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Search for cosmic strings and non -Gaussianity as evidence of symmetry breaking and inflation through cosmic microwave background

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Search for C osm ic Strings and N on -G au ssian ity as E vidence o f Sym m etry
B reaking and Inflation th rou gh C osm ic M icrow ave B ackground
by
Eunhwa Jeong
B.S. (Yonsei University, Seoul, Korea) 1997
A dissertation subm itted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Physics
in the
GRADUATE DIVISION
of the
UNIVERSITY OF CALIFORNIA at BERKELEY
Committee in charge:
Professor George F. Smoot, Chair
Professor M artin W hite
Professor Eugene Chiang
Spring 2007
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UMI Number: 3275459
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Search for C osm ic Strings and N on -G au ssian ity as E vidence o f S ym m etry
B reaking and Inflation th rou gh C osm ic M icrowave B ackground
Copyright 2007
by
Eunhwa Jeong
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1
A b stract
Search for Cosmic Strings and Non-Gaussianity as Evidence of Symmetry Breaking
and Inflation through Cosmic Microwave Background
by
Eunhwa Jeong
Doctor of Philosophy in Physics
University of California, Berkeley
Professor George F. Smoot, Chair
We have developed techniques to probe cosmic strings and non-Gaussianity through
Cosmic Microwave Background Radiation.
In the cosmic string search effort, we
devised techniques and applied them to identify the impediments th a t blur the signals
from cosmic strings and analyze their effects quantitatively by running simulations.
We have introduced a new method to detect non-Gaussian feature in the cosmic
microwave background anisotropy and calibrated its performance through computer
simulations.
Professor George F. Smoot
Dissertation Committee Chair
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To my parents,
my wife Jiuk Jang and
my daughter Erin Jeong
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ii
C ontents
List o f Figures
v
List o f Tables
vii
I
M o tiv a tio n
1 In trod u ction
1.1 Cosmology Today Where Two Extremes M e e t ......................................
1.2 Observational Windows for GUT, Super String and In f la tio n ............
1.2.1 Gauge Theory and Its Remnants .................................................
1.2.2 Brane Inflation and Cosmic F /D S t r i n g s ...................................
1.2.3 Non-Gaussianity of The Primordial F lu c tu a tio n ......................
II
2
C osm ic S trings
1
2
2
5
5
8
11
13
T h eoretical B ackground o f C osm ic Strings
2.1 Grand Unified Theory of Gauge Theory in C osm ology..........................
2.1.1 Spontaneous Symmetry B r e a k in g ................................................
2.1.2 Finite Temperature F i e l d s .............................................................
2.1.3 Kibble M e c h a n is m ..........................................................................
2.1.4 Topological D e fe c ts ..........................................................................
2.1.5 Defects and Cosm ology....................................................................
14
14
16
17
19
20
22
2.2
C osm ic S t r i n g s .............................................................................................................
24
2.2.1 Gauge S tr in g s ....................................................................................
2.2.2 Brane Inflation Scenario and F /D -S trin g s...................................
2.2.3 Cosmic Strings Density in Expanding U niverse..........................
Physical Effects by Cosmic Strings .........................................................
2.3.1 Space-time Around A Cosmic S t r i n g ..........................................
24
27
29
30
30
2.3
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iii
3
4
2.3.2 Lensing Effect and Double Im a g e s.................................................
2.3.3 Kaiser-Stebbins E f f e c t .....................................................................
2.3.4 Gravitational Radiations from Cosmic S t r i n g s ...........................
2.4 Summary on Predictions And Observations of Cosmic Strings . . . .
2.4.1 Double Images by Cosmic S tr in g s .................................................
2.4.2 Constraints on Cosmic String Param eter by C M B ....................
34
36
38
40
40
40
P a t t e r n S e a rc h o f C osm ic S trin g s w ith C M B
3.1 Introduction ..................................................................................................
3.2 Effect of Cosmic Strings on C M B ..............................................................
3.2.1 Signal from a moving Cosmic String ...........................................
3.2.2 Probability distributions for relevant p a ra m e te rs........................
3.2.3 Statistical F lu c tu a tio n s.....................................................................
3.2.4 Variance: Quadratic Estim ator Test for String Contribution .
3.2.5 Statistical Test for Temperature Step Expected from Random
S trin g s .................................................................................................
3.3 Search for Temperature S t e p s .....................................................................
3.3.1 Evaluating String P a t t e r n ..............................................................
3.4 C onclusion........................................................................................................
43
43
46
46
46
49
53
V a lid ity T e st o f P a t t e r n S e a rc h
4.1 Introduction and M odeling..........................................................................
4.2 Application to W M A P ..................................................................................
4.3 C o n clu sio n s.....................................................................................................
65
65
71
74
III
5
Inflation and N o n -G a u ssia n ity
N o n -G a u ss ia n ity fro m In fla tio n
5.1 A Brief Description of I n f l a t i o n ..................................................................
5.1.1 Horizon and Hubble R a d i u s ...........................................................
5.1.2 E n t r o p y ...............................................................................................
5.1.3 Accelerated e x p a n s io n .....................................................................
5.1.4 Inflation by generic scalar f ie ld s .....................................................
5.1.5 e-fo ld in g ...............................................................................................
5.1.6 P e rtu rb a tio n s .....................................................................................
5.2 Non-Gaussianity as A Test of I n f la ti o n .....................................................
5.3 Calculations of N on-G aussianity..................................................................
5.3.1 Maximum Entropy Method ............................................................
5.3.2 Minkowski F u n c tio n a ls .....................................................................
B ib lio g ra p h y
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56
57
58
64
75
76
76
77
79
79
81
85
86
87
90
90
97
101
iv
A Friedm ann-R obertson-W alker U niverse
118
A .l Sign Conventions and Notations in GeneralR e la tiv ity .......................... 118
A.2 Friedmann-Robertson-Walker (FRW) M e tr ic ......................................... 119
A.3 Friedmann E q u a t i o n ................................................................................... 120
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V
List o f Figures
1.1
1.2
CMB Anisotropy m a p s ................................................................................
Angular Power S p e c tr a ................................................................................
5
6
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
Evolution of gauge coupling c o n sta n ts......................................................
Temperature dependent SSB p o t e n t i a l ...................................................
Domain wall fo rm a tio n ................................................................................
Cosmic string f o r m a tio n .............................................................................
Conic space-time around a straight cosmic string ................................
Double image formed by a cosmic s t r i n g ................................................
Kaiser-Stebbins e ffe c t...................................................................................
Kaiser-Stebbins effect simulations in CMB a n iso tro p y .........................
Hubble Space Telescope image of CSL-1 ................................................
16
18
20
24
35
36
37
38
41
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
Probability distributions for tem perature s te p s ......................................
Temperature distribution of full sky from W M A P ..................................
Temperature distribution of the full sky,Galactic plane removed . . .
The x 2 distribution for the uniformly-distributed-strings model . . . .
Definition of L .............................................................................................
The L-field sky map ...................................................................................
Relative a n g l e s .............................................................................................
Temperature discontinuity pattern I ...........................................................
Temperature discontinuity pattern I I ........................................................
Distribution of 6Topt ....................................................................................
48
49
51
52
55
55
59
60
61
63
4.1 Decomposition of a CMB anisotropy sky map p a t c h .............................
4.2 Evolution of errors of characteristic parameters ...................................
4.3 Distributions of tem perature steps in W MA P 3-year W -b a n d ..............
66
70
73
5.1 Entropy for non-Gaussian realization a n a l y t i c .......................................
5.2 Entropy dispersion for non-Gaussian simulations 1...................................
5.3 Entropy dispersion for non-Gaussian simulations II.................................
93
95
96
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5.4
5.5
5.6
Entropy dispersion for non-Gaussian simulations III...............................
Minkowski functionals for a Gaussian random field in two-dimensional
space .................................................................................................................
Minkowski functionals for WMAP super-horizon perturbation . . . .
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v ii
List o f Tables
1.1 Cosmological p a ra m e te rs ..............................................................................
7
2.1 Optical I m a g e s ...............................................................................................
2.2 Constraints by C M B .....................................................................................
40
42
3.1 Variance Limits on Cosmic Strings ...........................................................
3.2 D ata sets used in pattern s e a rc h ..................................................................
53
62
4.1 Simulation results at the critical values of A above which <tas are small
enough and do not decrease any more.........................................................
72
5.1 Limits on non-Gaussianity p a r a m e te r ........................................................
5.2 Required magnitude of non-Gaussianity p a ra m ete r.................................
5.3 Optimal bin s i z e s ............................................................................................
89
90
94
A .l Notations in General R e la tiv ity ..................................................................
119
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A ck n ow led gm en ts
First of all, I would like to thank my academic adviser George F. Smoot for his con­
stant effort to nurture me with rich scientific motivation and his exemplary attitude
as a scientist. He has kept reminding me to stay unbiased, and this is especially
im portant for me because my research is involved with close interplay between the
observational certainty and speculation. It was a great excitement and also a big fun
to see how much intellectual works and painstaking experiments by a sheer number
of people were required to establish the big bang theory as the base camp to explore
the ultim ate origin of the Universe. George was at the center of the achievement and
his endeavor induced me to deeply respect the science.
My Parents have been the never-exhausted source of confidence th a t has been
the momentum th a t revived me when I was tired. They have always prayed for me,
and have shown great love for me. They are great teachers and the models of my
life. A huge credit for my successful completion of the Ph.D course should go to my
parents. My wife Jiuk joined me in the middle of my graduate course, and she has
been with me all the time ever since. She has been of great help and become the best
friend of mine. Her skill on documentation was a valuable help for me to prepare
presentations.
I owe many relatives for their warm encouragement and support. Especially, my
three sisters and their husbands are so nice people th a t I love them very much. They
are willing to give advices and hands whenever I need.
I should also thank my
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parents-in-law, my uncles and aunts for their constant encouragement.
I would like to thank my office m ate Dr. Bruce Grossan for valuable discussion
and help in everyday life, and also thank many unnamed scientific and administrative
personnel in the INPA group at Lawrence Berkeley National Laboratory and at the
department of physics at the University of California, Berkeley. I want to express my
gratitude to all of my friends here in Berkeley who made my graduate life richer.
Eunhwa Jeong
Berkeley, California
May, 2007
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1
P art I
M otivation
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2
C hapter 1
Introduction
1.1
C osm ology Today W here Tw o E xtrem es M eet
Since 1980’s, we have experienced a dram atic progress in our knowledge on the
Universe in both theory and observation. The prime driving force of the great success
in modern cosmology came from precision measurements incorporated with ingenious
analysis and hi-tech engineering. The discovery of Cosmic Microwave Background
(CMB) Anisotropy provided not only observational confirmation of various predic­
tions of the Big Bang but also an invaluable tool to explore further the early universe
and its evolution [123]. The supernovae observations revealed a startling aspects of
the energy contents and the status of today’s Universe [94] [103] and as a result, they
revitalized the cosmological constant problem which turned out to be a very tan ta­
lizing puzzle. Deeply rooted as a precision science, cosmology has started playing
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3
a unique role of a natural laboratory for the elementary particle physics. Particle
physics has made brilliant success in both experimental and theoretical sides, usu­
ally theoretical works have wandered way ahead at the frontier and experiments have
tested and paved the correct path. The Standard Model of particle physics opened
up a new paradigm in 20th century and was mostly confirmed by experiments th at
it successfully describes the interactions of the building blocks of the Universe at
least up to the Electroweak scale (~ lOOGeV). Now, the theoretical particle physics
at the frontier culminating with Superstring Theory is attacking the Planck scale
(~ 1019GeV) which is considered as the ultim ate energy scale or the smallest length
scale of the Nature th a t laws of physics can possibly reach up to date. Since it is very
unlikely th a t one would be able to obtain such a high energy state terrestrially, scien­
tists turned to the early universe where it is conjectured to be in such an extremely
high energy state th a t the unified interactions might have existed at th at era. It is
very fortunate th a t the high energy physics does have some remarkable predictions
which can be potentially tested with cosmological observations. Among those predic­
tions yet to be resolved are the topological defects and non-Gaussianity of primordial
fluctuations. Much information about the history of universe was washed out over
time and obscured by adverse effects and contamination, however there are still rec­
ognizable images of the early universe on which we rely to trace back to earlier eras.
The earliest light available today is the Cosmic Microwave Background Radiation
th at started its journey toward us when the universe was just 380,000 years old. The
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4
primordial gravitational waves or cosmic neutrino background could provide informa­
tion on younger universe than CMB does in the future, currently we are not equipped
with a technology developed to catch such evasive signals. The tiny ripples contained
in CMB sky map provide a snapshot of the landscape of the early universe at 380,000
years after the big bang and enable us to probe under what condition the big bang
initiated. The ripples in CMB anisotropy are so weak (approximately one part in
100,000) th at we can treat them perturbatively, which means, in Fourier space, each
mode is decoupled and not disturbed by other modes in a good approximation. This
linearized analysis makes one possible to interpret the physical meanings of the fluc­
tuations of CMB very clearly. The observations of CMB anisotropy strongly support
the ACDM model (see figure 1.2 and table 1.1), which predicts th at the total energy
density of the universe today is nearly th a t of critical density to make the space­
time almost flat with QA ~ 0.7 (dark energy), Qcdm ~ 0.25 (cold dark m atter) and
Qb ~ 0.05 (baryon). The ACDM model was prom pted by cosmological observations
and raised profound questions on the nature of apparently invisible but dominant
part of the energy of the cosmos. The CMB observation also supports the inflation
with nearly scale invariant power spectrum at the super horizon scale.
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5
Figure 1.1: CMB anisotropy maps from COBE DMR and WM AP . Top: Anisotropy
sky map from COBE DMR (53GHz+90GHz) [146]. Anisotropies in CMB was first
discovered in 1992 by COBE DMR Mission team [123]. It shows the super-horizon
scale (~ 10°). Bottom: CMB Anisotropy sky map from W MA P [148]. Resolution is
greatly improved from COBE DMR providing details down to ~ 0.3° scale.
