Coline MAUREL
GEN
2022-2023
Okayama University
1 Chome-1-1 Tsushimanaka, Kita Ward, Okayama, 700-8530, Japan

Estimation of pointing systematics offset for Cosmic
Microwave Background polarization analysis
du 07/05/2023 au 11/08/2023

Under the supervision of:
1. Internship Supervisor: Hirokazu ISHINO
email: scishino at s.okayama-u.ac.jp
2. School Supervisor: Nicolas CAPELLAN
email: nicolas.capellan@grenoble-inp.fr
Ecole nationalesupérieure de physique,électronique, matériaux
PhelmaBât. Grenoble INP - Minatec
3 Parvis Louis Néel - CS 50257
F-38016 Grenoble Cedex 01
Tél +33 (0)4 56 52 91 00
Fax +33 (0)4 56 52 91 03

Confidential: No

Acknowledgements
I would like to thank my supervisor Professor Hirokazu Ishino for giving me the opportunity towork in his laboratory. This internship has been very enriching for me. I discovered the world of
research in an international context in a field I had never studied before. Therefore, this experiencehas significantly improved my adaptability skills. Secondly, I am glad to had the chance to live inJapan because it is a wonderful country. I strongly appreciate traveling around Japan but what I
prefer the most is the Japanese people kindness.
I would also like to thank sincerely Yusuke Takase, a PhD student from the laboratory. Yusuke
helped me a lot during my internship. He shared to me his knowledge and his experience in cosmology.
Thanks to him, I could progress with confidence in my work. He is also behind a number of tools that
I used to get my results.
I am also thankful to Mr Bernard Chenevier and Ms Valery Parry for being the coordinators of myinternship. They allowed me to apply for this internship and then took care of getting me in touchwith Pr. Hirokazu Ishino. This experience would have never happened without them.
Finally, I would like to express my gratitude to all the students of the laboratory, especially Raffuzzi
Nicolò, Bortolami Marco, Nagano Yuya and Morinaga Mami, for their help. I have learned about
cosmology in a wonderful working environment thanks to them.

1

Contents
Glossary

2

List of figures

2

List of tables

2

Introduction

2

1 Introduction to Cosmology
1.1 Observation of the universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Cosmic Microwave Background (CMB) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Cosmic Inflation and LiteBIRD satellite . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 CMB anisotropies
8
2.1 Temperature and Polarization anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Representation of CMB maps and power spectrum: Healpix . . . . . . . . . . . . . . . . 10
3 LiteBIRD systematics
12
3.1 LiteBIRD systematics errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Introduction to pointing systematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.3 Reconstructed maps including pointing errors . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Foregrounds
17
4.1 Introduction to Foregrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
4.2 Preparation of Foregrounds and Component separation . . . . . . . . . . . . . . . . . . . 17
5 Estimation of ∆r
19
5.1 Resolution scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.1.1 Generation of Foregrounds and CMB maps . . . . . . . . . . . . . . . . . . . . . 19
5.1.2 Reconstruction of maps with pointing offsets . . . . . . . . . . . . . . . . . . . . 20
5.1.3 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1.4 Component separation and calculation of the residual CMB . . . . . . . . . . . . 20
5.1.5 Mask operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2.1 Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.2.2 ∆r as a function of the pointing systematic offset . . . . . . . . . . . . . . . . . . 23
5.2.3 Comparison to another resolution method . . . . . . . . . . . . . . . . . . . . . . 24
Conclusion

24

References

24Résumé24

Abstract24

2

Glossary
CMB: Cosmic Microwave Background
JAXA: Japan Aerospace Exploration Agency
NASA: National Aeronautics and Space Administration
ESA: European Space Agency
COBE: The Cosmic Background Explorer
WMAP: Wilikinson Microwave Anisotropy Probe
LiteBIRD: The Lite (Light) satellite for the study of B-mode polarization and Inflation from cosmic
background Radiation Detection
LFT: Low Frequency Telescope
MFT: Medium Frequency Telescope
HFT: High Frequency Telescope
TOD: Time Ordered DataPYTHON: programming language used for many applications such as cosmology
JULIA: programming language for computational science
FALCONS: Framework for Astrophysical Locus Computing ON the Sky
Psym3: Library used to create realistic maps of the sky at different frequency and different angularresolution
FGBuster: ForeGroundBuster library which provides easy-to-use functions to perform component
separation of sky maps into different frequency maps
FWHM: Full Width at Half Maximum3

List of Figures
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Horizon problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Flatness problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Angular Scale Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CMB map and Temperature power spectrum from Planck mission . . . . . . . . . . . .
Thomson scattering of quadrupole anisotropies create linear polarization . . . . . . . . .
Representation of E and B mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ring Order, visualization of the scheme for nside=2, nside=4, nside=8, nside=16 . . . .
CMB anistropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CMB power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Decomposition of δr between margins, statistics and systematics . . . . . . . . . . . . .
Smoothing operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Power spectrum for the smoothing operation . . . . . . . . . . . . . . . . . . . . . . . .
Low, Mid and High-Frequency Telescopes (left) and LFT detectors configuration (right)
LiteBIRD telescopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pointing systematic error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LiteBIRD scanning strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LiteBIRD scanning path for 5000s, 1 day, 1month and 1 year . . . . . . . . . . . . . . .
Reconstructed map program results with 1 arcmin offset (temperature) . . . . . . . . .
Distribution of Foregrounds across the frequency . . . . . . . . . . . . . . . . . . . . . .
Foregrounds at the frequency f = 78.0GHz . . . . . . . . . . . . . . . . . . . . . . . . .
Foregrounds and CMB at the frequency f = 78.0GHz . . . . . . . . . . . . . . . . . . .
Component separation result. Residual between the input CMB and CMB after thecomponent separation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Resolution scheme for the estimation of δr . . . . . . . . . . . . . . . . . . . . . . . . . .
Creation of CMB maps with Foregrounds (dust and synchrotron), smoothed according
to the 22 LiteBIRD beam size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Pointing systematics scan : 0 arcmin offset, 1 arcmin offset, no arcmin offset . . . . . . .
Temperature maps before and after the deconvolution operation . . . . . . . . . . . . .
Residual CMB = CM B0arcmin − CM B1arcmin . . . . . . . . . . . . . . . . . . . . . . .
Galactic plane mask with 0 (left) and 2 degrees apodization (right) . . . . . . . . . . . .
Residual CMB, Temperature and Polarization anistropies with the galactic plane mask .
Residual CMB power spectrum (with or without galactic plane mask) . . . . . . . . . .
Residual CMB power spectrum for several pointing offsets) . . . . . . . . . . . . . . . .

