estimates of r of incorrect foregound modelling. The tests

were aimed at a detection of a signal with r = 0.1, but the

goal of future missions is to reach r = 0.01 or lower, so we

also consider an r = 0 model. We conclude that neglect-

ing a non-power-law frequency dependence of foregrounds

may have a non-negligible effect on r; whereas neglecting a

small free-free or spinning dust component is likely not to.

We found that over-parameterizing the spectral indices had

significant consequences; in the limit of a low signal-to-noise

ratio the result can be highly prior-dependent.

We discussed methods of mitigating possible bias,

through model comparison as more complexity is added to

the foreground model, and through increasing the signal-to-

noise ratio on spectral parameters by reducing their num-

ber and using ancillary data. We did not cover all scenarios

of mismatch, but the approach of checking the goodness-

of-fit through model comparison, and checking for a de-

pendence of results on priors should be generally applica-

ble. We did not explore the effects of different masks al-

though this will be important to investigate with data (see

e.g., Dickinson et al. 2009). Data from Planck and ground-

based and balloon experiments will further elucidate the

nature of the polarized foregounds and allow their mod-

elling to be refined. For full-sky data from future ultra-

high sensitivity experiments such as CMBpol (Bock et al.

2009), COrE (The COrE Collaboration 2011), and LiteBird

(Hazumi et al. 2008), the effects studied here will be more

important as we push towards r = 10−2− 10−3levels.

We acknowledge the use of the Planck Sky Model, devel-

oped by the Component Separation Working Group (WG2)

of the Planck Collaboration. We thank Aurelien Fraisse,

Steven Gratton, and David Spergel for useful discussions.

This work was performed using the Darwin Supercomputer

of the University of Cambridge High Performance Comput-

ing Service (http://www.hpc.cam.ac.uk/), provided by Dell

Inc. using Strategic Research Infrastructure Funding from

the Higher Education Funding Council for England. JD ac-

knowledges support from ERC grant FPCMB-259505. CD

acknowledges an STFC Advanced Fellowship and ERC grant

under the FP7.

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