1.2
O bservational W indow s for G U T , Super String
and Inflation
1.2.1
G au ge T h eo ry and Its R em n an ts
Gauge theory and its spontaneous symmetry breaking (SSB) is the backbone of the
Standard Model of particle physics. It showed its predictions in excellent agreement
with what happened in the particle accelerators. Grand Unified Theory (GUT) is
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6
Angular scale
90®
6000
2*
0.5°
0,2*
WMAP
Acbar
Boom erang
5000
C8I
^
X 4000
\
1000
r
(§
L>
3000
2000
+
-
100
1000
100
500
1000
1500
400
ACBAR
B03
CB!
MAXIMA
VSA
600
Multlpole m o m e n t I
1000
2000
I
Figure 1.2: Angular power spectra from various CMB anisotropy observations. Left:
Angular power spectra covering both large and small scale perturbations [148]. The
solid curve represents the best fit curve of ACDM model. Right: Power spectra for
small scale perturbation with I > 300 [124]. Red line is the ACDM model best fit,
the orange and the light orange curves show the 68% and 95% confidence levels.
one of the unconfirmed predictions of the Standard Model, but it is very likely to be
true because of the great success in gauge interactions which was proved as a very
successful effective theory at electroweak scale or lower energy scale. According to
spontaneous symmetry breaking scenario, the Universe started with unified gauge
interactions i.e. higher symmetry and it was broken down to three different forms of
interactions as we observe today,
G
H
SU{3)c x SU(2) l x U{ l ) EM
(1.1)
where th e SU(2)L is a broken sym m etry a t to d a y ’s universe. G auge sym m etry break­
ing almost inevitably leaves topological defects behind. Possible types of topological
defects known are monopoles (O-dimension), cosmic strings (1-dimension), domain
walls (2-dimensions) and textures (3 dimensions). Topological defects except for tex-
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7
Cosmological Parameters for ACDM model
parameter
H 0(Hubble parameter)
^(reduced Hubble parameter)
n s (scalar spectral index)b
Dbfi2(baryon density)
Da (dark energy density)
Dm(m atter density)
<j8(m att. flue, on 8/i-1 Mpc)
to (age of the universe)
r (re-ionization optical depth)
WM AP 3-year
73.5 ± 3.2km /s/M pc
0.735 ± 0.032
0.951 ± 0.016
H 0 0 0 9 1 +0 00075
u .u z z ;o _ c _ o 00073
0.763 ± 0.034
0.237 ±0.034
0.742 ± 0.051
13.73l°;]bGyr
o .o s s tS
All combined8.
70.8t['okm /s/M pc
0.708+°;°^
0.938±S;8il
0.021931“
0.738 ± 0.016
0.262 ± 0.016
n 7 c i +0.032
U. 1 o r _ 0031
13.84 ± 0.14G yr
0 .U
070+°
U
/U _Q 027
028
aResults with combined data from the experiments WMAP , 2df, SDSS, ACBAR,
BOOMERanG, CBI, VSA, SN astier, SN gold, WL and BAO.
bAt k = 0.002/Mpc.
Table 1.1: Cosmological parameters for ACDM model computed using WMAP 3-year
data only (middle column) and various d ata sets combined (right column). Q refers
to the density relative to the critical density.
ture are concentrations of false vacuum configuration and they are produced when
symmetries are broken, true and false vacua coexist. At the points separated more
than particle horizon in the symmetry breaking era, field values including phase at
those points must be uncorrelated. As true vacuum configurations expand and fill
the space, false vacuum regions are squeezed between uncorrelated true vacua and
form topological defects, which is called Kibble mechanism [70]. Topological defects
make the field configuration avoid jumping discretely, which would lead to an infinite
gradient or equivalently infinite kinetic energy density of the field. A texture is not
a defect th a t contains higher vacuum energy density than surroundings but it is a
region of true vacuum with a continuously varying phase configuration as a function
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8
of position. Domain walls and monopoles can result in cosmological catastrophe if
sufficient diluting processes are not provided, for their energy density would come to
dominate the Universe otherwise. However, cosmic strings are known to evolve like
the radiation as the Universe expands which does not lead to disaster, because cosmic
strings reconnect and intercommute when they collide, so long strings are chopped to
form loops th a t can be readily evaporated by em itting gravitational waves, resulting
in faster dilution than collisionless m atter density in the expanding universe [38, 149].
Inflation provides an excellent explanation on the universe being not over-closed by
monopoles or domain walls. If the Universe underwent a phase transition in the in­
flationary big bang model and it produced the topological defects as the remnants,
monopoles and domain walls are explicitly required to be created before or during
inflation so they are sufficiently thinned out to be consistent with today’s observation
(nearly flat universe). However, cosmic strings are free from this constraint for their
non-over-closing evolution property. Thus, the production of cosmic strings at the
end of the inflation to avoid the devastating blow of inflation and thereby making
cosmic strings potentially available for cosmological observation remains as an open
possibility.
1.2.2
B ran e Inflation and C osm ic F / D S trings
Inflation naturally resolves several very im portant questions in the Big Bang model
such as large scale homogeneity, isotropy of CMB, flatness problem and entropy prob­
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9
lem. Many observations have shown remarkable consistency with the inflation sce­
nario.
Among a multitude of theoretical models of inflation, the brane inflation
scenario inspired by string theory and its extra dimensions was recently proposed [32]
and is currently taking great attention from both theoretical physicists and cosmologists. It proposes very attractive prediction on the production of cosmic strings whose
observability is positive [26, 41, 66, 96, 111]. According to brane inflation scenario,
the space we live is compared to a plane (called “brane”) in the bulk and inflation
is described as the collision of brane and anti-brane th at is connected to brane by
extra-dimensions. The brane/anti-brane configuration contains U( 1) x {/(l) sym­
metry initially and this symmetry disappears when the brane/anti-brane annihilates
(inflation ends) as one U( 1) gives D-strings and the other gives F-strings. Since the
cosmic F/D -strings are produced at the end of inflation, they do not suffer from the
dilution by inflation. No other cosmologically dangerous defects such as monopoles or
domain walls th a t can over-close the Universe are to be produced. The brane inflation
scenario is one of the few theoretical models th a t have potentially testable predictions
with cosmological observation d ata available currently or in the near future.
The existence of cosmic strings will be direct evidence for symmetry breaking
scenario or string theory which is full of mathematical beauties but struggling for
experimental supports. Cosmic strings are known to have some observable effects:
tem perature discontinuity in CMB [68], wakes of m atter, double images of a light
source and radiation of gravitational waves. Double image effect is one of the methods
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10
th a t can possibly locate a long cosmic string. There was a candidate of double image
by a cosmic string th a t eventually turned out to be two interacting galaxies [112, 113].
Though it was a failed case, it showed a good example of work how we approach
and test a suspected double image. Cosmic strings can move relativistically, form
cusps or kinks when they collide and reconnect. String cusps or kinks can radiate
gravitational waves (GW) as they move violently [24, 25]. There are various sources
of gravitational waves, showing different spectra. Gravitational waves can be detected
by observing the polarization of CMB or deformation of proper distance caused by
traveling gravitational waves. Laser interferometry gravitational wave detectors like
LIGO or LISA are trying to catch the signals from cosmic (super)strings. In this
dissertation, we focus on the direct search for cosmic strings of cosmological scale
using CMB anisotropy data. In chapter 3, we devise a technique th at searches long
patterns of discrete tem perature steps in a CMB anisotropy map. The pattern search
utilizes the Kaiser-Stebbins effect which is illustrated in detail in section 2.3.3. In
developing the pattern search technique, we had to estimate the effects of instrumental
noise, pixelization of data and other causes th a t can obscure the signals from cosmic
strings. We build a new algorithm to detect step-like structures in a CMB anisotropy
map in chapter 4. This algorithm explicitly shows how much a step-like signal from
a cosmic string would be blurred by finite pixelization of a CMB anisotropy d ata and
background fluctuations in a quantitative manner. We calibrate the detecting power
of the algorithm through computer simulations.
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11
1.2.3
N o n -G a u ssia n ity o f T h e P rim ord ial F lu ctu a tio n
Deviation from the Gaussian distribution of the primordial perturbation is a
generic feature of inflation models, and the non-Gaussianity is one of the most impor­
tant observational tests of inflation. In the inflationary big bang model, the quantum
fluctuation of inflaton (the scalar field(s) responsible for inflation) is the seed for
primeval inhomogeneity of m atter distribution. Then, the inhomogeneity of m atter
distribution is transferred to CMB photons at the last scattering. The CMB temper­
ature fluctuation is related to the curvature perturbations, <3?,by
~ Vt [$ (x ) + /<j>$2(x)]
(1.2)
where t]t is the radiation transfer function and it is a scale dependent parameter. The
relation (1.2) is a perturbation expansion up to the quadratic term. The curvature
perturbation $ is again generated by the inflation fluctuation <j), and the relation is
not necessarily linear [107],
$ [ 0 + /^ 0 2] .
(1.3)
We may go further to express the inflaton fluctuation in terms of more fundamental
field derived from quantum fluctuation in a similar way as (1.3), if the relation is nearly
linear, keeping up to the second order. After rescaling the fields and truncating the
higher order terms, the tem perature fluctuation of CMB photons can be expressed as
[76]
~ nr
[<Mx) + f NL (« !(x ) - (4>|(x)» + 0(4>|(x)]
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(1.4)
12
where $ l ( x ) is an auxiliary Gaussian random field and
/jv l
is the non-Gaussianity
param eter th a t accounts for the non-linear contributions from quadratic terms. Thus,
any nonzero value of f NL would be mean the deviation from the Gaussian distribution.
Among the representative techniques to probe non-Gaussianity are one-point tem­
perature distribution function fitting, Minkowski functionals, bispectrum (two-point
correlation function) and skewness (three-point correlation function) methods. All
the efforts so far found the CMB anisotropy data ( COBE DMR , MAXIMA, WMAP
) are consistent with Gaussian fluctuation and no evidence of non-Gaussianity up to
date. Instead, these works have set upper limits on the non-Gaussianity parameter
I n l i \ f n l | ^ O (102) at 95% CL with the most refined data available. It is esti­
mated th a t the P L A N C K data will allow one to probe the non-Gaussianity feature at
|f n l | < 5 at 95% [76] while viable theoretical models on inflation predict \ / n l \ ^ 1
[23]. In chapter 5, a new technique to detect non-Gaussianity is introduced. It utilizes
one of the unique feature of Gaussian distribution th a t a Gaussian distribution has
the maximum possible entropy among continuous probability distributions with fixed
variance. Current calibration of the new method shows th at it requires the minimum
detectable magnitude of
/jv l
as
|/ j v l |
~ 57 at 95% CL.
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13
P art II
C osm ic Strings
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14
C hapter 2
T heoretical Background o f C osm ic
Strings
2.1
Grand U nified T heory o f G auge T heory in C os­
m ology
The Standard Model of particle physics explains the three gauge interactions of
the nature, strong, weak and electromagnetic interactions, very successfully based
on the mathematical frame of quantum field theory imbued with gauge theory. At
today’s energy scale (~ 10~4eV), the gauge interactions are described as S U (3)c x
SU(2)l x U ( 1 ) y with three families of quarks and leptons (the electroweak symmetry
was broken at the energy scale ~ 102GeV, SCI (2)L x U ( l ) y —> U ( I ) e m )-
Each
family consists of [Q,uc,dc; L,e°\ states where Q = (u ,d ), L = {y,e) and uc, dc,
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15
ec are charge conjugate SU (2)L singlets. The transformations of the components
under the gauge groups are (3, 2,1 /3 ), (3,1, —4/3), (3,1,2/3), (1,2, —1) and (1,1,2)
respectively where hyper-charge Y is related to electric charge Q e m and isospin T:iL
by Q e m — Y /2 + T^l [33]. When this scheme is combined with the Big Bang theory,
it is very likely th a t the Universe was very hot in earlier times, possibly approaching
the Planck scale (~ 1019GeV), and the higher symmetries now broken remained
unbroken. Moreover, the Minimal Supersymmetric Standard Model (MSSM) predicts
that all three gauge coupling constants merge into one force at the energy scale around
1016GeV. Even though we don’t have direct evidence of GUT, it is now a common
belief that the Universe started with higher symmetries and they were broken down
to smaller symmetry groups such as SU (3)c x SU (2) l x U(1) y structure. The energy
scale at which the symmetries of GUT is expected to be restored is well beyond the
capability of current technology. However, the situation is not so hopeless because
when symmetries are spontaneously broken, production of certain types of topological
defects is almost inevitable. Topological defects are byproducts of the finiteness of
speed of interaction. Some types of topological defects must be suppressed or diluted
and some other types are allowed to survive till today to be consistent with the
constraints from current cosmological observations.
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16
60
60 - a '\
0
0
5
10
15
20
l og, „Q ( Ge V)
Figure 2.1: The evolution of three gauge coupling constants. Solid lines are extrap­
olations of non-SUSY GUT and dotted lines represent those of SUSY GUT. In the
SUSY GUT, three coupling constants merge at the order of 1016GeV [33] [95].
2.1.1
S p o n ta n eo u s S y m m etry B reaking
A simple model of complex scalar field <f> which is responsible for spontaneous
symmetry breaking such as a Higgs field has the following form of Lagrangian
£=
where D
- A(4>'* -
g x )2
(2 .1)
and F/w are defined by
iyi> = (a, -
)$
( 2 .2)
(2.3)
This system has gauge symmetry under the transformation
(2.4)
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17
Symmetry-breaking can occur in two ways either by coupling to a time-dependent
scalar field or by thermal transitions.
(1) Coupling to time-dependent scalar field:
X rolls slowly from negative values (while \ < 0, the potential has its minimum at
$ = 0 ) and when it has positive values, the potential has new minima at |<f>|2 = gx,
thus $ = 0 becomes false minimum.
(2) Thermal effects
When the scalar field is in contact with thermal bath, the vacuum is found by minimiz­
ing the free energy E —F S . Entropy is higher when symmetry is not broken because
of greater number of massless degree of freedom and thus the unbroken phase is fa­
vored in higher temperature. As the universe cools down, the system approaches the
zero-temperature potential and the symmetry about $ = 0 is now broken. So, $
starts at zero, and it rolls down to one of the true minima, $ —►e%e^ g x where the
symmetry under the transformation (2.4) is not apparent about the true minima.