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List of Tables
1

LiteBIRD frequency bands and beam sizes . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4

Introduction
The aim of Cosmology is to observe and explain the universe we live in. Currently, the model thatbest describes the evolution of the universe is the “Big Bang”. It proposes that the universe is expanding from an initial hot and dense state. This theory helps explain several phenomena that we observe
such as the presence of the Cosmic Microwave Background (CMB), a microwave radiation that fills all
the space around us. In the late 1970s, physicists such as Alexei Alexandrovich Starobinsky introduced
the theory of Cosmic Inflation to complete the Big Bang model. It suggests an exponential expansionof the early universe. This theory could solve many unsolved problems such as the extreme uniformityof the universe, known as the horizon problem. In fact, the universe could have been homogenized
before this inflationary period.
However, the theory of Cosmic Inflation still needs to be proven, and the CMB provides a unique
way to do it. In fact, if we had to summarize the progress of research to prove cosmic inflation, we
would have said that we still have to calculate a parameter, noted r, hidden in this micrometric radiation. More precisely, to calculate “r”, we need to study the polarization anisotropies of the CMB.
That is why, the JAXA (Japan Aerospace Exploration Agency) created the LiteBIRD project whichaims to launch an international satellite around 2030 to study the polarization anisotropies of the CMB.
The satellite will have to meet many specifications to minimize the error made in the calculation of“r”. Some errors called “systematics” are often made by the instrument and should be controlled early
before launch. For example, pointing systematics errors occurs when we believe we are pointing in acertain point in space when in fact, we are slightly off-target. Pointing errors could totally bias theestimates on r. In this report, we will ask how much pointing error is acceptable to obtain a sufficientestimation on r, to validate or not the theory of cosmic inflation.
Therefore, the objective of my work is to define a specification for the acceptable range of pointing
error for the LiteBIRD instrument. This specification is essential to construct the satellite, without
which we will not be able to validate the cosmic inflation theory. This report will cover several cosmological parameters and it will present the results I obtained through simulations which I obtainedwith two different programming languages, Julia and Python.
This report contains five sections. The first one will present the essential cosmological insights tounderstand the content of my work. Then, I will focus on CMB anisotropies and more specifically, I
will depict the different types of polarization. Afterwards, I will define systematic errors and present
the LiteBIRD satellite. Subsequently, I will talk about foregrounds, one of the most important perturbations to study the CMB polarization and finally, I will present my resolution scheme and my resultsfor estimating the pointing systematic offset.

5

1

Introduction to Cosmology

1.1

Observation of the universe

Cosmology is a scientific discipline whose aims to study the Universe. Usually, it deals with large
objects so that the SI units are not really convenient [1]. For instance, instead of using meters, wewould prefer to use system solar distances such as astronomical unit AU (distance between Earthand Sun), interstellar distances like the parsec pc (distance between Earth and Proxima Centauri, thenearest star of the Sun) or intergalactic distances as Megaparsec M pc (distance between us and M31,
the Andromeda galaxy). We also use the mass of the Sun M0 as the unit for mass and the Gyr as theunit for time (The age of the Earth is 4.6 Gyr).
The study of the universe began with observations. One of the main observations we have madeis certainly the redshift of galaxies which is a very important clue to prove the expansion of theuniverse. The Big Bang model is an obvious phenomenon which comes the expansion of the universe: ”it expands from an initially highly dense and hot state to its current low density state” [1]. In the
middle of the 20th century, another cosmological model was considered : the Steady State Model. It
provides that there are ”no privileged locations in space or privileged moments in time” [1] . However,recent observations and especially the Cosmic Microwave Background (CMB) discovery, have led many
scientists to reject the Steady State Model.

1.2

Cosmic Microwave Background (CMB)

If the universe was as hot as the Big Bang model suggests, the baryonic matter (made up of
protons and neutrons) was ionized so that the numerous free electrons created by ionizations madethe universe opaque. Therefore, the primordial plasma emitted blackbody radiation. However, as theuniverse expanded, the universe cooled. And, when its temperature dropped to 3000 K, electrons andprotons could come together to create atoms. This phase is called the ”recombination”. As photons canpass through atoms, when we dropped to 3000 K, the universe was no longer opaque but transparent.
For many scientists at the time, if the Big Bang model was true, there had to be a relic of theradiation of the Blackbody at 3000 K. It was only in 1964, that two radio-astronomers, Arno Penzias
and Robert Wilson, pointed out the background radiation without knowing what they were doing. The
radiation they detected was close to 3K instead of 3000K because of the expansion of the universe: theexpansion tends to stretch the wavelengths. Hence, the current temperature of the CMB is equivalent
to a microwave radiation :T = 2, 25 ± 5 K(1)Numerous projects have been led since this major breakthrough. For instance, NASA launched
COBE (The Cosmic Background Explorer) in 1989 which yielded important insights about the radiation. For instance, it has shown that the CMB is well fitted by a blackbody spectrum, which is inline with its presumed origin. COBE observations also revealed CMB temperature anisotropies that
permit us to know, among other things, the composition and the age of the universe. And last but notleast, the CMB presents polarization anisotropies made by Compton scattering. The study of theselatter anisotropies could help us to understand what happened at the beginning of the universe -forinstance, validating of not theories such as Cosmic Inflation.

1.3

Cosmic Inflation and LiteBIRD satellite

The theory of Cosmic Inflation was introduced in the late 1970s. It permits to solve many issues
that the Big Bang theory alone cannot address, although it still requires to be demonstrated. The
Cosmic Inflation theory suggests that the Big Bang includes a period of exponential expansion of the
Universe. In fact, it attempted that the size of the universe increased by a factor of 1030 during 10−30
seconds. This inflation would have created primordial fluctuations, the origin of the CMB anisotropies.
Inflation is really convenient because it permits us to solve two main issues that cosmologists used todeal with.
The first one is called “the horizon problem”. Distant regions of space are so far away that,
according to the expansion of the universe, they have never been able to be in contact. Nevertheless,these regions have similar properties. For instance, the temperature of the CMB is approximately thesame in every direction. How to explain this homogeneity if these regions have never seen each other?
6

This is the horizon problem. Inflation could solve it because it suggests that the universe had time tohomogenize before it exploded.

Figure 1: Horizon problem
The second issue that inflation could solve is “the flatness problem”. In fact, every previous
calculation has shown that the Universe has zero curvature which means that it is flat. These results
imply that the density of the universe must have a really specific value. Since we do not believe incoincidence, there has to be a particular reason to explain why the universe is so flat [2]. Inflation canbe the one. Actually, if the universe exponentially expanded during a very short time, it would haveflattened the universe.

