Developing a new template foregroundcleaning method for LiteBIRD
122K. Ichiki , T. Yamashita , H. Kanai ,34N. Katayama , E. Komatsu and the LiteBIRD phase A1 team
1

Kobayashi-Maskawa Institute, Nagoya University, Japan
2
Graduate School of Engineering, Yokohama National University, Japan
3
Kavli IPMU, Japan 4Max-Planck-Institute for Astrophysics, Germany

ABSTRACT
Detecting gravitational wave background generated during inflation through B-mode polarization of the cosmic microwave background (CMB) is oneof the main scientific goals of future CMB experiments. However, it has already become clear that synchrotron and thermal dust emissions fromour galaxy hinder the cosmological B-mode signal across the sky. Here, assuming only Gaussianity and statistical isotropy and homogeneity of theCMB map, we develop a new template cleaning method taking into account spatial variations of spectral indices of the foreground emissions, andapply it to various foreground models using Monte-Carlo simulation technique. We find that the method successfully remove the unwanted bias
in determining the tensor-to-scalar ratio r that was found in the simple template cleaning method presented in Katayama and Komatsu 2011. In
this poster, we present how small r we will be able to reach according to the frequency bands and noise specification proposed for the LiteBIRD
experiment.
To subtract it let us consider the linear combination

Introduction[Q, U ]ν0 (n̂) +Inflation is an accelerating expansion phase in the early universe
and offers a solution to the nagging problems in Big-Bang cosmology, such as the horizon problem, flatness problem, monopole
problem and so on. Many models of inflation have been proposed so far, and one observational consequence that most inflation models commonly predict is the existence of gravitational
wave background in the universe. Whilst a direct measurementof the gravitational wave background from inflation using laserinterferometers seems very difficult with the current technology,
an indirect detection through the B-mode polarization of the Cosmic Microwave Background (CMB) may be possible in the near
future, provided that the energy scale of inflation is high enough,as high as the grand unification scale of particle physics. The
amplitude of the inflationary gravitational waves is usually characterized by a parameter r, so called the “tensor-to-scalar ratio”,
which is defined by

αi[Q, U ]νi (n̂) =i=12X1+2Xi=1!

d1(νi)αi [Q, U ]νd (n̂) +i=1

αi CMB(n̂) + Nν0 (n̂) +ννdβ̄dd2(ν0) +, d2(ν) = gν2
X

αiNνi (n̂)

i=12X!

d2(νi)αi [Q, U ]νd (n̂)δβd(n̂) (4),i=1

whered1(ν) = gν!

ννdβ̄dνlogνd.

(5)

We can remove the dust emission if the coefficients αi are set tothe solution of the equations
dj (ν0) +2X

dj (νi)αi = 0 ,(6)i=1

for j = 1, 2. In Method (2), we consider a modified black bodyspectrum for dust emission instead of a power law one. In thiscase, the above equation is modified to, with x ≡ hν/kT ,ν[Q, U ]ν (n̂) ≈ CMB(n̂) + gν [Q, U ]d(n̂)νd+ Nν (n̂)

β¯d ν1xex δT (n̂)
1 + logδβd(n̂) + xex − 1νde − 1 T̄(7)

and dj (ν) are modified to,(1)

From recent observations of the CMB, r is constrained as r .
0.12. At this level, the B-mode polarization signal is dominatedby foreground emissions, namely synchrotron and thermal dust
emissions, from our galaxy. Therefore, we have to estimate theforegrounds and subtract them to reach the cosmological signal
and many methods have been proposed and tested (for a review,see [1]). In Katayama and Komatsu ([2]; hereafter KK11) we usedthe Planck Sky Model (PSM) to show that the residual foregrounddue to spatial variation of synchrotron spectral indices yields abias in the tensor-to-scalar ratio that is as small as δr = 6 ×
10−3. The spatial variation of dust spectral indices was, however,unknown at that time. Now that we have some information about
spatial variation of the dust indices from the Planck temperature
data, we can revisit the bias in the tensor-to-scalar ratio fromdust indices.
In this work we develop new template cleaning methods to mitigate the biases that will be introduced by spatial variation of thespectral indices of foreground emissions. We consider two methods: Method (1) – we assume that both synchrotron and dust
emissions show power law spectrum with spatially varying spectral index and Method (2) – we assume that synchrotron emission is modeled as a power-law component while dust emission
is described by the modified black body spectrum with spatiallyvarying spectral indices and dust temperature. Below we shalldescribe our template cleaning methods.

d1(ν) = gνννdβ̄dν1(ν)=d(ν)
log,d21ex − 1νd.