2.1.2
F in ite T em p eratu re F ield s
Consider a real scalar field $ and the system has simpler discrete symmetry at
the beginning. Given the Lagrangian as
£ = ~ d ^ d ^ - ^ (4 >2 - <r2)2,
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(2.5)
18
T>T.
T<T,
Figure 2.2: The finite-temperature effective potential of a scalar field in contact with
thermal bath. The red (dashed) curve shows Vt with T > Tc where the parity
symmetry is restored while the blue (dotted) curve represents Vr with T < T C and
the parity symmetry about the absolute minima at 4>c = ±77 (rj = >/<x2 —T 2/4) is
broken. The black (solid) plot is the potential at critical tem perature.
the system has a discrete symmetry under the parity inversion
about $ = 0 .
Then, the effective potential at finite tem perature becomes [143]
W
= - ^ T 4- ^
2r 2
+ j ( T 2 - 4ff2) $ 2 +
+ 0(T )
(2.6)
where 4>c is the classical field or ensemble average (4>) and 0 ( T ) represents the terms
with linear or lower powers of T. Tc is defined by the critical tem perature at which
[<92 Vr/5<I>2]$c=o changes its sign. It is evident th a t Tc = 2<r and for T > Tc, VT has
its global minimum at 4>c = 0 and the symmetry is preserved. For T < Tc, 4>c = 0
is a local maximum and Vt has true minima at 4>c = ±^/<r 2 —T 2/4. The discrete
symmetry about the true minima is broken. As the tem perature of the Universe
continuously decreases across Tc , the phase transition occurs smoothly and it is called
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19
the second-order phase transition. Figure 2.2 shows the tem perature dependence of
the effective potential.
2.1.3
K ib b le M ech anism
The fact th a t the interaction speed is finite and the notion of the existence of
the moment of the beginning of the Big Bang Universe combine to give a finite
correlation length, £ (t), beyond which any physical event is causally uncorrelated.
The energy scale of the GUT symmetry breaking is presumably around 1016GeV
which is evidently the radiation dominated era.
The upper bound of correlation
length can be set by the particle horizon. The particle horizon at th at early moment
of the Big Bang can be calculated most conservatively by setting the speed of particle
as the speed of light c, and the correlation length can be limited to
r^c (jf
f (tc) < dH(tc) = a(tc) / —t t t ^ c H ~ \ Q .
Jo ca(t)
(2.7)
A (Higgs) field th a t is responsible for the spontaneous symmetry breaking must be
uncorrelated beyond this length scale and therefore, at the time of phase transition,
the field has randomly distributed vacuum energy landscape over the super-horizon
scale. This entails inevitably the formation of the stable non-minimal vacuum energy
structure in space and called “topological defects” . This mechanism th at produces
the topological defects is known as the Kibble mechanism [70].
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20
surface of <&fc*0
Figure 2.3: Formation of a domain wall. Light blobs represent the region of true
minima ($ c = ±?y) of vacuum energy while gray region has non-minimum vacuum
energy.The points where $ c = 0 between x_ and x + constitute a surface called
“domain wall” . The thickness of a domain wall stays finite as a result of minimization
of energy density (kinetic + potential) as is depicted in orange stripe.
2.1.4
T op ological D efects
During the cosmological phase transition, if two minimum potential configurations
are separated by more than particle horizon, the field values at these points should
be uncorrelated because there is not enough time for them to interact and have
continuous field values relative to each other. As these minimum configurations grow
over time and collide each other, the field value will suddenly change from one th at
gives the minimum of the potential to another across the boundary.
If the field
changes truly in a discrete manner then, it would blow up the kinetic energy term in
the Lagrangian (~ (V<f>)2) at the boundary. Thus, in order to avoid this unphysical
configuration, there must be bumping region th a t connects two minima smoothly
though it is not a minimum energy configuration. This fact inevitably leaves the
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21
topological defects. Topological defects can have various forms depending on the
configuration of the vacuum manifolds. A point-like (zero-dimensional) defect is called
“monopole” , a line-like (one-dimensional) defect is “cosmic string” and a surface
(two-dimensional) defect is called “domain wall” . As an example, figure 2.3 visually
illustrates how a domain wall is formed when a discrete symmetry is broken. When
a field of which the effective potential is given by (2 .6 ) undergoes a spontaneous
symmetry-breaking, regions of minima (light spots in figure 2.3) are stochastically
formed over the super-correlation scale in which the field values are either +77 or —77.
Consider two positions x_ and x + th a t are separated more than correlation length
and they have field values such th a t $ c(x_) = —77 and $ c(x+) = + 77. Imagine a
continuous path x _ x ix + th at connects x_ and x + then, the field <f>c makes a smooth
transition from —77 to +77 along this path. It is evident th a t there must be at least
a point where <1>C becomes 0 and it is denoted as x t . Likewise, any continuous path
th at links the two points has a point where $ c = 0. The collection of all the $ c = 0
points makes up a surface or domain wall on which the potential is of no minimum.
H om otopy group
Mathematically, the criterion for the existence of stable topological defects for a
spontaneous symmetry breaking process Q —> 7i can be expressed in terms of the
homotopy group:
IIo(A'f) 7^ T
—> domain wall
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22
n l{ M ) ^ X
—> cosmic string
Ii2{M ) ^ X
—> monopole
n 3 (.M) 7^ X
—> texture
where Q is the unbroken symmetry group, H is the symmetry subgroup of Q left
behind the symmetry breaking and irn (Ad) is the n th homotopy group or mapping
from the n-dimensional sphere S n (Sl0 =point, 5 1= circle, Sl2 =ball,- ■•) into the vacuum
manifold M. = G/H.. X is the identity group. The texture does not depart from
minimum vacuum energy configuration but the gradient of the field is not zero since
the phase of the field is continuously varying over the space. Thus, the kinetic term
contributes to energy while the potential energy stays zero. The textures are unstable
to collapse from the scaling argument called “Derrick’s theorem” .
2.1.5
D efects and C osm ology
That the nature has SU (3)c x U ( 1 ) Em symmetries today has been experimen­
tally tested and well consistent with the experiments. Therefore, it is very natural to
conceive th a t greater symmetries (or possibly Grand Unified Theory) existed in the
early and hot universe. Then, it looks almost inevitable th a t the GUT of the gauge
theory would produce topological defects from its spontaneous symmetry breaking
in early universe. However, there are remarkable observational constraints on the
existence or population of defects. For instance, so far all the experiments trying
to detect magnetic monopoles have failed to discover them but set upper limits on
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23
their population. While cosmic strings do not necessarily lead to cosmological dis­
aster [136, 38], monopoles and domain walls are destined to over-close the universe,
which is clearly against observation if there were no significant annihilation or de­
caying processes. For example, monopoles should have over-closed the universe so
badly if their population didn’t decrease dramatically since their production at GUT
symmetry breaking. Let’s choose the symmetry breaking scale Tc = 1014GeV and we
don’t assume Inflation nor systematic annihilation of monopole-anti-monopole pairs.
We also assume th at the universe has undergone adiabatic expansion since th a t sym­
metry breaking time. Then, because of Kibble Mechanism, monopoles are produced
roughly one per horizon volume at th a t time. Co-moving horizon at th a t time be­
comes, assuming th a t the phase transition took place during the radiation (photon +
massless neutrino) dominated era,
( 2 .8 )
where ac ~ 2.35 x 10~27, Hubble distance c H ^ 1 — 2997.9fl_1Mpc = 9.2503h~l x
1027cm and Qr = 4.15 x 10~5h~2(T /T 0)4. So, the physical size of the horizon X h
today is a(t0)xH = 33.7m. Thus, the monopole density today would be
»» (io ) =
),V//) 3 = 2.61 x 10 5m 3
(2.9)
and this would result in a disastrous monopole energy density
PM(to) =
) ~ 10n GeVm 3
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( 2 . 10 )
24
with rriM ~ 1016GeV [73], while the critical density today is roughly 5GeVm-3 . This
huge discrepancy between observation and prediction forces one to introduce new con­
cepts on or modify our background knowledge of cosmology or particle physics. The
original motivation of inflation was to account for the null results from observational
efforts of GUT monopole search.
2.2
C osm ic Strings
2.2.1
G au ge S trings
<bc=ne'
i>c=ne'"
<I>C=0
# *«
x,
•
v
4>c=0 ,
" ■».■
/ xA
!>=>
<t>c=T\ein‘"
m
4>c=ne'"'0
m
(a)
4
(b)
Figure 2.4: Formation of cosmic strings by f/(l)-sym m etry breaking, (a): bubbles
of true minima (light spots) are stochastically created over the space larger th at
correlation length £(£). (b): the shaded island inevitably has a point (x0) where
$ = 0 . At x 0, the potential
P rod u ction o f cosm ic strin gs can b e illu strated m ost easily by a sp on tan eou sly
broken U{ 1) symmetry. We can write a generic Lagrangian for a complex scalar field
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25
$ invariant under a U{ 1) transformation $ —> em04>,
C=
- \ F fU,F ,u' - A($f4> - r f f
Z
(2.11)
~E
where 9 (x) is the phase factor, n is an integer winding number,
and
are
the covariant derivative and the stress-energy tensor for the gauge field A Mdefined in
(2.2) and (2.3). When the system is invariant under the global phase translation i.e,
9 = constant, the gauge field A )X vanishes and accordingly, the covariant derivative
reduces to a partial derivative and F)lv vanishes. The figure 2.4 visually illustrates
the production of cosmic strings in the course of phase transition. At the early stage
of symmetry breaking, regions of true minima (light spots) are stochastically created
in space as sketched in figure 2.4 (a) with uncorrelated phases because of the lack
of causality by th at time. For instance, consider the phase values are assigned as
9 = 0,7t/3,27t/3 at x i , x 2 ,X3 respectively. The red-dotted contour in figure 2.4 (a)
represents the path along which the phase continuously changes to 0 —> 7r/ 3 —> 27r/3.
As the tem perature of the universe decreases, the minima regions merge and the
bubble th at maintains non-minimum potential is isolated as shown in 2.4 (b). The
blue-dotted curve th a t connects x i and x 0 represents the path on which the phase
of $ is kept 0 while the other two blue curves linking x 2 — x 0 and x 3 —x 0 are also
p ath s w ith con stan t p hases n j 3 and 27t/3 respectively. It is evid en t th a t there must
be at least one point where $ vanishes in the region enclosed by this contour to avoid
discrete jumps in <J>and let x 0 denote this location. Otherwise, the phase has to jum p
from 0 to some finite value at a certain point and it would cause the kinetic energy of
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26
the system (~ (V $ )2) to diverge. This non-minimum vacuum energy configuration is
topologically stable and forms a cosmic string. The linear energy density of a gauge
cosmic string depends on the structure of the string and coupling of the scalar field
and the gauge fields involved. An Abelian Higgs model illustrates the cosmic string
solution in a simple way. The model has approximate solutions for <J> and
at large
distance from the string called “Nielsen-Olesen solution”
$
-► rjeine
(2 . 12 )
A. -
<213)
and its linear energy density of an infinite cosmic string lying along the z-axis is given
by [136]
p =
^
=
—
j
Cdx 2 ~
J
+ V (4>)^ d x2 ~ r f
(2.14)
where uj is the effective cross-section of the string and the radii 5$ and 5a within
which the potential V (<f>) and the magnetic field term B 2/ 2 stay non-negligible are
approximately given by
~
and 5a ~ (er/)_1. For the global U (l) strings,
there is no gauge field and the energy density is approximated as
p ~ 2'Krf In
(2-15)
where <5is the core width and R is a radius th a t must be cut-off at some large distance
by the curvature radius of the string or by the distance to the nearest string in the
string network. Thus, the cosmic string param eter Gfi is often expressed as, in the
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27
natural unit,
G/x ~ (
\ m pi j
(2.16)
where mp; = 1.2211 x 1012GeV is the Planck mass. Since r) can be regarded as the
energy scale of the symmetry breaking, G/x directly contains the information on the
symmetry breaking scale. For rj ~ 1016GeV which is theoretically presumed to be the
GUT scale, we can estimate G/x ~ 10-6.
2.2.2
B ran e Inflation Scenario and F /D -S tr in g s
Superstring theory has been considered as a purely speculative theory in the sense
th at there has been no experimental/observational evidence for the theory. However,
its mathematical beauty and remarkable consistency provide very strong motivation
for physicists to explore the theory. Recently, brane inflation model stemming from
string theory predicted the production of cosmic strings at the end of the inflation.
Historically, the idea th at a fundamental string in Planck scale would appear in
cosmological size was proposed and discarded after finding many problems with that
possibility [140]. Cosmic strings with Planck scale (G/x > 10-3) is clearly inconsistent
with current observations (G/x < 10~5). But new models from brane inflation fixed
this disease by introducing more sophisticated compactifications of extra-dimensions
and suggested new predictions on G/x which do not conflict with observational bound
[67, 6 6 , 111].
Brane inflation assumes the existence of extra-dimensions and the
universe in which we live is a slice of the hyper-space. According to the brane inflation
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28
scenario [32], there are brane and anti-brane linked by extra-dimensional coordinates,
all the ordinary m atter and radiations (baryons, leptons and gauge fields) are confined
to branes and do not propagate into extra-dimensions. Prom an observer’s point of
view who is bounded in the brane, each point on the brane appears to have internal
degrees of freedom. The brane/anti-brane structure initially contains U{ 1) x U( 1)
symmetry. Inflation is described by the dynamics of fields defined in the bulk. As
inflation proceeds, the D3-brane and D3-anti brane move towards each other. When
they collide, two branes annihilate and radiation dominated era starts as the F l / D l strings from the breaking of {7(1) x {/(l) symmetry and radiation left behind [22, 101,
27]. Some remarkable features in this scenario are th a t there are only cosmic string
production but no monopoles or domain walls and the cosmic strings are produced at
the end of inflation so th a t it can avoid magnificent dilution of cosmic string network
by the exponential expansion th at could blow out any hope of observing cosmic
strings. Viable models on brane inflation [111, 6 6 ] find the cosmic string parameter
G/ jl in the range
lO" 12 < G f i < 10-6.