Figure 2: Flatness problem
Therefore, Cosmic Inflation offers many explanations of the evolution of the universe that cannot be
resolved by the Big Bang theory alone. However, it still has to be proved. That is why the LiteBIRD
project was initiated by JAXA (Japan Aerospace Exploration Agency). This project aims to launch
an international satellite around 2030, to search for evidence of Cosmic Inflation by studying CMB
polarization. In fact, inflation theory suggests that the primordial gravitational waves 1 created during
inflation would have imprinted a specific pattern in CMB polarization called “B-mode”. Besides, thestudy of this pattern could raise valuable evidence that supports the occurrence of Cosmic Inflation
during the first time of the universe.

1 Deformation waves of space-time that are usually very weak. To detect them, we need very massive objects undergoing a strong acceleration, such as the Inflation period72

CMB anisotropies

2.1

Temperature and Polarization anisotropies

The temperature of the CMB is extremely uniform across the sky with an average temperature of
2.7 Kelvin. However, it presents tiny fluctuations around a hundred microkelvins. To represent thesefluctuations over the sky we use a tool called the “power spectrum” which allows the decompositionof temperature fluctuations into different angular scales. To understand what is an angular scale, we
present the figure 3 extracted from [2]. In this figure, we have sinusoidal fluctuations at two different
angular scales : at 5 and 60 degrees.

Figure 3: Angular Scale Fluctuations
To obtain the power spectrum, we describe temperature data with spherical harmonics which aremathematical functions used to describe variations on a sphere. Spherical harmonics are characterizedby two parameters : m and l. For example, if we consider a random vector on the sphere, denoted asn, the variation of temperature at a specific point defined by n can be expressed as :X
∆T (n) =aTlm Ylm (n)(2)
l,mWhere Ylm are spherical harmonics and alm is a coefficient which characterizes the intensity of thefluctuation. The power spectrum is given by :
Cl =

1 X
alm a∗lm
2l + 1 m(3)

By applying this decomposition, we get access to large scale fluctuations (corresponding to low
value of l) and finer details (high value of l) of temperature fluctuations.
In 2009, the European Space Agency (ESA), with the collaboration of NASA, launched “Planck”,
a satellite conceived to measure CMB temperature anisotropies with much higher sensitivity thanits predecessors COBE and WMAP. Figure 4 gives Planck results, the most precise results we havecurrently. In particular, this power spectrum provided many information about the universe, such asits composition or its age.

Figure 4: CMB map and Temperature power spectrum from Planck mission

8

So far, the focus of CMB studies has been on temperature anisotropies. The next challenge is tomeasure CMB polarization which would also provide additional information about our universe. For
instance, the polarization of the CMB provides a way to study the gravitational waves produced during
Inflation. When light is polarized, the electric field oscillates in a particular direction, perpendicular
to the direction of propagation. Therefore, polarization provides numerous information about the electromagnetic wave we are studying, but light is not necessarily polarized. By chance, CMB is polarized
due to Thomson scattering (figure 5, an elastic scattering of electromagnetic waves by free-charge
particles. In the CMB case, photons were constantly scattered by electrons until the recombination
phase where photons were no longer trapped and were free to spread throughout the universe. So, theCMB polarization we see now is the last scattering of photons by electrons during the recombination.

Figure 5: Thomson scattering of quadrupole anisotropies create linear polarization
To describe the distribution of radiation polarization on the celestial sphere, we use the Stokes
parameters Q and U defined as:(
Q = Ex2 − Ey2(4)
U = Ea2 − Eb2
Where Ea is the 45-degree direction in the x-y plane and Eb the perpendicular direction. Of course,
Q and U depend on the way we choose the coordinate system but for the CMB analysis, we prefer touse variables that are independent of the rotation. Hence, we use two quantities 2 called “E-mode”
and “B-mode”3 as shown in figure 6.

Figure 6: Representation of E and B mode
The main contribution to the CMB polarization is density perturbations but these have only created
E-mode patterns whereas gravitational waves from inflation have both printed E-mode and B-mode
polarization. That is why, LiteBIRD aims to detect the B-mode polarization, which has never beenclearly detected before because it remains difficult. In fact, E-mode amplitude is larger than B-mode’s.
Secondly, there are additional sources of B modes such as gravitational lensing of E modes. The latter2 specific linear combinations of Q and U insensitive to coordinate rotations
3 More specifically, when discussing the power spectrum, we will refer to E-mode and B-mode. However, when working
with maps, we are going to use Q and U parameters9occurs when light is deflected by massive celestial bodies. Therefore, gravitational lensing of the
CMB can change the polarization pattern from E-mode to B-mode and also from B-mode to E-mode.
However, this latter conversion does not affect E-mode that much as B-mode amplitude is lower thanE-mode.

2.2

Representation of CMB maps and power spectrum: Healpix

To study CMB anisotropies, I used different libraries, such as Healpix, a library that allows the
manipulation of data on a spherical map. It can be used with both Python and Julia. Maps are
divided into several pieces called pixels. The number of pixels is directly linked to the resolution of
the maps. Most of the time, we first define the resolution nside, which is usually a power of 2, forinstance nside = 512. Each pixels have its own value, represented by a numpy array, and its own
position defined by the Healpix pixelization schemes. The scheme I used is called ”Ring”, it describes
rotations around the North-South axis from the top to the bottom of the sphere. For example, if weassign each pixels its position number, we can see on the map that the top is at 0 while the bottom
has the highest value according to the number of pixel -and therefore, the resolution (figure 7)

Figure 7: Ring Order, visualization of the scheme for nside=2, nside=4, nside=8, nside=16
Healpix provides many functions. For example, it permits to read maps in the “fits” file format,
write maps based on well-structured matrices and display maps with “hp.mollview” function. For
instance, below you can see three maps of CMB anisotropies. In figure 8, maps were read from a fits
file which contained CMB data - temperature and polarization Q and U.

Figure 8: CMB anistropies

10

As we can see, temperature anisotropies vary on the scale of hundreds of microkelvins while polarization varies on the range of tens of microkelvins. This data can come from previous CMB surveys,
such as the Planck mission, or from map making algorithms created to simulate sky maps with CMB
components. These latter algorithms typically use the Monte Carlo method.
After observing these maps, we use the hp.anafast function, use to convert a map into a power
spectrum. We usually plot all the components in the same figure to make the comparison easier. For
instance, figure 9 shows the power spectrum of the previous CMB maps.