[Q, U ]ν (n̂) = CMB(n̂) + gνβd(n̂)[Q, U ]νd (n̂) + Nν (n̂) ,(2)where νd is some reference frequency for dust emission. ThePlanck data suggest that the spatial variation of βd(n̂) is small
compared with the mean index β̄d. Therefore, we can Taylor expand the spatial dependence of βd(n̂) and write the above equation as[Q, U ]ν (n̂) ≈ CMB(n̂) + gνννdβ̄d 

1 + lnννdδβd(n̂) [Q, U ]νd (n̂) + Nν (n̂) .

(3)

It is the term, δβd(n̂)[Q, U ]νd (n̂), that biases the estimate of thetensor-to-scalar ratio when using the internal template method
assuming a constant spectral index βd in KK11. Therefore, wecan reduce the bias by estimating δβd(n̂)[Q, U ]νd (n̂) and subtracting it.

0.1

0.1

0.01

0.01

0.001

Modified BB with noise
noiseless0.001

onecomp
PSM
PySM

0.001

0.01

0.1

rinputForeground maps for thesimulation
In this work we consider two foreground sky models. One is thePlanck Sky Model (PSM), in which the synchrotron and thermal
dust emissions are modeled as power laws in antenna temperature with spectral indices varying in sky directions. The other isthe one component dust model (onecomp) in which thermal dust
emission is modeled by the modified black body spectrum with
spectral indices and dust temperature varying in sky directions.

0.001

0.01

0.1

rinputFigure 1: (Left) Estimated tensor-to-scalar ratio r against
the input, using the method (1), for various foreground models. PySM is another foreground model which we do not discuss in this poster. (Right) Estimated tensor-to-scalar ratio
r for the onecomp foreground model, using the method (2).
In this case, the significant bias found in the left panel isreduced, but the boosted noise hinder the detection of thecosmological signal (blue). If we would switch off the noise,
we recover the input signal using the method (2) (magenta).

Instrumental noise
In the Monte-Carlo simulations we add instrumental noises expected in the LiteBIRD experiment to each observing band. The
noise levels are listed in Table 1.

Conclusion• We have developed new template cleaning methods
for the foreground removal problem in future CMB
B-mode experiments.

Let us begin with Method (1), in which we assume that the spectrum of dust emission obeys a power-law. Ignoring synchrotron
for the moment for simplicity, we model the Stokes parameters Qand U at a given frequency ν asν
νdmodified BB (method (2))(8)Methodology

deltamap methodrest

2 |h(k)+|2 + |h(k)x |2r≡.
h|R(k)|2 i+ d1(ν0) +2X

The result is shown in Figure 1, where the estimated tensor-toscalar ratio rest is shown against the true (input) one rinput. In
the left panel of fig. 1, we show results using the method (1) (deltamap method in the figure). We find that the method successfully
removes the foreground and gives unbiased estimations if the foreground emissions are simple power-laws (PSM; blue). However, if
the dust emission is modeled by a modified black body spectrum,
the method (1) fails to remove the foreground and introduces asignificant bias as large as δr ≈ 0.1 (onecomp; red).
We can do better to the onecomp model if we use the method(2), where we model the dust component by a modified black
body spectrum accordingly. However, in this case, we find that
instrumental noise starts to dominate at the foreground cleaned
band because template cleaning method ’cleans’ the part of theCMB as well as the foreground, leading to an unwanted noise
boost (right panel of Fig. 1, blue). To check this is caused by the
instrumental noise, we perform noiseless simulations for which weshow the result in magenta in the same panel.

Simulation and Result
We generate hundred mock stokes Q and U maps of the CMB ateach LiteBIRD band based on the theoretical power spectra (Cℓ),
where the B-mode amplitude CℓBB is characterized by the freeparameter r. The foreground maps and instrumental noise areadded on them at each band. We then use the method discussed
above and maximize the likelihood:ihexp − 12 [Q, U ]T C −1[Q, U ]p(9)L(r, β̄d, β̄s) ∝|C(r, β̄d, β̄s)|

where [Q, U ] is a vector of a template-cleaned map and C is acovariance of the same map.

• We have found that method (1) can remove the significant bias that was found in KK11 due to thespatial variation of spectral indices of the foregroundemissions, and r ≈ 0.001 is within reach, if the emissions are simple power-laws given by the Planck SkyModel.
• For the more complicated foreground model(onecomp), method (1) brings significant bias in estimating r as large as δr ≈ 0.1. Method (2) canremove the bias, but in this case instrumental noise
dominates in the estimation of r at the level ofδr ≈ 0.01.

References[1] K. Ichiki, Progress of Theoretical and Experimental Physics
2014(6), 06B109, 06B109 (Jun. 2014).
[2] N. Katayama and E. Komatsu, ApJ 737, 78, 78 (Aug. 2011),
1101.5210.