(2.17)
Though the brane inflation scenario provides a natural account for inflation, it remains
as a purely speculative theory to date and the production of cosmic strings and their
stability are highly model-dependent, which imperatively necessitates the ultim ate
arbitration by experiments/observations.
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29
2.2.3
C osm ic S trings D e n sity in E xp an d in g U n iverse
To see the behavior of cosmic string density in the expanding universe, we can
conceive a simple model of cosmic string network as a cosmic string gas. The stressenergy tensor for a cosmic string gas has the following form [73],
H
=
JL
0
1
o
o
0
- P il
0
0
(2.18)
Ml
0
o
0
0
i - «
0
0
l - / ? 27 2
where d9 is the mean separation between cosmic strings, /?,, = vs/c is the speed of
cosmic string segment and
= (1 —/3|) 1^2 is the Lorentz factor. Since ps = (T°0)
and ps = (—T \ ), we can easily find the equation of state for the cosmic string gas
given by
v , = (2ft2 - 1 )
(2.19)
Using the relation p oc a r 3^l+w\ the evolution of cosmic string energy density becomes
ps oc a
-2(l+/?s2)
(2 .20)
For non-relativistic cosmic strings (/3S <C 1), the cosmic string energy density is
p rop ortion al to ~ a -2 and it w ould quickly d om in ate th e universe over oth er sp ecies
of energy. The energy density of ultra-relativistic cosmic strings behave the same
way as radiation (ps oc a -4) and this case would not lead to cosmic catastrophe. But
this argument does not include the inter-commutation and gravitational radiation of
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30
cosmic strings. When these interactions of cosmic strings area included, it is found
th at even the cosmic strings moving at non-relativistic strings are free from over­
closure of the universe (ps oc a -4). String reconnection probability is a key param eter
th at affects greatly to the cosmic string density. It is shown th a t local/semi local
strings in Abelian/non-Abelian gauge theories with critical couplings always reconnect
classically in collision [38] while the reconnection probability for cosmic F/D -strings
are not necessarily unity.
2.3
P hysical Effects by C osm ic Strings
2.3.1
S p a ce-tim e A rou nd A C osm ic String
For an infinitely long linear energy distribution along the 2 -axis, we can write the
metric for this system as
ds2 = eA{r)(dt2 - dz2) - dr 2 - eB{r)d02.
( 2 .21 )
W ith the convention x M= (t, r, 9, z), the metric tensor reads
= diag [eA(r), - 1 , - e B(r), - e ^ ] .
(2.22)
Then, from the definition of Christoffel symbols
(2.23)
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31
the nonzero components list (no summation implied here)
Ci
= r “0 = \ g mgm,i
(2.24)
rjo
= -) /'» ,,■
(2.25)
v )k
= \ 9 li(9 ij,k + g i k , j - 9 jk,i)
(2.26)
and in terms of A ( r ) and B(r),
= r »0=
C
1a',
r i, =
(2.27)
= 1 eBB \ r j , = 1
r?2 = il, =
(2.28)
-/r , r), = r), = l/i'
(2.29)
where / denotes J;. Nonzero components of the Riccitensor are diagonal,
Rfiv
_
1
Roo
Rn
-pa
26
=
[A!' + { A ' f + \ a !B']
Ld
A ” + ;\ b " + \ ( A Y + \{B<?
+ A 'B ' + 1 (B ')2]
R 22
R 33 =
A
Roo-
(2.30)
(2.31)
(2.32)
(2.33)
The Ricci scalar is easily derived,
R = g ^ R n v = - A " - \ { A ' f + I A 'B ’.
(2.34)
Now, we can plug the curvature tensors and scalar in the Einstein field equations,
R ^ - \ % uR = ~ ^ G T ^
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(2.35)
32
using the fact R = SnG T where T = g ^ T ^ ,
A" + (A!)2 + l- N B '
8nG(2T°0 - T)
(2.36)
8itG ( 2 T \ - T )
(2.37)
8ttG(2T 22 - T).
(2.38)
Z
2A" + B" + (A')2 + ^ { B ' f
B" +
1
=
+ A! B'
Z
When an infinite straight cosmic string is lying along the z —axis, the generic form of
stress-energy tensor becomes, in cylindrical coordinates,
(2.39)
T “„ = a ( r )
where a (r) is the string energy density and r = \ / x 2 + y2 in cylindrical coordinates.
Since, T \ = T \ = 0 and T °0 = T% = \ T , the equation (2.36) is actually homogeneous
and we can combine (2.37) and (2.38) to derive an equation th at is independent of
the stress-energy tensor
A!' + (A')2 + \ a 'B' = 0
(2.40)
Z
2A!' + (A1)2 — A!B'
= 0.
(2.41)
By eliminating B' from the above equations, we have
A" + -(A !)2 = 0
4
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(2.42)
33
and it has two solutions for A (r ) and B(r),
A (r0) + I3 l n r- =a ^ ,>
a =
^
3A' ( r0 )
A (r ) =
(2.43)
constant.
(i) A ( r ) = A ( r „ ) + | l n ^ :
A'(r) + 2B'(r) — 0 => B ( r ) = B ( r 0) - ^ In - — r ° + Q .
O
OL
(2.44)
(ii) v4(r)=constant:
B" + U b 'Y = —8irGT(r) = - 167rG a ( r ) .
z
(2.45)
The first case potentially makes the metric blow up at certain r and thus dismiss these
solutions as being unphysical. For the case A{r) is constant,wecan consistently set
A(r) = 0 by rescaling the coordinates t and r to rewrite the metric (2.21) as
ds2 = dt2 —dz2 — dr2 —eB^ d d 2.
W ith the substitution C(r) =
(2.46)
we have (2.45) in terms of C(r)
C "(r) + 8 tt Ga(r)C(r) = 0.
(2.47)
In flat space-time, the energy density function of a local gauge string, <r(r), decays
exponentially at r
r
5 where 6 is the string thickness. Thus, we can set
= 0 at
8 and C(r) has an asymptotic form from (2.47),
C(r) = C'{r —> oo)r + b,
r
<5.
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(2.48)
34
where b is a constant. We can first calculate the linear string energy density along
the z-axis with the help of (2.47),
poo p2ir
^
Jo
Jo
i
n V S ® « - ^ ( ° ) - ^ c o ) ]
(2.49)
where c/2'1 = C 2(r) is the metric determ inant of x-y plane. To avoid singularity in the
metric (2.46), the region around the string core (z-axis) has to be flat, so the area of
a circle centered at z-axis with radius r <C 5 becomes
Acirc(r) = f
Jo
\Jg^dr'dO = / C{r')dr'dO — n r 2.
Jo
(2.50)
Thus, we can readily find from the condition (2.50) th at
lim C"(r) = 1
(2.51)
r —>0
The circumference of a circle with radius r
5 under this metric is given by
/■27T
l(r) = /
Jo
C(r)d0 = 2nC(r) = 2irC'(r —> oo)r + b.
(2.52)
Then, the deficit (surplus) angle compared to flat space becomes, in terms of the
linear energy density of string /x,
A = 2n — lim
r —>00
2.3.2
r
= 87tG/jl.
(2.53)
L ensing Effect and D o u b le Im ages
Even though the conic space-time built around a straight cosmic string is locally
flat, it can effectively deflect the path of light ray. A straightforward and potentially
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35
co sm ic string
(a)
(b)
Figure 2.5: Figure (a) shows the conic space-time built by a straight cosmic string
along the z-axis. Figure (b) illustrates the deficit angle due to conic nature of the
metric. When we redefine the angular coordinates in the way 9 —> (1 —4Gfi)9 then,
the space-time becomes flat except the deficit angle A = SnG/i which is equivalent
to removal of a wedge angled 8ttGfi.
observable effect of the conic space-time is the double images illustrated in the figure
2.6. Under the approximation A = 8-k G h
1, the opening angle between two images
measured by the observer at A or B can be easily derived from the geometry shown
in figure 2.6 (a),
I sin Z S Q A = d sin a
I sin Z S Q B = d sin (3
Z A Q B + a -I- P = A sin 9
and using the small approximation, we get
(/>= a + (3 —
,A sin 9
I+ a
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(2.54)
36
<|>= a + P
image 1
Q
•
S ®CT) A sin0
9I
#
light source
light source
o
observer
d
image 2
cosmic string
(b)
(a)
Figure 2.6: Double image formed by conic space-time. The opening angle of the
image is derived under the small angle approximation.
where I = QS, d = S A = S B and 9 is the angle between the sting and x-y plane
(we choose the coordinates so th a t the observer and two images lie on the x-y plane)
[136],
2.3.3
K a iser-S teb b in s Effect
A moving cosmic string can wake up objects sitting nearby it and make them move.
Figure 2.7 shows a simplified case of the Kaiser-Stebbins effect [6 8 , 136, 131, 125]. At
the moment the string (moving at the velocity —vsy) passes the line connecting the
observer and the light source, the observer and the light source start moving toward
each other. In the rest frame of the string, the observer and the light source have
the relative velocity in the x-y plane vrei ~ w.,Asin0 to the 1st order of A. Thus, if
the light source emits photons with frequency
u jq
before the string passes between the
light source and the observer, the photons after the string passes will be blue-shifted
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observer
"
9ir
"
'
"
"
-
-
v . =(v, Asin0)/2
v rel =(vsAsin0)/2
(b)
(a)
Figure 2.7: Blue(red)-shift of photons by a moving cosmic string (Kaiser-Stebbins
effect). Photons are relatively blue-shifted after a cosmic string passed between the
light source and the observer due to relative speed between them invoked by the
string.
because of the relative speed between them. After transforming the reference frame
to the rest frame of the observer, using the relation between the energy (frequency)
of photon and the thermodynamic tem perature
Suj_ _ S T
(2.55)
we get the tem perature shift relation by a moving conic space-time,
ST
— = S n G fijv , • (t x k )
(2.56)
where 7 = 1 / y / l —v% the Lorentz factor, t and k are the unit vectors along the
direction of the string and the line of sight respectively.
The co-moving particle
horizon at the last scattering is
/ *a d e c
X H i t de c) =
/
JO
da
o?H
2c^Gj&^c
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(2.57)
___________
0 .9 0 m K
(1 2 0 .0 , SO.O) G a la c tic
- 0 . 9 0 ______ ___
(1 2 0 .0 , 5 0 .0 ) G a la c tic
(1 2 0 .0 , 5 0 .0 ) G a la c tic
Figure 2.8: Simulation of blue(red)-shift of CMB photons. It is assumed th a t va^ s ~ 1
here, (a) a patch of real sky map from WMAP lst-year centered 6 = 50°, 4> — 120°
(Galactic coordinates), (b) a string with G/i = 10- 6 and moving to the right on the
plane of the page is added in the middle. Tiny shift is barely recognizable, (c) 10 times
more powerful string is simulated (G/i = 10-5). The Kaiser-Stebbins effect is now
dominant over all other sources of perturbations. Here, reddening means blue-shift
i.e. shift to higher thermodynamic temperature.
and the average distance between the strings are ~ x h / 3 [136], its angular size we
can observe today is then,
Qss =
X n (W )/3
X H y'd ec
_ o.Olrad = 0.58°.
(2.58)
to)
Thus, we expect th a t the size of tem perature steps produced by Kaiser-Stebbins Effect
is of order
2.3.4
1 °.
G ravitation al R ad iation s from C osm ic S trings
Like a cosmic string segment oscillates at relativistic speed due to its extremely
large energy density and therefore its tension, a cosmic string loop also oscillates
relativistically. Radiation of gravitational waves is the most effective way to lose
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39
its energy for a cosmic string loop. Its typical oscillation frequency is estimated as
/ ~ L -1 , where L is the length of the loop. Then, the energy rate radiated away in
the form of gravitational wave by the time-varying quadrupole moment is given by
[132]
~ G M 2L‘ f
^
~ IG /r
(2.59)
(XL
where M is the mass of the loop, \i = M / L and T is numerical constant with magni­
tude of order O (102). Then, the characteristic time r of the loop in which the loop
loses most of its mass energy is approximately
t
~
M
~
TGpfl
L
---------- .
TG n
(2.60)
K
'
Gravitational waves emitted in early universe by the violent oscillations of cosmic
strings can be stretched by the expansion of space to wavelength order of light-year
scale. This long wavelength gravitational waves can disturb the regularity of pulsar
timings when the wave passes between the pulsar and the observer. This effect may
be used to detect the gravitational waves produced by cosmic strings. A conservative
estimation on the upper limit of the cosmic string parameter Gfi is made by pulsar
timing as G/r < 4 x 10-5 [136].
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40
2.4
Sum m ary on P red iction s A nd O bservations o f
C osm ic Strings
2.4.1
D o u b le Im ages by C osm ic S trings
Optical Images + Gamma rays
Gn
~ 4 x 10“ 7
~ 7 x n r7
~ 1 (T 14
(EW or SUSY scale)
Method
CSL-1 Double image (CapodimonteSternberg-Lens Candidate No.l)
Double image of
Q0957+561 A,B
Possible source of Galactic
511 keV gamma rays
Reference
[112 ]
[114]
[40]
Table 2.1: Works on optical images discovered which are suspected as ones produced
by cosmic strings.
The CSL-1 was a good example of double image search as a means of cosmic
string detection in that it provided a good practice to investigate a candidate of
double image by cosmic strings. The CSL-1 double image apparently had nearly the
same spectroscopic curves. However, the Hubble space telescope confirmed th a t they
are a pair of interacting elliptical galaxies.
2.4.2
C on strain ts on C osm ic S trin g P aram eter b y C M B
Cosmic microwave background radiation is also a good tool to search for cosmic
strings. There are two methods widely used, estimation on the contribution to power
spectrum and direct pattern search using Kaiser-Stebbins effect. It is observationally
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41
Figure 2.9: Hubble Space Telescope image of CSL-1 which was suspected as a possible
double image produced by cosmic string. It turned out to be a pair of interacting
elliptical galaxies.
confirmed th a t the topological defects including cosmic string did not played major
roll in creating primordial fluctuation and, if any, they did marginally. It is agreed th at
the contribution to power spectrum by cosmic string is at most 10% [55, 96, 142, 42].