Figure 9: CMB power spectrum
As you can see, the temperature component is higher than the polarization ones, especially for low
value of l : four orders of magnitude more than mode E and nine orders more than B-mode -indeed, we
saw that E-mode amplitude is higher than B mode. Moreover, the temperature component decreases
for larger values of l which means that temperature anisotropies are lower for large angular scales.
We can also see that the B-mode component is flat. This is because, in this case, we removed theprimordial B-mode. Therefore, what we see is only lensing B-mode. In fact, gravitational lensing
converts E-modes into B-modes, so B-mode can be seen as a leakage of E-mode. Since the lensing
process is random and that it occurs many times, the distribution of B-modes does not show any
specific pattern. Therefore, the power spectrum, which represents the variance, does not show any
particular fluctuations along the angular scale, so it is flat.
Power spectrum is a useful tool to explain CMB anisotropies. It is also very important for us
because ClBB is proportional to “r” the ratio between scalar and tensorial fluctuations and, basically,
if we calculate this ratio, we will be able to validate whether or not the Cosmic Inflation theory iscorrect. More precisely, it has been set that if we want LiteBIRD to validate cosmic inflation, thecalculation of r should be accurate to ten thousandths. Today, many works are conducted in order toprovide such precision.

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3

LiteBIRD systematics

3.1

LiteBIRD systematics errors

The error δr < 0.001 set for LiteBIRD has been divided into three equal parts : statistics, margins
and systematics. The statistics are easy to calculate because we know their numerical expressions.
Then, margins are also taken into account because of random errors that we cannot predict. Finally,systematic errors are caused by a deviation in data values which means that all the values are shiftedin the same direction with the same offset. For example, we must zero a voltmeter before doing any
measure otherwise the value will be biased. For LiteBIRD measurements, we count approximately aδr, including
hundred of systematic errors that need to be controlled with an error less than ∆r < 3∗100
factors such as beam, noise, and pointing (figure 10)

Figure 10: Decomposition of δr between margins, statistics and systematics
For example, beam systematic refers to the error caused by the limited telescope’s observation
area. To take into account telescopes beam effect, Healpix provides a smoothing function that consistsof applying a Gaussian convolution to the Healpix map which averages the values of neighboring
pixels. However, the deconvolution operation is not provided by Healpix. During my work, I used adeconvolution function created by Yuya Nagano, a PhD student from the laboratory. This function
takes into parameters the alm coefficients, the FWHM and the maximum value of l -Spherical Harmonic
parameter- lmax . It consists in dividing each group of alm coefficients by another coefficient bl that
depends on the l parameter and the component we are studying -T,Q or U. The function also set
to zero every coefficient over l = 191 because previous works have shown that the two part we areinteresting in are around l = 4 and l = 100 and, since nside = 32 only provides l values unitl lmax = 95,
we choose nside = 64 which means lmax = 191.
Figure 11 shows four CMB temperature maps : the first one is without smoothing, the second is thesame map after smoothing operation with a Gaussian with a Full Width at Half Maximum (FWHM)
of 5 degrees, the next one is the same map after deconvolution with the same FWHM as before andthe last one is the residual temperature CMB beteween the map not smoothed and the deconvolued one.
The residual map shows that the deconvolution operation is not ideal because otherwise the map
would be equal to zero. To evaluate the deconvolution function, we prefer to work with the power
spectrum. As shown in figure 12, the difference between the not smoothed map we start with (bluecurve) and the deconvolved map (green curve) is hardly perceptible until l=191. However, the residualshows a straight increase at l=191 since the deconvolution function has set the alm coefficients to zero
(green curve in the bottom). In consequence, the residual curve converges to the not smoothed map(blue curve) which continues to slowly decrease as shown in figure 9. As for the smoothed map (orange
curve), the shape is the same as the others for small values of l but it starts to decrease as a quadratic
function for higher values since, in power spectrum space, we have the relation Ĉl = Cl ∗ b2l . You canalso see that even though it is smoothed, the component does not completely fallen off, but it describes
large fluctuations. This is due to the pixelization effect. In fact, the pixelization of the spherical map
cannot represent perfectly the data, it often introduces uncertainties, especially for small angular scales.

12Figure 11: Smoothing operation

Figure 12: Power spectrum for the smoothing operation
Smoothing operations are really useful for LiteBIRD simulations as telescopes have beams. LiteBIRD satellite aims to detect CMB polarization with the best sensitivity we never had before thanks
to ”3 telescopes detecting across 15 frequency bands from 34 to 448 GHz with thousands of detectors” (LiteBIRD). In total we count 22 beam sizes (table 1 for LiteBIRD depending on the telescope (LFT,MFT,HFT), the frequency band (15 in total) and the size of the detector (red/yellow VS
green/blue ones in LFT as you can see in figure 13).
Therefore, beam effects are not negligible when studying satellite measurements. In the following
section, we will present another type of systematics called ”pointing systematics”. My internship has
focused especially on pointing systematics errors but we will continue to include beam effects in thecalculations because beam systematics are the one of the most challenging errors.

13Telescope
LFT
LFT
LFT
LFT
LFT
LFT
LFT/MFTLFT/MFTLFT/MFT
MFT
MFT/HFT
HFT
HFT
HFT
HFT

Center Frequency [Hz]
40
50
60
68
78
89
100
119
140
166
195
235
280
337
402

Beam size [arcmin]
70.5
58.5
51.1
(41.6,47.1)(36.9,43.8)(33.0,41.5)(30.2,37.8)(26.3,33.6)(23.7,30.8)
28.9(28.0,28.6)
24.7
22.5
20.9
17.9

Table 1: LiteBIRD frequency bands and beam sizes

Figure 13: Low, Mid and High-Frequency Telescopes (left) and LFT detectors configuration (right)

Figure 14: LiteBIRD telescopes

3.2

Introduction to pointing systematics

Pointing systematics are errors that occur when we think we are pointing at a certain point inspace when in fact we are slightly off target. These errors are automatically made by the satellite but
we can still work to minimize them to a certain threshold. As you can see in figure 15, the pointing

14

should be at the origin but we are pointing at the red arrowhead. The ρ parameter gives the distance
between the two points and χ is the angle between the offset and the scan direction.