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42
Limits on G/x from CMB data
G/x
Method (data)
G/x < 3 x 10“ 6
Power spectrum
(90% CL)
(OVRO)
G/x(& 7s> ~ i x 10-e
Kaiser-Stebbins Effect
( COBE DMR 4yr 53GHz)
G/x < 1.3 x 10~ 7
Power spectrum Contribution
(99% CL)
( W M AP lst-yr & SDSS)
Kaiser-Stebbins Effect
G»<J3S7s) < 7 x 1 0 - 6
(95% CL)
( W M AP lst-yr DATA)
G/x < 1.07 x 10“ 5
P attern Finding
(95% CL)
( W M AP lst-yr DATA)
G/x < 3.4(5) x lO" 7
Power Spectrum contribution
(68(95)% CL)
( W M AP lst-yr DATA)
G/x < 3.2 x 10- 7
Power Spectrum contribution
( WM AP lst-yr DATA)
(95% CL)
Kaiser-Stebbins Effect
G/x < 1.6 x 10“ 5
( W M AP 3 yera DATA)
(95% CL)
Reference
[55]
[93]
[96]
[63]
[82]
[142]
[42]
[64]
Table 2.2: Constraints on cosmic string param eter G/x set by Kaiser-Stebbins effect
and Power spectrum using CMB data.
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43
C hapter 3
P a ttern Search o f C osm ic Strings
w ith CM B
Based on E. Jeong & G. F. Smoot, “Search fo r Cosmic Strings in CMB Anisotropies”,
Astrophys. J. 624: 21-27, 2005.
3.1
Introduction
Current theories of particle physics predict th a t topological defects would almost
certainly be formed during the early evolution of the universe [60]. Just as liquids
turn to solids w h en th e tem p eratu re drops, so th e interactions b etw een elem en tary
particles run through distinct phases as the typical energy of those particles decreases
with the expansion of the universe. When conditions favor the appearance of a new
phase, the new phase crops up in many places at the same time, and when separate
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44
regions of the new phase run into each other, topological defects are the result. The
detection of defects in the modern universe would provide precious information on
events in the earliest moments after the big bang. Their absence, on the other hand,
would force a major revision of current physics theories about the energy scale of
symmetry breaking or scenarios for phase transitions or both.
The potential role of cosmic topological defects in the evolution of our universe has
interested the astrophysical community for many years. Combined theoretical and
experimental work has led to the development of observational signatures in a variety
of diverse d ata sets. These have, on one hand, removed one m ajor motivation for the
primary role in structure formation and, on the other hand, raised new motivations.
These have also narrowed the range of allowable topological defects, so th a t most
viable models are likely to produce some variety of cosmic strings.
Cosmic Microwave Background (CMB) anisotropy power spectrum observations
rule out topological defects as the primary source of structure in the universe [89,
96]. Observations strongly favor adiabatic random fluctuations from something like
inflation rather than the structure formation from topological defects (e.g., cosmic
strings). These CMB results, while supporting inflation, do not rule out lower level
contributions from topological defects. Interest in cosmic strings has, in fact, been
renewed with recent theoretical work on hybrid inflation, D-brane inflation, and SUSY
grand unified theories (GUTs).
The idea th a t inflationary cosmology might lead
to cosmic-string production is not new; however, it has received new impetus from
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45
the brane world scenario suggested by superstring theory. A seemingly unavoidable
outcome of brane inflation is the production of a network of cosmic strings [111 ], whose
effects on cosmological observables range from negligible to substantial, depending on
the specific brane inflationary scenario [6 6 ]. A number of theory papers [130, 137, 26],
anticipating th a t searches for cosmic strings will be negative and set significant limits,
have begun developing modified theories so as not to produce them. At a minimum,
these modified models must introduce a new field (also a warping brane).
Since a moving string would produce a step-like discontinuity in the CMB, it will
cause the tem perature distribution to deviate from a Gaussian. We may be able to
detect non-Gaussian aspects of the tem perature distribution with sufficient resolu­
tion. We search for strings in two ways: statistical and pattern-discovery methods.
Statistical analysis determines how much the distribution of tem perature fluctuation
deviates from the Gaussian distribution or what fraction of the fluctuations might be
due to strings. In the second approach, we can search for cosmic strings directly from
the tem perature map by their distinctive pattern of anisotropy. The latter approach
is much easier and more straightforward when the CMB signal is not contaminated se­
riously. Similar ideas on pattern searching preceded ours a decade ago on COBE and
Owens Valley Radio observatory (OVRO) data [85, 55]. However, with the release of
Wilkinson Microwave Anisotropy Probe ( W M AP )* data, it is appropriate time now
to apply a pattern search to WM AP . These methods are distinctly different from a
1h tt p :/ /l a m b d a .g s f c .n a s a .g o v / p r o d u c t / m a p /
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46
simple fitting to the angular power spectrum [96].
3.2
3.2.1
Effect o f Cosm ic Strings on CM B
S ignal from a m ovin g C osm ic String
—>
Consider a cosmic string with mass per unit length /x,velocity (3,and direction
s, both ofwhich are perpendicular to the line of sight,and which is backlit by a
uniform blackbody radiation background of tem perature T. Because of its angular
defect 6 — 8itG/j,, there is a Doppler effect on one side of the string relative to the
other, which causes a tem perature step across the string of [6 8 , 48]
Y
= 8 txG ^(3 .
(3.1)
where 7 = 1/ (1 —fp ) 1^2. This expression was generalized for arbitrary angles be­
tween the string direction s, string velocity /3, and the line of sight n as [131, 134]
= 87rGfj,j(3h ■( / ? x s ) .
3.2.2
(3.2)
P ro b a b ility d istrib u tio n s for relevant p aram eters
The tem perature discontinuity 8 T /T in equation (3.2) is determined by three
factors G/x, 7 (3 and h • (/3 x s) which arise from different sources. The factor Gfi
is related to the symmetry-breaking scale and is the key param eter th a t describes
various properties of strings. We can assume 7 /J ~ 1, except near cusps and kinks,
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47
where one can have 7 > 1 , and n ■0 x s) depends on the combination of relative
angles among /3, s and n. We can also assume th a t [1 x ,s is random in direction to
the line of sight n in three-dimensional space. If we denote fl x s as u sin 4>, then
h • (/3 x s) = h ■u sin 4>= cos 9 sin 4>.
(3.3)
The infinitesimal probability th a t cos 9 has an arbitrary value is 2tt sin 9d9/An =
|d c o s0 , and thus its probability distribution is uniform in cos 9. If the string velocity
(3 and its direction n are uncorrelated, the probability distribution for sin 4> = y is
y / (1 —y2)1^2. Then, the probability distribution for k = h - 0 x s) being an arbitrary
value —1 < k < 1 becomes, substituting cos 9 = x,
^ / l ^ y 26(k ~ x y }dy
=
kdx
-
\1 J\k\
f |fc| x \J x 2 - k 2
=
^ cos-1 |A:|.
(3.4)
Often, string velocity and direction will be perpendicular to each other, so th at
sin <j) ~ 1, and in this case the probability reduces to P (k) =
These probabil­
ity distributions are plotted in Figure 3.1.
In a matter-dom inated universe, the projected angular length of a string in the
redshift interval [zi,z2) scales as z ^ 2 — z ^ 2 [16]. The standard cosmological model
(Q,de ~ 0.7,
~ 0.3) gives zia ~ 1100 [16]. The apparent angular size of the horizon
at the CMB last scattering redshift zia is
9h = - L
y/Z ls
= 1.8° ( —
V
z ls
) V2 ~ 1.7°.
)
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(3.5)
48
0.8
0.6
^
0 .4
CL
0.2
0.0
1.0
- 0 .5
0.0
0.5
1. 0
k
Figure 3.1: Probability distributions for tem perature steps in equation (3.2). The
solid curve represents the probability with both string direction s and string velocity
/3 having random directions while the flat dotted line assumes P-Ls (sin (f> = 1 ) but
that the direction of ft x § is random relative to the line of sight (cos 6 random). For
much of our analysis, these two distributions give simpler results.
The average distance between strings is roughly d n /3 .
Thus the typical angular
distance between the discontinuities on the sky is of the order of Oh • The rough
expected magnitude of the jumps in tem perature is of the order of S T /T ~ 136"// —
196//. W ith an angular resolution of 0°.6, there could be sharp jumps up to about
S T jT ~ 406// (i.e., the peak range of tem perature steps in a distribution of possible
steps). If the angular resolution is poorer, then the blurring effectively smooths the
steps so th a t the maximum range of steps is at a somewhat smaller level. For the
case of W M AP first-year data, the angular size of a pixel with the highest resolution
is Opixei ~ 0 ° .ll, and it is reasonable to assume th a t there are not enough strings per
pixel to apply the central limit theorem and make the distribution Gaussian.
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49
sotid
do sh ed
: 2.6M G a u s s 'o n s fit .
: sin g le G a u s s ia n fit.
O
z
Ld
Ld
m
Ll .
1.0 x104
-
1.0
-
0 .5
0 .0
0.5
1.0
A T (m K )
Figure 3.2: Temperature distribution of the full sky from W M A P . Pixels with | A T| >
lm K or |Galactic latitude| < 10° are excluded, because these signals are mainly due
to the Galaxy or bright sources. The dashed curve is the best-fit Gaussian with
fj, = 23.6iiK and a = 201.25jiK . The y ^ /D O F for the Gaussian fit is 4085.87/398
and the solid curve represents 2.6M-Gaussians fit with fi = 23.6/xiT, er0 = 6682.22fiK
and (Jcmb = 80.25fiK . The x 1jD O F for the 2.6M-Gaussians fit is 435.76/398.
3.2.3
S ta tistica l F lu ctu a tio n s
The signal in a microwave sky map will be made of several components: noise
from the instrument, foreground signals (expected to be small away from the galactic
plane and with strong sources punched out), CMB fluctuations from adiabatic (or
appropriate) fluctuations, and potential signals from the strings.The signal observed
at any pixel is then
'Fpixel = T OlSGpixel T Tforeground T 8FcMB
(3-6 )
where we can decompose STcmb into 5Tadiabatic + 6Tstring- Since the nature of the
signal is the superposition of a random Gaussian signal (noise, 5Tadiabatic) and a non-
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50
Gaussian contribution by strings (8Tstring), there are two basic forms of distribution
functions (foreground will be removed from the beginning).
Gaussian signal - The major portion of the CMB signal obeys a Gaussian distribution
(Figure 3.2),
(3.7)
°"2
where
e rg
and
acmb
—
alJ ni
+
a CMB
are variances from noise per observation and CMB fluctuations,
respectively and rq is the number of observations for the ith pixel. We make use of
the information th a t the noise is Gaussian (actually Gaussian per observation, and
thus variance is inversely proportional to the number of observations per pixel) and
the intrinsic CMB fluctuations are distributed as a Gaussian.
Non-Gaussian signal from strings.- W hen a straight-moving string is added to a
region, we can expect a tem perature distribution th at is the sum of two Gaussians
rather than single Gaussian because of the blue- and redshift by a transversely moving
string. Then the probability distribution in equation (3.7) is modified to
/ 2 (AT)
=
g± =
ayTn L
(3.8)
(A T - p ± 6 T / 2 ) / a
where p = N blue/N total, the ratio of blue-shifted pixels to total pixels, q = 1 —p, and
ST/ 2 is half of the effective height of the step across the string, as given in equation
(3.2). If we denote v and aobs as the mean and standard deviation calculated from
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51
3 .0 x 1 0 “
d o tte d
solid
: WMAP 1 s t - y r DATA
uniform distribution
of strings
O
z
UJ
a=5
Ld
O'
-1 0 0 0
-5 0 0
0
500
1000
A T (/*K )
Figure 3.3: Temperature distribution of the full sky excluding the pixels with |Galactic
latitude| < 10° or |A T| > lm K . The solid curve represents the best-fit curve of the
uniformly-distributed-strings model, and the dotted curve shows actual W M A P firstyear data.
observation, then they are related to p and a as
v = v + p 6 T /2 + ( l - p ) ( - 6 T / 2 )
= n + (2p — l)<JT/2
<&. =
(3.9)
< ( A r - t /)2 >
/ oo
(A T - u)2} 2{ ^ T ) d A T
•00
= cr2 + 4p(l —p)(6 T /2 )2 .
Here weused an approximation on the integration interval as —lm K
lm K — ►—oo < A T
<
(3.10)
< AT <
oo. This is a good approximation because a ~ 200fxK
and thus lm K ~ 5 cr, and the variance integral in equation (3.10) over [—5 cr, 5<r] cov-
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52
205
200
b
195
90
0
50
100
150
200
250
<5T(,uK)
Figure 3.4: The x 2 distribution for the uniformly-distributed-strings model. The
minimum xLin ~ 435.43 with 398 degrees of freedom. Parameters span 0 < ST <
200.3 pK and 192.1 /xK < a < 200.8 /xK at the 95% CL. Here a 2 represents the
2 __
total variance, except the string contribution, a 2 = ^
1/rii + (rldiabatic. The outer
contour is for the 95% CL, which is almost overlapped with the 6 8 % CL contour in
the vertical direction.
ers 99.9985% of th at over [—oo, oo]. This approximation is also implied in equation
(3.17). The tem perature distribution of the sky can be explained with appropriate
combinations of fi( A T ) and / 2 (AT). All the models introduced in the following
sections are based on these two primary patterns of tem perature distribution.
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53
Signal
Limit
&string (/^K)
S T /T
S = 87rGp7/3
7.4xlO “ B
2 .9 x l0 “ 5
7 .3 x l0 - 5
Limita Gfi
(xlO 6)
2.9
nsBh
( x l 0 16GeV)
Total
2 .0
201.25
1.2
1.3
Total - Noise 6.7mK/obs
80.26
Uniform Distribution
57.82
2.9
2 .0
of Strings
P attern Search
N /A
1 .7 x l0 ~ 5
0.7
1.0
aThe Lorentz factor and velocity are set (7 0) ~ 1.
bSymmetry breaking scale responsible for the production of cosmic strings.
Table 3.1: Variance Limits on Cosmic Strings. All the values presented are at 95%
CL.The Uniform-Distribution-of-Strings model on the fourth row is the distribution
given in equation (3.16). The GUT symmetry-breaking scales, t)s b , in the last column
are calculated using the relation tjsb ~
(^A4) ^ 2 with m pi = 1.2 x 1019GeV.
3.2.4
V ariance: Q uadratic E stim a to r T est for S tring C ontri­
b u tio n
Assuming th a t the signals are all statistically independent, we can find the variance
in the map
*7?
®noise U ®foreground
"b ®CJMB
(3.11)
We remove the significant foreground contribution from Galactic emission by dropping
pixels with |Galactic latitude| < 10° or |A T| > lmK. The first condition is to exclude
galactic area in which the non-CMB signal is dominant. We lose 217 more pixels by
imposing the condition |A T| < lm K , 4 pixels are less than —lm K and 213 pixels
are greater than lm K . Since the Gaussian tail probability allows ~ 1.7 pixels in the
|A T| > lm K region, and they mostly form clusters, we can drop those pixels, as they
are from unusual bright sources. We introduce a tem perature distribution based on
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54
the Gaussianity of the signal for each pixel in the WMAP first-year d ata and calculate
&c
m b
,
which contains all the contributions except instrumental noise, by
(3,12)
+ a CMB
where N is the total number of pixels, 2,598,695, i is the pixel number with the
maximum resolution of W M AP first-year data,
is variance due to noise per mea­
surement and Hi is the effective number of measurements for the fth pixel. The dis­
tribution / 2.6m(AT) in equation (3.12) has a minimum y 2 for ucm b = 80.25^°;g5//K
and cr0 = 6682.22^31228/rK, and thus ag = 200 .68 ^g!74AtK , each with the 95% con­
fidence level (CL). This observation in turn allows an upper limit of the order of
astring ^ 200p K and thus providing a limit of 87rGfj, < 2 0 0 //K /2 .7 3 K = 73.5 x 10- 6 .
Figure 3.3 shows the plots of / 2.6m(AT) and W M AP first-year data.
We obtain a better upper limit, if we know the mean contribution of instrumental
noise and other signals. According to the W M AP first-year d ata release, the sum over
pixels of (number of measurements per pixel)-1 for 2,598,695 pixels included here is
1968.96. Thus, the effective variance due to noise is
noise
(183.93lS:Ss V K )2, 95% CL
(3.13)
where n* is the effective number of measurements for the zth pixel. Under the as-
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55
sumption th a t the signals other than noise and from strings are negligible,
S trin g ~
a CMB
^
a S ~
a noise
(3 -1 4 )
=
(80.25^65 I^K)2, 95% CL,
(3.15)
and we are able, by taking the instrum ent noise into account, to set a lower upper
limit.
line o f discontinuity
v i a
| ? 3 1 ||
T32
14
/ ^
.V T
?41
hotter
1^ 1 3 | | ^ 1 4
T23
lih E
T
Ti
y
cooler
t2
Figure 3.5: Generating the vector field L. Pixels on the left are demoted to the middle
pixels, and going from the middle to the right shows the definition of the gradient of
tem perature V T and L, which lies along the tem perature step.
Figure 3.6: The L field for the full sky with 64-pixel demotion. The center of the map
is [6 — 0°, 4>= 0°) in Galactic coordinates. Each arrow represents a local tem perature
step, and the tem perature steps greater than 100fiK are set to 100fxK. The pattern
along the equator is due to the Galaxy.
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56
3.2.5
S ta tistica l T est for T em p eratu re S tep E x p ec te d from
R an d om S trings
For the anticipated distribution of strings, one would expect a random distribution
of tem perature steps that has roughly equal probability between plus and minus the
maximum tem perature step amplitude. We also fitted to a distribution th at was a
Gaussian for the other signals convolved with a uniform distribution of tem perature
steps. Let ST0 = 87rG/ry/3 T0 be the characteristic value for a string; then the actual
effect of the tem perature step left on the measured CMB is kST0/2, where —1 < k =
cos9 simp < 1 from equation (3.4) and T0 = 2.73K. Assuming sin 4> ~ 1, i.e., the
direction of a string and its velocity are most likely perpendicular, we have P (k ) ~ | .
Then we obtain the distribution function of tem perature
(3.16)
where / ^ ( A T , kSTo/2, a) is the two-Gaussians form with p = 0.5 and ST = kSTo of
the zth Gaussian in equation (3.12). Then, the apparent variance cr2nif orm for this
distribution becomes, in terms of a and ST0,
= < ( A T - ft)2 >
/
OO
(A T - p,)2f T(A T )d A T
•o o
- E
-
+
adiabatic
(3.17)
where the first two terms are directly from equation (3.12) and the last term is the
contribution from strings. Here we have decomposed o 2 into a iiabatic
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string and
57
equation (3.17) gives the explicit form of <J%ring, cr%ring = STq/12.
From the x 2
distribution in Figure 3.4, we have 192.1 fjK < a < 200.8/iK and 0 < ST0 < 200.3/iK
at the 95% CL. Thus, tem perature variation by a moving string can be limited to, in
terms of deficit angle,
cm
0 < — - = 8irG/ny/3 < 7.34 x 10~5, 95%CL.
Tq
(3.18)
The results for the variance limit on cosmic strings are summarized in Table 3.1.
3.3
Search for T em perature Steps
One can directly search for tem perature steps from the CMB sky map by their
topological configuration or pattern.
We define an algorithm to search for CMB
tem perature steps produced by cosmic strings in a background of CMB adiabatic
fluctuations and (at present even more dominant) receiver noise. (For the WM AP
first-year data release, the signal-to-noise ratio for an average W -band pixel is about
0.5.) As a smoothing process, we demote 2 2” pixels of maximum resolution to one
pixel with a representative tem perature value (average) assigned. This demoted pixel
can be obtained by repeating the demotion process shown in Figure 3.5 until we reach
the desired value of signal-to-noise ratio. Since a s ~ 201/xK and anoise ~ 184/iK
for the W M AP first-year data, it is reasonable to take 16-pixels demotion or more,
because we have e r ^ ~ 104fiK and anole = anoise! (16)1/2 ~ 46/xK, and thus the
contribution from noise becomes sub-dominant. Then a vector can be derived out of
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58
four neighboring demoted pixels. The param eter V T is defined as
(VT)* =
i ( T 2 + T4 - T : - T 3)
(3.19)
(VT% =
\ ( T s + T 4 - T x - T 2).
(3.20)
We build up a vector field for the full sky th a t contains 12 x 2 16-2n V T values where
22” is the demotion factor. For each V T, the perpendicular direction to V T represents
the direction of the step discontinuity. Thus, if there is a consistent elongated line of
discontinuity, we can interpret it as the signal due to moving string. After covering
the full sky with the VT-field, we rotate the vector field by —7t / 2 to define the L-field
so that each arrow lies along the local isothermal line, with the left-hand side of the
arrow being higher tem perature. Figure 3.6 shows the L-field map for the entire sky.
The length of each arrow represents the magnitude of the tem perature gradient on
th at point, and any tem perature step greater than 100/rK is set to 100/rK to make
steps visible th a t are less than
100/iK .
It shows a long coherent structure along
the galactic plane, this is because there is a steep tem perature riseapproaching the
galactic plane.
3.3.1
E valu atin g S trin g P a tter n
W e define th e con n ected n ess of tw o neighboring tem p eratu re step s with tw o sm o o th ­
ness conditions for the heights of steps and the curve that links neighboring L values.
Component definition o f connectedness. We assume th a t the tem perature distribution of W M AP first-year d ata is approxi-
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59
*+i
Figure 3.7: Relative angles defined for connectedness. Here
is a unit vector along
the line th a t connects Li -1 and L,. The angle 6iti+1 is defined by the angle between
Di and Di+1, 9iti+1 = cos_1( A ■A + i)-
-l
mately Gaussian, / t (AT) =
as (2ty)1^2
exp { | [(AT - //) /a s ]2}, with fj, ~ 25/iK
and a s — 200//K. Starting with this tem perature distribution, fr ( A T ) , after taking
n -pixel demotion, we can derive the distributions of L x and Ly,
L\
/K= e 2
a„V27r
1
fcomp{Lx) —
(3-21)
where o \ is the variance for the tem perature distribution for n-pixel-demoted pixels,
for example, cri6— 104/rK.The component L y also has almost the same distribution
as equation (3.21), so we use equation (3.21) for both L x and Ly.
This gives the
probability distribution function for change of the x-component, A x = Llx —L1^ 1, by
/ a ,
( 4 .) =
(3.22)
and the same function for the y-component. Since a pattern formed by a moving
string should have a relatively constant height of step along the curve of pattern, we
can impose a condition for connectedness as
|A X| < A LC, | A J < A L C.
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(3.23)
60
Galactic Coo rdinate s
52.3
48 . 7
45.1
3 7 .i
34.2
31.2
36.8
47.9
42.3
53.5
59.0
l(deg)
Figure 3.8: Long tem perature discontinuity pattern found at (I = 45°, b = 43°) in
Galactic coordinates. The vector field is derived from the 16-pixel-demoted tem per­
ature sky map. This sequence has its maximum likelihood defined in equation (3.26)
at 5T = 12/jK .
Then, the probability th at two adjacent vectors L* and Li+1 meet these conditions is
given by
J
pALc
Pi,i + 1
=
1 2
f ^ ( A x)d A t
I—ALC
t(
(3.24)
where A L c is the maximum value allowed for A x and A y. We set another condition
for connectedness
0i>i+i = cos-1 [ p i ■A + i) < 0c
(3.25)
for a sequence to avoid too sharp turns (Figure 3.7), where 9C is the maximum angle
—*
allowed for 9iyi+1 to claim th a t
3 .9
and L i+1 are connected. Figure 3.8 and Figure
show examples of patterns th a t comply with the definition described in equations
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61
Galactic Co ordina tes
2 8.0
24.3
?
20.5
13.0
121.9
124.6
127.4
l(deg)
130.1
132.9
Figure 3.9: Short pattern with 16-pixel demotion from the W M AP first-year data
located at (I = 127°, b = 20°) in Galactic coordinates. The 5Topt for this pattern
is 0/xK, i.e., no preferred bias is found th a t is against its apparent feature. This is
because the tem perature-step vectors on the right half of the curve have the wrong
direction relative to the expected string velocity at this part, while those on the left
half are aligned correctly, and these two opposite contributions cancel out the bias.
(3.23) and (3.25) found in the W M AP first-year data. Here, we set A L c = 83.3/zK
for each data set and 0C = 7t / 6 . The <7 i 6 values for selected d ata sets are shown in
Table 3.2.
Likelihood o f sequenceGiven a sequence of tem perature steps defined in the previous paragraph, we can
estimate its likelihood for a signal due to a moving string. If a long tem perature
step is formed solely by a moving string, the L values assigned on the curve should be
tangent to the curve. After allowing contamination by noise, by adiabatic fluctuation,
or by other possible sources, each L on the curve will be off from the local tangent.
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62
DATA set
W M A P lst-year
simulation STCm b + STnoise
simulation S 1 white noise
a i(n K )
200.52
200.57
200.51
a i(n K )
126.61
122.50
100.41
<Tie(p>K)
96.28
83.30
50.15
064 (/^AT)
78.84
55.60
25.04
Table 3.2: D ata sets used in pattern search.Here an represents (variance)1/2 for the npixel demoted sky map. The simulation d ata STcMB+STnoise have the same variance,
mean, power spectrum and the same amount of noise as W M A P first-year data, while
the simulation data <5Twhite noise contain the same variance and mean as the W M A P
first-year data.
However, if the contamination is not overwhelming, there should be still a bias seeded
by the string. We define the bias of a sequence with N connected arrows quantitatively
in terms of a relative likelihood function as
£$2,(8T)
=
fi)2/2^
(3.26)
»=1
: left —handed curve
+1
4>i ~
0
—1
: straight line
: right —handed curve
where STi is a tangent vector at the ith grid point on the curve at which
is located
and \5Ti\ = ST. Here fa is a phase factor defined at the ith grid point on the curve
to give the correct direction of the string velocity on the curve.
If the sequence
turns left (right) locally at the ith grid, then 0 ; = + 1 (—1 ), and it is zero when the
sequence is straight at th a t point. On both the head and tail of a sequence, the phase
factors are set to zero. This is an approximate prescription to describe a realistic
—
¥
^
model of string motion. When Lj values are perfectly coincident with faSTi, then
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63
1000
800
s o lid : WMAP 1 s t - y e a r d a t a
d o tted
: SIMULATED SKY MAP
d a s h e d : WHITE NOISE
a.
o
I—
600
o
O
LU
200
0
100
50
<5T
(H e ig h t of S te p ,
150
fiK)
Figure 3.10: Distribution of STopt at which the likelihood function defined in equation
(3.26) becomes maximum for a sequence. Each d ata set takes 16-pixel demotion. We
included sequences th a t contain five or more arrows connected. The simulated sky
map has the same number of pixels, power spectrum, and variance as those of W M AP
first-year data. W hite noise contains the same number of pixels and variance as those
of W M AP first-year data.
C^Js(ST) becomes 1 and if L* values are off by either direction or magnitude or both
from faSTi values, then C^Js(ST) decays exponentially. Thus, if there is a nonzero
ST th a t gives a maximum C^Js(ST) and we define it as STopt, then it is relatively
m ore likely th a t there is a constant tem perature step with height 8Topt, embedded
in the sequence. Figure 3.10 shows the comparison of results between actual data
( W M AP first-year data) and a simulated sky map. The simulated data contain the
same number of pixels, power spectrum and variance as those of W M AP first-year
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64
data. We can estimate from the curve the height of the tem perature discontinuity in
equation (3.1),
6 T < 4 5 fjiK , 95%CL
which is equivalent to the symmetry-breaking scale t)sb
3.4
(3.27)
0.94 x 1016GeV.