Figure 15: Pointing systematic error
LiteBIRD will contain a star tracker to control this error. Before launching, the angle between thestar tracker and the telescope pointing direction, let’s say θ0 , will be adjusted to a highly precise value.
But, this value is certainly going to change once in space. If we not θ̂ the new angle, we obtain (5).
θ̂ = θ0 + θb + σθ

(5)Where θb is the bias error and σθ is the statistical error. The statistical error σb can be eliminate
if we repeat the measurements many times (6) :< θ̂ − θ0 >= θb(6)

The biased error θb is also called the ”pointing offset”. It can be calculated with CMB previous
measurements and pointing sources, but the way it will be calculated for LiteBIRD is still under investigation. So, we may wonder how much pointing offset θb is allowed to keep δr < 0.001?. To
answer at this question, I had to calculate ∆r as a function of the pointing offset in order to establish
a suitable threshold for the pointing error. To do so, I used a program which permits to calculate
reconstructed maps which take into account the pointing errors based on satellite measurements andthe scanning strategy of LiteBIRD.

3.3

Reconstructed maps including pointing errors

In order to reconstruct CMB maps including the pointing systematics errors, I used the FALCONS
library created by Yusuke Tasake, a PhD student from the laboratory where I worked. FALCONS
provides the calculation of pointing TOD (Time Ordering Data) of LiteBIRD knowing the scanning
strategy. Besides, the library provides a ScanningStrategy structure, already filled for LiteBIRD.
LiteBIRD will be launched at Lagrange Point 2 (”L2 point”), a point of equilibrium between the
Sun and the Earth for low-mass objects thanks to the gravitational influence between the two massive
bodies. From this point, LiteBIRD will describe two rotations (figure 16). The first rotation is aroundthe green axis called the “spin axis” with an angle of β = 30◦ 0.1rpm and the second rotation is theone of the spin axis around the Sun-Earth direction with an angle of α = 65◦ 90min.
Therefore, the Scanning Strategy structure of Falcons requires α and β values but also, the rotation
speeds, the sampling rate, the scan duration, the starting point etc. Falcons is already set for the LiteBIRD instrument but of course we can change the values if we need to. For instance, scanning durationof LiteBIRD is three years, but figure 17 shows LiteBIRD’s scan path during 5000s, 1 day, 1 month and1 year. We used to call these maps ”hitmaps” because they show how often we hit the pixels as we scan.
Once we know the scanning strategy, we are able to create a reconstructed map including thepointing errors, based on satellite measurements. For instance, in figure 18, you can see that on theleft we have a temperature map without any offset based on previous measurements and on the right
we displayed the reconstructed map with 1 arcmin pointing offset.
15

Figure 16: LiteBIRD scanning strategy

Figure 17: LiteBIRD scanning path for 5000s, 1 day, 1month and 1 year

Figure 18: Reconstructed map program results with 1 arcmin offset (temperature)
To conclude, there are many errors that must be controlled in order to determine the ratio r. In
this work, we are going to take into account pointing systematics and LiteBIRD beam sizes. We will
also consider the additional radiation called ”foregrounds” to provide a more accurate CMB analysis.
164Foregrounds

4.1

Introduction to Foregrounds

In addition to the CMB, LiteBIRD satellite observes many other types of radiation from celestial
objects. These are called foreground radiation. One of these is called “Synchrotron” radiation. Thislatter is caused by the spiral motion of relativistic electrons in the galactic magnetic field. It is a
non-thermal radiation but it has a strong linear polarization so it is very important to us. The otherforeground we are interested in is dust radiation. In fact, dust absorbs visible and ultraviolet lightemitted by stars and re-emits it in the radio and infrared regions. It contains thermal and linear
polarization. Figure 19 is extracted from [11], it describes the distribution of these latter foregrounds
across the frequencies.

Figure 19: Distribution of Foregrounds across the frequency
Since we want to extract only the CMB polarization from the observed data, we need to remove
these foregrounds in an optimal way. In the following, we will focus especially on synchrotron anddust, because they are the main components to remove for CMB analysis, although they are notthe only ones. The operation to isolate the signal is called “component separation”. It aims, as thename suggests, to separate the different components of the sky -for instance, synchrotron, dust and
CMB components. The component separation is based on the distribution of the radiations acrossthe frequency which we got from previous mission like Planck. We can introduce the S/N ratio : theratio of signal power to noise power (units in decibels [dB]). In this case, the signal refers to the CMB
radiation while the noise refers to the dust and the synchrotron.
Foreground radiation has a particularly strong signal in the region of the galactic plane in the sky.
The signal-to-noise ratio, S/N, is poor which can introduce large errors due to the uncertain nature of
the radiation model. Therefore, a process called “mask” is used to reduce the component separationof the CMB in the galactic plane. We selectively mask regions in the galactic plane where foregrounds
dominate in order to minimize their influence on the component separation. This allows us to focus
more on the regions of the sky where the CMB signal is more dominant.

4.2

Preparation of Foregrounds and Component separation

FGBuster is the package we use to generate foregrounds and apply the component separation. It
needs information about the instrument we are using. The following instruction assigns many details
about LiteBIRD to “instrument” such as instrument set up, frequency bands and beam size.
instrument=get_instrument("LiteBIRD_IMOvl_full")Then, we use FGBuster to get access to the data, whether it be from observation or simulation.
The following instruction assigns to freq-maps a matrix which containing the dust and synchrotron
map data -where d0s0 refers is a predefined sky map with dust and synchrotron:17freq-maps= get_observation(instrument,"d0s0",nside=512, unit="K_CMB")
The dimensions of the resulting matrix are 22 x 3 x 3145728. The value “22” corresponds to thenumber of maps with different beam sizes. Within these 22 maps, there are 15 different frequency
bands. The value “3” refers to the three components of the map -temperature and polarization Q and
U- and “3145728” is the data along the pixels according to nside = 512. For example, in figure 20 you
can see that temperature and polarization maps for the 6th component which means the 6th beam
size and the 5th frequency band - 36.9 arcmin beam size and 78.0 GHz frequency band.

Figure 20: Foregrounds at the frequency f = 78.0GHz
Let’s add the CMB component given by the predefined CMB data, the results are given figure 21.
We can see that the presence of foregrounds is more important for polarization than for temperature
which makes the LiteBIRD mission more complicated.

Figure 21: Foregrounds and CMB at the frequency f = 78.0GHz
If we remove the CMB from the previous map using the “basic comp sep” function and that westudy the residual between the CMB extracted and the input CMB we first uploaded, we obtain amap practically equal to zero µK with some errors of the order of 10−5 µK for polarization (figure 22).
Therefore, the results of the component separation are acceptable since this function has also its own
systematics.

Figure 22: Component separation result. Residual between the input CMB and CMB after the component separation

18

5

Estimation of ∆r

5.1

Resolution scheme

In this section, we are trying to determine a threshold value of the pointing systematic offset inδrorder to provide a precision of δr = 0.001 which means approximately ∆r = 3∗100= 3.3 ∗ 10−6 forpointing systematic. Therefore, we need to establish a meticulous method in order to take into accountboth the characteristics of each LiteBIRD telescope and the foregrounds. Below, we present the scheme
for a pointing systematic error of 1 arcmin , but further down in this report we will present the results
for many other pointing offsets.
[F oregrounds + CM B]smoothed (22 maps)

1 arcmin pointing offset scan

0 arcmin pointing offset scan

Deconvolution operation

Deconvolution operation

Component separation

Component separation

Residual CMB

Mask operation

Calculation of ClBB

Estimation of δr

Figure 23: Resolution scheme for the estimation of δr

5.1.1

Generation of Foregrounds and CMB maps

First, we create foreground maps using the FGBuster package. In this case, we will only generate
dust and synchrotron because they have the strongest polarization component. We obtain 22 maps,
with a resolution nside = 512, that need to be smoothed according to the LiteBIRD beam sizes. So,
next step is to smooth the 22 maps using the Table 13 while adding the CMB component previously uploaded. Of course, we also smoothed the CMB component with the same beam size as the foregrounds.
Figure 24 shows the T,Q and U maps of foregrounds and CMB smoothed with a beam size equal
to 70.5 arcmin before pointing systematics. We note Map 1 because these map corresponds to the ones
with the first beam size in Table 13 with the lowest frequency band. As we can see, foregrounds polarization varies on similar scales as CMB ones, roughly on the order of tens of micro kelvins. However,
the temperature of foregrounds varies between micro kelvins and thousands of micro kelvins, which is
why the temperature scale is larger than before in figure 24.

19

Figure 24: Creation of CMB maps with Foregrounds (dust and synchrotron), smoothed according to
the 22 LiteBIRD beam size
5.1.2

Reconstruction of maps with pointing offsets

The next step is to generate the reconstructed maps including the pointing offset. This step hastaken a long time to run because we choose a sampling rate of 19 for each map, which means that we
made 19 measurements within 1 second during 3 years. We present here the results for 1 arcmin offset.
Note that we are also doing the calculations for 0 arcmin even though there should be no difference
between the input and output maps since the offset is zero, but it allows us to take into account theerror made by the reconstructed map program itself. Figure 25 shows one of the temperature maps
4
for 0 arcmin pointing offset, 1 arcmin and no offset at all and as you can see the difference is verysmall : on the order of micro kelvin.

Figure 25: Pointing systematics scan : 0 arcmin offset, 1 arcmin offset, no arcmin offset

5.1.3

Deconvolution

Once we generated the 22 reconstructed maps for each pointing offset, we do the deconvolution
operation. In fact, we remove the smoothing effect while applying a deconvolution, using a gaussianfunction, to each of the 22 maps with their respective beam size coefficients. Figure 26 shows one of
the temperature maps before and after the deconvolution operation. As you can see, the map is lesssmoothed after the deconvolution, we can see more clearly the details of the map.
5.1.4

Component separation and calculation of the residual CMB

After the deconvolution operation, we move on to the component separation. The aim of this latter
operation is to extract the CMB component from the 22 maps while removing foreground parts. As forthe result, we obtain a 3 ∗ 3 ∗ 3145728 matrix where the first three dimensions refer to the component
we want to extract -CMB, dust or synchrotron-, the next three dimensions is for temperature and polarization -Q and U- and then the 3145728 dimensions contain the data. Figure 27 shows the residual
CMB between 1 and 0 arcmin offset. Note that the residual varies on the order of micro kelvins whichis quite acceptable considering the errors made along the calculations.
4 Note that we also did the same thing for polarization maps

20

Figure 26: Temperature maps before and after the deconvolution operation

Figure 27: Residual CMB = CM B0arcmin − CM B1arcmin
5.1.5

Mask operation

We can continue with these residual maps, but the results are unlikely to be satisfactory because
we have missed a phenomenon: foreground radiations are particularly strong in the galactic plane.
Therefore, to minimize this effect, we now use a mask, provided by the Planck Legacy Archive. The
mask is simply a map set to zero in the galactic plane region and equal to one in the rest of themap5 . The Planck Legacy Archive is very convenient because it permits to choose the percentage ofthe map we want to mask. Another particularity is that the frontier between the two regions can bea very distinct line or it can be smoothed. We call this procedure the ”apodization”. For instancefigure 28 shows galactic plane masks, with 40 percent of the map hidden, and two different degrees of
apodization. This procedure allows to avoid the ringing artifacts which occur when we transform asignal with a sharp transition into frequency space. In fact, the Fourier transform of a square function
is a sinc function. That is why we prefer to convert a signal with a Gaussian shape because the Fouriertransform remains a Gaussian function. Therefore, in this work we use a 2 degree apodization.

Figure 28: Galactic plane mask with 0 (left) and 2 degrees apodization (right)
Figure 29 shows the residual CMB after applying the mask. As you can see the scale has changed
but mostly because the galactic plane contained the largest values in absolute terms. So, when we
5 So, to apply the mask, we simply have to do a multiplication of the residual map and the mask

21

remove the galactic plane region, we remove the largest values that is why the scale has been reduced.

Figure 29: Residual CMB, Temperature and Polarization anistropies with the galactic plane mask
Now that our maps are ready, we can calculate the power spectrum and the ∆r value.

5.2

Outcomes

5.2.1

Power spectrum

Figure 30 shows the power spectrum of the residual CMB between 0 and 1 arcmin pointing offset.
These figures are obtained with the healpix function hp.anaf ast which can also permits to differentiate
temperature and polarization components, known as E mode and B mode.

Figure 30: Residual CMB power spectrum (with or without galactic plane mask)
First of all, we see that the mask has reduced the magnitude of the power spectrum. This is mainly
due to the reduction of the variation scale when we applied the mask. Secondly, you can see that thetemperature component is two orders of magnitude larger than the polarization component which isnormal because the power spectrum is a square function of the fluctuation intensity so the previous
one-order difference between temperature and polarization components has become two orders withlogarithmic function. Then, we can see that the curves of the power spectrum are flat until l ≈ 191.
Actually, when we are looking at the residual CMB, we look at the pointing systematic offset since
we have eliminated the CMB component as itself. The offset is constant so there is no reason to havea power spectrum fluctuation across the angular scales. We can also see that when we calculate thepower spectrum for different offsets. For instance, figure 31 shows the power spectrum of the residual
CMB, with the galactic plane mask, for several pointing offsets.
As you can see, all power spectrum are flat with different magnitudes. The larger the pointing
offset is, the larger is the magnitude of the power spectrum. In fact, if we increase the pointing offset,
the difference between the data and the zero arcmin pointing offset data also increases so the residual
increases. Now that we have the power spectrum, we can determine the ∆r as a function of the pointing offset ρ and set a suitable threshold for the poitning offset.