C onclusion
We have investigated W M AP first-year d ata to search directly or set a limit on
cosmic strings. For the statistics of the full sky map, we set the limit of contribution
to variance by the strings as 0 < astring < 80.26/uK which gives upper limit on
the deficit angle of 0 < S = SttG h j P < 2.9 x 10“5. In addition, we set up a model
with a random distribution of strings with random orientation with which the relation
between o string and the height of the tem perature step SnG/i'yfJ can be found. A limit
on the deficit angle is obtained from the model of uniform distribution of strings, 0 <
5 — 8nGiij)3 < 7.3 x 10-5 . This corresponds to the symmetry-breaking energy scale
tjsb
~ rripi (Gfi ) ^ 2 ~ 2 x 1016GeV. We developed a pattern search algorithm th a t can
visualize the landscape of the CMB tem perature variation of the sky. there were some
fairly long tem perature rows, but we did not find any compelling pattern of cosmic
strings predicted by theory. Instead, by considering the distribution of the heights
of the tem perature discontinuities, we estimated 0 < 5 = 8irGfi'y(3 < 1.5 x 10~5, or
equivalently tjsb ~ 0.9 x 1016GeV.
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65
C hapter 4
V alidity Test o f P attern Search
Based on E. Jeong & G. F. Smoot, “The Validity of The Cosmic String Pattern
Search with The CosmicMicrowave Background”, Astrophys. J. Lett. 661: L I, 2007.
4.1
Introd uction and M odeling
Cosmic strings are one of the relic structures th a t are predicted to be produced
in the course of symmetry breaking in the hot, early universe. Their discovery would
be an im portant landmark for high energy physics. The quest for cosmic strings has
been conducted in two ways: a statistical method and a direct search for individual
cosmic string. On the statistical side, many studies agreed th a t the contribution
from cosmic strings to statistical observables such as the power spectrum is at most
10% [96, 141, 42, 27, 142], reconfirming th a t cosmic strings played a minor role, if
any, in making the universe. Other workers have set upper limits on the cosmic string
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66
Cg
square patch of
CMB sky map
constant background Gaussian fluctuation
temperature
(CMB + noise)
temperature step
by cosmic string
Figure 4.1: Decomposition of a CMB anisotropy sky map patch. A square patch is
decomposed into the uniform background tem perature (T0) + Gaussian fluctuation
(° g ) + discrete tem perature step ( ±A) The number of pixels in a patch is N 2.
parameter G/t [55, 93, 63, 82, 117, 98, 43]. In this Letter, we introduce a technique for
estimating how strong a signal from cosmic strings has to be in order to be identified
unequivocally. The detecting power of this technique is calibrated by applying it to
simulated anisotropy maps based on reasonable modeling. In the last part of this
Letter, we apply this technique to W M AP 3 year W -band d ata set 1 and analyze the
implication of the results. A cosmic string can leave discrete tem perature steps in a
CMB anisotropy map due to Kaiser-Stebbins effect [6 8 ] with the height of a step ST
given by
ST = 8TvGfj,jspsT h ■(v x s)
where
(4.1)
= (1 —/ ^ ) -1/2, (3S — vs/c and T — 2.725K is the universal background
tem perature of CMB. However, those step patterns are probably obscured by instru­
mental noises and other physical structures of anisotropies. We consider a square
patch of the CMB anisotropy sky map th a t extends to a size of the horizon at the
1S ee h t t p : / /la m b d a .g s f c .n a s a .g o v / p r o d u c t /m a p /
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67
time of recombination so th a t the conic space-time formed by a cosmic string can
be in effect within the region. Then another square patch of the CMB anisotropy
sky map th a t contains a segment of moving cosmic string can be decomposed into
three parts, as illustrated in Figure 4.1 (1) the uniform background tem perature (T0),
(2) the Gaussian fluctuation (variance <7q ), and (3) the discrete tem perature step
(±A) from a moving cosmic string. Here, A is related to ST in (4.1) by A = S T j2.
The uniform background tem perature T0 comes from superhorizon-scale primordial
fluctuations. Since we pick a horizon-sized region, the super-horizon fluctuations will
appear to be a constant tem perature shift for the whole patch. Smaller scale (sub­
horizon) fluctuations and instrumental noise add up incoherently to form a Gaussian
fluctuation with the variance
— aCMB + anoise> where o'cMB and (y\oise are vari­
ances for fluctuations of primordial origin and instrumental noise, respectively. We
introduce five parameters th a t characterize a square patch of a string-embedded sky
map, T0, ctg, A, p (blueshifted pixels/total pixels), and 8 (orientation of step). We
concoct a simulated patch of the CMB anisotropy map by assigning arbitrary legiti­
mate values to the parameters and adding the three components illustrated in Figure
4.1. To recover these parameters from the CMB anisotropy map where the step p at­
tern is intermixed, we employ five observables (p, A, 8, ac, T0) th a t can be expressed
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68
in terms of
mean
: r = Ar“ 2 ^ T i
(4.2)
i
dipole moment
: d = TV” 3
2 ]^
(4-3)
i
A —inertia : A = 4iV-3 ^
T) |d • r*| / |d|
(4.4)
(2) - r ) 2
(4.5)
i
variance : er2 = N ~ 2 ^
i
where T) represents the signal at the ith pixel and r* is the two-dimensional gridded
position vector for the ith pixel defined in a square patch.
In the absence of the background fluctuation (oq = 0), the following analytic
expressions for the characteristic parameters return the exact input values:
p =
2d — 4 (|A —r | + A —r)
Ad — |A —t \
(4.6)
A
( | A - t | - 4 d )2
2 (2d — |A —r|)
(4.7)
=
9 = ta n - 1 (dyjd x)
oG =
To =
[rr2 - 4 A 2p ( l - p ) ] 1/2
t
—A (2p — 1).
(4.8)
(4.9)
(4.10)
W ith the background fluctuation turned on, an observed characteristic param eter
Qi,obs is expressed as an unbiased estimate Qi plus a Gaussian error oq1
Qi,obs = Qi±CTQi-
= p, A, 9, crG, T0),
(4-11)
The errors (op, oA , oe, oaa, aTo) present in equation (4.11) are measures of how pre­
cisely the information on a tem perature step screened by background fluctuation is
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69
recovered. Figure 4.2 shows the behaviors of errors as functions of A for three different
pixel numbers for a patch. The standard deviation of the background fluctuation cho­
sen here is the mean standard deviation of patches with angular radius Or = 1.8° (the
angular size of the horizon at the recombination for the ACDM model) for WMAP
3 year W -band data set with the Kp2 mask applied. The most pronounced feature
of the graphs is th at the error bars are wildly undulating for small A compared to
(Tg with attenuating envelopes as A increases. The algorithm works poorly for low
A compared to aa such as the ±90° error on the orientation estimate. Even the
breakdowns of the algorithm do happen in the orange (shaded) region, resulting in
erroneous parameter values such as negative A or p not in the range between 0 and 1.
Simulations with collapsed results are not included in the statistics shown in Figure
4.2 since those cases are evidently the ones with insufficiently strong signals against
background fluctuation. As A increases, we begin obtaining the computed parame­
ters th a t are very close to the true values with acceptable errors (dubbed the “good”
results), and it allows us to recover the tem perature step parameters faithfully. We
also find from Figure 4.2 th a t a patch with more pixels works better with faster a t­
tenuation of uncertainties. Table 4.1 displays the performances of the algorithm for
the values of A at the borders, above which errors for the tem perature step (cta) do
not get any better. We use circular patches for WMAP data analysis rather than
square patches because of the computational advantage. Algebraic expressions for
the observables given in equations (4.6) - (4.10) work very well for circular patches
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70
8 x 8 pix e ls
0
50
100
A (/*K)
14x 1 4 p ix e ls
150
200
>0
10 0
A (MK)
150
2 8 x 2 8 p ix e ls
50
100
A (yuK)
ISO
100
A (/*K>
150
200
50
100
A i fi K)
150
200
50
100
A 0*K)
150
200
o.oMliniimi
[ l l l H -H-H i i i i i i i i
0
50
200
200
linnmirrrirr
0
50
10 0
A (fxK)
150
200
50
A(^K)
100
15 0
200
50
10 0
A (/*K)
150
200
50
100
A (/zK)
15 0
200
0
60 g
401
20Wm
oM
ii ii iin ii iii iiiim ii iiim iiim
-2 0 B
50
10 0
A {,*K)
150
200
50
100
A {fxK)
15 0
200
50
100
A {fxK)
150
200
Figure 4.2: Evolution of errors of characteristic parameters as functions of A with
input values of parameters (To = 10.0\xK, p = 0.5, 9 = 0, era = 116.0p K ). Left pan­
els: 8 x 8 pixels in a square.Middle panels: 14 x 14 pixels. Right panels: 28 x 28
pixels. For each size of patch, the number of simulations performed is 10,000. Orange
(shaded) regions indicate the ranges of A for which the step feature in a patch is not
obvious compared to its background fluctuation (oq = 116.0p K ). The result on A c(,s
is biased upward because the simulations with erroneous values of A ohs (A 0bs < 0)
were not included in the statistics.
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71
with negligible numerical differences for the case of square patches. A square patch
with 28 x 28 pixels at the normal resolution of WM AP W -band data covers nearly
the same area as the circular region with angular radius Or = 1.8°. We performed a
further detecting power test with 28 x 28 pixels patches and found empirically th at
the relation between the critical value of A, A c, above which “good” results start to
come out, and oq is
Ac ~ 0.25aG.
(4.12)
The different choices of input values for the parameters p, 0, or T0 showed no notice­
able difference in the evolutions of errors.
4.2
A pp lication to WMAP
An observed CMB anisotropy map is an aggregate of various independent modes of
perturbation ranging from tiny sub-horizon scales to super-horizon scales well beyond
the correlation length. As illustrated in Figure 4.1, fluctuations with larger or smaller
scales compared to the size of a test patch are neatly prescribed, but the intermediate
scales whose wavelengths are comparable to the size of test patch will appear to be
continuous tem perature tilts th a t also give plausible values of the step parameters.
One drawback of this algorithm is th at it does not distinguish between a discrete
tem perature step and a smooth tem perature slope. However, this shortcoming can
be easily fixed: If an apparent tem perature step is detected at a spot on the map,
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72
Param eter
p ..................................
A ( fiK )
9 (deg)
aG{ p K )
T0( p K )___________
p ..................................
A (/xK)
9 (deg)
(tg {(jK )
T0( p K )___________
p ..................................
A (/xK)
0 (deg)
aG{ p K )
T0 (/xK)___________
Input
O utput (1 a)
Patch with 8 x 8 pixels
05
0.5 ± 0.1
80.0
83.4 ± 16.6
0.0
-0.1 ± 12.3
116.0
113.0 ± 12.0
100___________________ 9.7 ± 18.9
Patch with 14 x 14 pixels
05
0.5 ± 0.1
45.0
47.1 ± 9.5
0 .0
- 0 .2 ± 12.6
116.0
115.0 ± 6.2
100___________________ 9.6 ± 10.9
Patch with 28 x 28 pixels
05
0.5 ± 0.1
25.0
25.8 ± 4.7
0 .0
0 .2 ± 11.2
116.0
115.7 ± 3.0
1O0___________________ 9.9 ± 5.3
Table 4.1: Simulation results at the critical values of A above which a As are small
enough and do not decrease any more.
we repeat the analysis with a half-sized patch at the same spot. If the structure
is a continuous slope, then it would return half the value of A 0{,s than the previous
result, while for the signal from a discrete step, the returned A o6s should stay the
same within the error. We conducted the step signal search through the WMAP
3 year W -band data set and found 193,160 unqualified signals against background
fluctuation (the constraint in eq. [4.12] is not applied), 129,049 qualified steps+tilts
(signals th a t meet the constraint eq. [4.12] i.e., A 0(,s > 0.25<tg), and 12,330 qualified
discriminated steps. The A 0bs value of qualified steps is in the range
(18.3 ± 3.0)p K < A o6s < (115.4 ± 6.0) p K
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(4.13)
73
WMAP 3year W -band
io r
.... U nqu alifie d si gnal
._
Qualified s t e p s + t i l t s
Qualified s te p s
6
-
4-
2
-
50
100
150
200
A (MK)
Figure 4.3: Distributions of A 0bs detected in the WM AP 3 year W -band data set. The
number of step signals found are 193,160 unqualified signals ( dotted curve), 129,049
qualified steps+tilts (dashed curve), and 12,330 qualified steps (solid curve).
as the curves show in Figure 4.3. Patches with radius 0R = 1.8° in the W M AP 3
year W -band have a background fluctuation og less than 176 //K in the Kp2-maskcleared region. This means, at the worst case, we can identify a step signal as low as
176/u/C'/4 = 44f i K with moderate errors. Therefore, if there are cosmic string signals
with A > 115.4 /jK and if they are located in the available region (out of the Galactic
plane or the Kp2 masked region), they would not be missed. Thus, we can set an
upper limit on the cosmic string signal
A = 4irGneffjsPsT\ cos 0| < 127.4/xA",
95% CL
(4.14)
where 0 is arbitrary. Thus, with (7 S(3S) ~ 0.15 and fief f « 1.5// [136], the upper limit
of the cosmic string param eter can be estimated as Gfi < 1.6 x 10“ 5 at 95% CL.
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74
4.3
C onclusions
We have presented and tested a new technique to directly discover cosmic strings
via the patterns they would produce in a CMB anisotropy map. We found th a t the
minimum magnitude of the step signal th a t is required to be unequivocally identified
is 44 fiK for the WMAP 3 year W -band data set. This algorithm can be used to
crop the reliable step signals from the CMB anisotropy data, and it will serve as
the valuable ground work for future pattern searches with more refined data, such as
further W M AP data releases or P L A N C K data.
Acknowledgments. Computer simulations and data analysis with WMAP d ata set
were done using the HEALPix 2 [47]. This work was supported by LBNL and the
Department of Physics at the University of California, Berkeley.
2See http://healpix.jpl.nasa.gov
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75
Part III
Inflation and N on-G aussianity
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76
C hapter 5
N on -G au ssian ity from Inflation
5.1
A B rief D escrip tion o f Inflation
Modern cosmology has made great progress in both theoretical and experimental
sides. Its prime power th a t has led to such a phenomenal success will be the Relativity.