22

Figure 31: Residual CMB power spectrum for several pointing offsets)
5.2.2∆r as a function of the pointing systematic offset

To calculate the bias ∆r as a function of the pointing offset ρ, I used a predefined function basedon ”the Formalism of the Errors” in LiteBIRD collaboration (equations detailled in Appendix 1). Thisfunction took into parameter the B-mode component of the power spectrum and returns the ∆r value
as well as the likelihood function whose maximum is ∆r.
(Results not given here)
The resulting function has a square behaviour. Let’s calculate the Taylor development on a sphereof the polarization between the desired and the truly pointing point (7). This decomposition is possible
because the two points are really close in space: let’s assume that the desired point is indicated by
the vector n̂ and that the real pointing point is identified by the vector n̂ + ρ̂. where ρ̂ is the pointing
offset.
∂P
1 ∂P
1
∂P
P (n̂ + ρ̂) ≈ P (n̂) +
ρ+
ρθ +
ρϕ
(7)
∂r n̂
r ∂θ n̂r sin θ ∂ϕ n̂
By dimensional analysis, equation (7) gives that the pointing systematic offset is proportional topolarization units (8).
ρ∝P

(8)

Moreoever, earlier in this report, we assume that ∆r is proportional to the B-mode power spectrum
ClBB which units is Kevlin square (9).
∆r ∝ ClBB [µK 2 ] ∝ P 2 [µK 2 ](9)2

Therefore, if we apply a square operation on equation (8), ∆r should be proportional to ρ (10).
23∆r ∝ ρ2

(10)

So, it seems that the shape of the curve is quite acceptable even though it is not perfectly square.
Nevertheless, this representation is not optimal because it seems difficult to estimate precisely a suitable threshold for ∆r. We prefer using a logarithmic scale for ∆r as shown in the figure. This figure
also shown the threshold ∆r = 3.3 ∗ 10−6 that must not be exceed in order to validate the Cosmic
Inflation theory.
(Results not given here)
According to these results, the needed calibration accuracy should be less than (Not given here).
This value seems reasonable. However, it is quite small, so it is difficult to implement. We rather
expect a value of the order of several arcmin. In the following section we will compare our resolution
method with another one, really close, made by a former student of the laboratory.
5.2.3

Comparison to another resolution method

In this section, we compare our results with those we obtained by slightly changing the resolution
method. In fact, I used a study of a former student of the laboratory. His method consists of applying
an undergrading operation to the maps before the deconvolution and the component separation withthe healpix command :MAP_under_graded=hp.ud_grade(MAP,nside_new)
In this comparison we under grade the map so that the resolution is no more nside = 512 but is
now nside new = 64. We show ∆r as a function of ρ for the both method, with or without under
grading. We also added the limit ∆r < 3.3 ∗ 10−6 that must not be exceed.
(Results not given here)
You can see that with the points we calculated, the estimation of the threshold is the same: (Notgiven here). However, we can see that this slightly change resulted in to two very distinct curves.
Furthermore, the power spectrum are not the same. For example, in the figure, we display the difference
between the power spectrum for 1,2 and 3 arcmin pointing offsets.
You can see that, the more we increase the pointing offset, the more we see a difference for low
values of l: the undergrading curve shows a bump while no undergrading curve does not. We believe
this effect is due to the fact that, even though the resolution is better without undergrading, the statistics are better with undergrading operation because it averages the data in the same neighborhood of
pixels. That is why we think that we saw the bump in one case and not in the other.
So, the threshold we set is really important for the design of LiteBIRD because it allows to respect
the error we set for the estimation of the ratio r. If we do not exceed the threshold ∆r = 3.3 ∗ 10−6 ,we will be able to validate or not the Cosmic Inflation theory thanks to the LiteBIRD measurements.
However, there are still a lot of additional parameters we must take into account in order to estimate
an accurate value for ρ. For example, this comparison has shown the importance of statistical effects.
We could also have calculated more points around ∆r = 3.3 ∗ 10−6 . Another parameter we could havethought about is to the direction of the offset. In fact, during this work, we shifted the pointing offset
in the same direction for each of the 22 maps but we could have randomly changed the direction of
the offset by changing the value of χ (see figure 15) in the LiteBIRD scanning strategy.
Nevertheless, we can admit that the threshold (Not given here) is an acceptable first estimationof the pointing offset ρ for the purpose of the LiteBIRD project. These results could be useful for
eventual comparisons to future estimations of ∆r using tools other than Falcons or FGBuster, such as
litebird sim (see litebird sim github) which is currently used by many researchers working on the
LiteBIRD project.

24

Conclusion
My work during this internship was to estimate a threshold for the pointing offset ρ of the LiteBIRD instrument to be able to validate or not the theory of cosmic inflation. First, I learned how
to use the common tools to manipulate the CMB data in the sky. Then, I familiarized myself withthe pointing systematic errors by practicing with the program that allows to reconstruct a map withpointing errors. After that, I started my resolution of the pointing error threshold, taking into accountthe instrument beams and the foregrounds that need to be removed from the data. This resolution
led me to the conclusion that we need a pointing offset of (Not given here) to meet the specification∆r = 3.3 ∗ 10−6 , needed to validate the cosmic inflation theory. This value is small, but it does notseem too unreasonable either since we are looking for an unprecedented precision to explain the very
first moments of the universe. However, some technical improvements can be made. For example, afuture approach could be to change the direction of the pointing offset between the different telescopes.
Finally, this internship enabled me to make a good progress in computer simulation. It also permits
me to learn how to fit into a team and gain in autonomy. But above all, it was a great experience towork in an international environment, whether it was to learn how to communicate in English or to
get to know a new culture and a new way to live.

25

References[1] ”Introduction to Cosmology” by Barbara Ryden, 2006
http://carina.fcaglp.unlp.edu.ar/extragalactica/Bibliografia/Ryden IntroCosmo.pdf[2] ”The Cosmic Microwave”, Science astonishing, 2017https://www.youtube.com/watch?v=8WkBP2RU5nw&list=RDCMUCaNlbnghtwlsGFKzAFThqA&start radio=1&rv=8WkBP2RU5nw&t=6[3] ”Modern Cosmology” Second edition, Scott Dodelson and Fabian Schmidt, 2020[4] ”Probing Cosmic Inflation with the LiteBIRD Cosmic Microwave Background Polarization Survey”, Progress of Theoretical and Experimental Physics, Volume 2023, Issue 4, April 2023,
042F01, Published on November 21th 2022
https://academic.oup.com/ptep/article/2023/4/042F01/6835420[5] ”Lensing B-modes in the Cosmic Microwave Background polarization”, 2013https://astrobites.org/2013/07/24/lensing-b-modes-in-the-cosmic-microwave-backgroundpolarization/[6] ”From Anisotropies to Polarization”, waynehu
http://background.uchicago.edu/ whu/polar/webversion/node3.html[7] ”CMB Polarization”, 2016
https://lweb.cfa.harvard.edu/ cbischoff/cmb/[8] ”LiteBIRD simulation pipeline,Scanning Strategy”
https://litebird-sim.readthedocs.io/en/lightducc/scanning.html[9] ”Techniques for Nuclear and Particle Physics Experiments”, Second Revised Edition, SpringerVerlag, 1993[10] ”Planck Legacy Archive”
https://pla.esac.esa.int/#home[11] ”LAMBDA (Legacy Archive for Microwave Background Data Analysis) data products, foregrounds”
https://lambda.gsfc.nasa.gov/product/foreground/[12] ”litebird sim github”, Simulation tools for LiteBIRD
https://github.com/litebird/litebird sim[13] ”Empreinte carbone : combien de CO2 émet un Français chaque année ?”, Clément Fournier,2018
https://ferme.yeswiki.net/Empreinte/?PagePrincipale[14] ”Recueil de mesures unitaires pour calculer l’empreinte environnementale des activités de recherche”
https://youmatter.world/fr/emissions-co2-francais-empreinte-carbone/26

RésuméCe rapport présente le travail que j’ai effectué lors de mon stage dans le laboratoire de cosmologie àl’université d’Okayama. Il traite de l’inflation cosmique, une théorie qui s’insère dans celle du Big Banget selon laquelle l’univers se serait expandu exponentiellement pendant une période très courte, aux
tous premiers instants de l’univers. Cependant, cette théorie reste encore à être prouver. Pour cela,
il nous faut étudier les anisotropies de polarisation du fond diffus cosmologique (aussi appelé ”rayonnement fossile”), un rayonnement micrométrique qui remplit tout l’espace dans lequel nous vivons. En
effet, ce rayonnement aurait été émis par le plasma primordial et présente aujourd’hui des fluctuationsde température et de polarisation. Ces fluctuations auraient été créées par les ondes gravitationnelles
primordiales émises lors de l’inflation cosmique.
Dans ce rapport, j’explique comment les scientifiques procèdent pour étudier les anisotropies dufond diffus cosmologique. Puis, je présente le projet LiteBIRD qui vise à envoyer d’ici 2030 un satellite
pour mesurer les anisotropies de polarisation du rayonnement fossile. Mon travail s’est principalement
focalisé sur les erreurs de pointage du satellite. Plus particulièrement, j’ai tenté de déterminer l’erreurde pointage maximale autorisée pour pouvoir valider ou non la théorie de l’inflation cosmique. Ce
travail m’a demandé un effort bibliographique très important pour me familiariser au domaine dela cosmologie. Ainsi, par la suite, j’ai pu prendre en compte de nombreux paramètres tels que leserreurs de faisceaux et les rayonnements d’avant-plans qui sont primordiaux pour étudier le fonddiffus cosmologique. Finalement, avec ma méthode de résolution et les bibliothèques que j’ai utilisées,
j’obtiens un seuil de l’ordre de l’arc minute. Cette valeur semble convenable mais peut encore être
affinée, notamment en changeant la direction de l’offset selon les différents télescopes que contient
LiteBIRD. Le but étant in fine, de permettre la plus grande erreur de pointage tout en conservant lameilleure précision possible des mesures de polarisation afin de valider la théorie de l’inflation cosmique.

Abstract
This report presents the work I did during my internship at the Laboratory of Cosmology at
Okayama University. It deals with cosmic inflation, a theory that is part of the big bang, and according to which the universe expanded exponentially during a very short period of time in the first
moments of the universe. However, this theory still need to be proven. To do so, we need to study
the polarization anisotropies of the cosmic microwave background (CMB), a microwave radiation thatfills all the space in which we live. In fact, this radiation would have been emitted by the primordial
plasma and now shows temperature and polarization fluctuations caused by the gravitational waves
generated during cosmic inflation.
In this report, I explain how scientists proceed to study CMB anisotropies. Then I present the
LiteBIRD project, which aims to launch a satellite around 2030 to measure the polarization anisotropiesof the CMB. My work has mainly focused on the pointing errors of the satellite. More specifically, I tried
to determine the maximum pointing error allowed to validate or not the cosmic inflation theory. This
work required me to make a very important bibliographic effort to familiarize myself with cosmology.
This allowed me to take into account many parameters, such as beam errors and foregrounds, which
are essential for the study of the CMB. Finally, with my resolution method and the libraries I used,
I obtained a threshold of the order of arcminutes. This value seems reasonable, but can be improvedby changing the direction of the offset according to the different telescopes included in the LiteBIRD
instrument. Finally, the goal was to allow the larger value for the pointing error while keeping thebest precision for the polarization measurements in order to validate the cosmic inflation theory.

27

Appendix 1
This appendix is extracted from ”the Formalism of the Errors” in Probing Cosmic Inflation with the
LiteBIRD Cosmic Microwave Background Polarization Survey. We describe the link between the Bmode power spectrum ClBB and ∆r.
We define the likelihood function L(r) expressed in the multipole domain :logL(r) =lX
max

logPl (r)(11)

l=lminWhere lmin = 2 and lmax = 191. We also define :"#2l + 1 Ĉl
2l − 1log Pl (r) = −fsky+ log Cl −log Ĉl2Cl
2l + 1(12)

Where fsky is a sky map with fsky = 0.495. Ĉl is the measured B-mode, we represented it as thefollowing decomposition, assuming there are no primordial B modes (r=0) :Ĉl = Clsys + Cllens + Nl

(13)

Where Clsys is the estimated systematic effect, Cllens the lensing B-mode and Nl is the expected noise
power spectrum. The modeled power spectrum is:
Cl = rCltens + Cllens + Nl

(14)

Where Cltens is the tensor mode with r=1. The bias ∆r is defined as the value giving the maximumof the likelihood function:
dL
=0
(15)
dr r=∆r
In section 5.2.2, I use a predefined function based on these equations which allowed me to calculate
the ratio r from the B-mode power spectrum.

28