But, interestingly enough, relativity raised another big question which is crucial and
must be resolved: Large Scale Smoothness (also known as Horizon Problem.) The
observable universe today is so large th at even the whole age of universe provides
not enough time for any particle to travel across the universe. However, CMB and
many other observations consistently show th a t the universe in large scale seems to be
practically homogeneous even though it doesn’t have to be. As an example, today’s
sky contains about ~ 1 0 5 causally disconnected patches at the last scattering surface
while they have same tem perature down to 10-5 level. Though we don’t have any
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77
physical law th a t bans this accidental coincidence, it is very unlikely and unnatural
to happen just by chance. So, we should ask what is the underlying mechanism for
universal (at least within the current horizon) synchronization. The idea o f’’Inflation”
provides a great big picture to break the mystery [50]. It may also solve two big
V,
questions such as Flatness Problem and Unwanted Relics Problem. Inflation gives
a natural explanation how the universe acquired smaller scale inhomogeneity which
played the role of the seed for structures such as stars, galaxies, clusters, voids and
larger structures present in the universe today. Inflation has its own problems and
many works need to be done or we may have to introduce new concepts th at we do
not understand yet to complete the inflation scenario.
5.1.1
H orizon and H u b b le R adius
There are two useful meter sticks th a t can tell us whether two particles are close
enough to interact and stay in therm al equilibrium or not.
1. Co-moving Horizon
Co-moving horizon, 77, is defined by the maximum co-moving distance th a t a photon
can travel by the time t,
(5.1)
where a is the scale factor in FRW-metric and H is the Hubble parameter. The
implication of co-moving horizon is straightforward: when two particles are separated
by a co-moving distance x and if X > V then, they could never have communicated
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78
each other.
2. Co-moving Hubble radius
In an expanding universe, every point recedes from all other points by the expansion
of space. So, for two particles separated by a co-moving distance y and if, for a2 > ai,
c(t(a2) - t(ai)) < y(o 2 - 01 )
(5.2)
then, photon cannot catch up the expansion of the space and two particles are causally
separated i.e, cannot communicate at this time a\. When (5.2) goes to differential
form, it becomes cdt < yc/a or c ~ < y and we can define the co-moving Hubble
radius R h as
at
da
c
a
c
aH
(5.3)
Using the Friedmann equation, R h can be expressed as
1
aH
where f
H oy/tlm a-1 + Q,ra ~ 2 + Oa a 2 + (1 —Qq)
PC today and fl 0 = Y i ^ i -
(5.4)
^ nothing happens to the vacuum energy
n A, as the universe expands R h will increase for a while and start decreasing and
eventually goes to zero i.e, we will be more and more isolated from other parts of the
universe and won’t be able to encounter creatures who might be living somewhere far
away. For the ACDM model with energy contents (Q,m — 0.3, Ua — 0.7, fU ~ 10-5 ),
we are already in shrinking phase (in co-moving coordinates).
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79
5.1.2
E ntropy
Entropy density s is defined by
•= £ = ^
(« )
where S is the total entropy contained in volume V. Since relativistic particles give
most of contribution to entropy density, we can take the approximation p 4- p —
Pr + P
r
— | Pr = ^&9 s T s , where g s is the effective degree of freedom for entropy.
Since s oc T 3 oc a -3 , .s" 1 is proportional to volume and we will use s _1 as volume
when it is convenient. Entropy contained in the horizon is given by
SH OR =
471" 3
3
S— t
0 .0 5 ^ 1/2 ( ^ ) 3
~
, t < t EQ
(5.6)
<
3 x 1087(fio52) - 3/ 2(l 4- z)~z!2 ,
t > t EQ
where tsQ is the m atter-radiation equality time. The amount of entropy within the
horizon today is S h o r ^ o) ~ 1088, while S HoR(tdec) ~ 1083. This means th a t if
entropy is conserved or the universe undergoes adiabatic expansion, today’s horizon
contains ~ 105 causally disconnected patches at tdec. Moreover, at nucleosynthesis
time, SnoRitnuc) ~ 10 63, roughly 1025 times less than today’s, there are ~ 1025
uncorrelated patches at tnuc in our horizon. But, they all look almost identical.
5.1.3
A ccelera ted exp an sion
One way (or maybe the only way) to solve the Large Scale Smoothness problem
is to put all the causally disconnected regions until today within the horizon into
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80
a single horizon and make them reach in thermal equilibrium at some earlier time
and then let them lose contact one another. To do this, we need to decrease R H so
th at particles in causal contact can be disconnected later. This can be realized by
changing the
in (5.4). Q \a 2 in (5.4) is the only term th at makes R H smaller as a
increases. If
has enormous value compared to all other energy contents in earlier
universe, R H will be decreasing at th at time because
>//
1
oc a 1
.
(5.7)
The condition for decreasing Hubble radius is equivalent to acceleration of expansion,
dRn
dt
d ( 1 \
'
1
dt \ a H J
d ( 1\
dt \ a )
h
<0
a2
<->
- n
a>0
.
/ ro\
(5.8)
When Qa is dominant, scale factor increases exponentially while H becomes almost
constant,
tt2
R
SnG ,
.
k
= - g - ( P m + Pr + P a) - ^2 -
8ttG
,
~ 3 ~ Pa = c o n s t ■
(5 -9 )
.
For this idea to work to solve the smoothness problem, there must be a huge drop
in Hubble radius. Let a* be the scale factor at the onset of exponentialexpansion
(inflation), then Rn(ai) must be greater than today’s co-moving observable universe
(= H o 1) so th at all the uncorrelated regions until today were in causal contact with
one another at the time ai. This inflation cannot take place in radiation or m atter
dominated era because, from the Friedmann equations,
a
47tG ,
„ .
- = ---- ^ - ( p + 3p) =
47rG
— p (l + 3w)
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(5.10)
81
and for radiation and m atter, w = 1/3, 0 respectively and they always give de­
celeration. But vacuum energy can have w < —1/3 and give rise to acceleration.
Cosmological constant has the equation of state w = —1 and it can cause accelera­
tion.
5.1.4
Inflation by generic scalar fields
Among the various ideas, scalar field is by far the most favorite candidate for
inflaton who is responsible for the inflation. Regardless of its identity or specific form
of its potential, there are several common features th a t an inflaton must have in order
to explain the cosmological questions discussed above : slow-roll, coherent oscillation
and reheating process.
In analyzing a scalar field as an inflaton, we make the following assumptions:
(1)Vacuum energy is the dominant energy during the inflation,
tt2
8ttG
k
all other forms of energies will redshift away quickly as inflation goes on.
(2)Spatially homogeneous 0-field, 0 ^ a where a = global minimum of potential V.
(j)i
is the initial field value when inflation starts.
(3) The scalar field 0 can be written as
0 (f) =
(j)cl + A 0 Q M ,
A 0 q M
0cZ)
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82
where <pci is the classical part of the field and A (f>QM is the small quantum correction.
1. Slow-roll
Action for a scalar field is given by
(5.11)
where y '—g is the determinant of metric tensor, it becomes
for a flat FRW-
metric. Then we can get the Lagrangian density and energy-stress tensor directly
from (5.11),
(5.12)
3* "
=
(5.13)
i t o l f e i - fi""C
The energy density and pressure of </>-field are respectively
P<!> — 2*^2 ^
p$
=
\ i 2 - V ( 4 ’)
I
- \ m
(5.14)
2
(5.15)
The terms in the square bracket will decrease exponentially as a(t) undergoes expo­
nential expansion. Should (V 0 ) 2 be dominant over cf? / 2 and V(<j>), P4, =
and
a(t) ~ t, no inflation occurs. The equation of motion involved with (5.12) is
4>+ 3H(f) + F^
where
—0
(5.16)
is the decay width of (^-particle which is expected to be produced at the
end of the slow-roll. The decay term F^cj) is not operative during slow-rolling. And
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83
condition for slow-roll forces to drop (p term because the friction term and V ' are
dominant. This requirement can be written as
\V"\
«
V\
|mpi — |
(3i / ) 2 = 2Airp(j>/Tn2pl
(5.17)
\/487r .
(5.18)
During the slow-rolling, a ~ eHAt and we define the e-folding as
ft2
N((pi -»• fc )
H dt =
J
p u
r
q r /2
-jd<p = J — — d(j>
=
jf
~
8 txGV
3H 2
-------- = --------> 1 .
V"
V"
in ,
(5.19)
In the literature, it is custom to define the slow-roll parameters which are required
to be much less than unity as follows 1
V '\
16?xG \ V J ’
1 V"
8irG V
(5.20)
2. Coherent oscillations
After the slow-roll ends, 0-field starts oscillating around global minimum of the po­
tential
<j> = o .
The decay term
T ^cp
the time interval tosc ~ H ~ l ~ ^
is turned on and
(p
is no more negligible. During
< t < ~ I ^ -1, ^-particle is mass-produced and it
becomes brief matter-dom inated era and a ~ f2/3. By multiplying
the differential equation for
=
\{(p 'z )
(p
to (5.16), we get
as
+ (3 H + T*)p* = 0
(5.21)
l Some authors define the slow-roll parameters slightly differently. In [29], Ssr = esr —f]sr is used.
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84
and its well-known solution is
P<t>
0—
&
tosc')
(5.22)
where aosc and tosc are the scale factor and time at which coherent oscillation begins.
3. Reheating process
Assuming th a t ^-particle decays into light (relativistic) particles which can be treated
as radiation, we introduce two more equations along with (5.21),
(5.23)
Pr + 4 H pn — r > ^
Hz =
8 ttG
(5.24)
+ Pr )-
The solution to (5.23) is
-5/3'
m h T4>
107rt
- 5 /2 '
-3 /2
J 7 l p j I ' fp ( J
Q*n
1
(5.25)
-
Pn increases rapidly to mpiT^a2 soon after aosc and then, it gradually decreases as
a -3/2. So, the maximum tem perature reached here is Tmax ~ (m ^r^cr2)1/4. This
process is highly non-adiabatic and entropy increases as
(5.26)
5 = a 3T 3 ~ a \ p f f oc a3" ! ! = a 15/8.
At t ~ T^,
, 0 -particle starts decaying quickly and ordinary radiation-dominated era
begins. Reheating tem perature is usually referred to as the tem perature at t ~ T^, 1,
t rh
= T (t = r v 1) ~ o.55^" 1/ 4(mPi r ^ )1/ 2
.
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(5.27)
85
T r h depends on
rather than the inflation scale a. Initial vacuum energy is ~
and radiation energy left behind is ~ T r H. Thus, for a
cr4
TRH, most of the initial
vacuum energy is red-shifted away and only a small portion of it is converted into
radiation.
5.1.5
e-folding
There are two stages of expansion between the onset of inflation and reheating:
(1) a{ —» aosc, A-dominated era
n
— n ,pH(tosc—U) — n p Ntot
**OSC -----
(2 ) aosc —►a,RH, matter-dominated era
3
P o s c a o sc ~
3
_
PRH a RH
> a RH ~
°
V /S
— a o sc I rn
j
\
RH /
Q>OSC I
J
\ PRH J
So, the total change of scale factor is given by
4/3
aRH = dieNtot
■
(5 -2 8 )
/
Thus, the initial inflating patch with physical size ~ JT_1 is expanded to eNtot I
\ 4/3
J
H ~ l . Using this result, we can compute the total entropy contained in the inflating
patch at T r h ,
3
4/3
'patch
eNtot (
I rh
1
H-1
3
7^3
^
1rh~ 6
Ntot
m Pl
u ^Trh
( c OQ)
•
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(5 ’29j
x
86
Spatch must be at least equal to the amount of entropy inside today’s horizon (~ 10 88)
to solve the horizon problem. Then, we can find the necessary e-folding,
Spatch > 1088 -*■ N tot > 68 + \ In — + \ In
3 m pi 3
m pi
(5 .3 0 )
As an extremely large e-folding case, let’s consider a Planck scale inflation and efficient
reheating process,
a
~
T
r h
~ 1019GeV, necessary e-folding becomes N tot > 6 8 . For
the other extreme case, when cr ~ T r h ~ 102MeV, then N tot > 22.
To solve the flatness problem, total entropy within radius of curvature must be greater
than th at of horizon today: S curv > S h0• Since the radius of curvature at inflation is
{R c u rv )i
=
| n i _ 1|l/2 5
S cu rv
(R
cu^
r h
T
r h
~
/ rr N\ 4/3
H
eNtot(
\ T h h ,I
,3Ntot
K
rpS
1 RH
1
(5.31)
ct^Trh | a - l | 3/2 '
So, the minimum e-folding required for flatness problem is
N min = 6 8 + | l n — + i l n ^ +
3 TTlpi 3
TTlpi
2
5.1.6
- 1|.
(5.32)
P ertu rb a tio n s
A perturbation mode with scale A generally crosses out of the horizon at
N x = 45 + In —
+ - In 1 0 i 4Gey + 3 ln i 0ioG eV
'
(5‘33)
During the inflation, <^-field can be regarded as a massless particle because m 2 =
\V"\
H 2. Then, for a minimally coupled massless scalar field, quantum fluctuation
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87
(in Fourier mode) can be given by
< ^ s - i'- ,S
|* 1’ - ( £ ) ’ ■
<5-34>
Fluctuations are described by the equation
6(f>k + 3H5<fik + k 25<f)k/ a 2 = 0
and its super horizon solution is 8(j>k = constant i.e, as a mode k crosses out of the
horizon, the fluctuation freezes in. For modes 1Mpc ~ 3000Mpc, they give nearly
scale invariant spectrum. Amplitude of perturbation should be limited to 10-4 or less
to be consistent with CMB measurements. Gravitational fluctuation gives interesting
constraints on inflation scale. Metric fluctuation can be given by (A h)k — yyy and it
must be less than 10- 4 from the observational constraints. Then, H < 10_4mp; and
from H 2 =
energy scale of inflation is ~ p ) ^ < 10~2m pi, moderately below
the Planck scale.
5.2
N on-G au ssian ity as A Test o f Inflation
Inflation has provided excellent answers to loose ends of the standard hot big
bang model and it is now a crucial feature of the hot big bang scenario. W ith its
compelling consequences, inflation has become one of the hottest topics in both theory
and observation. A number of viable theoretical models of scalar field dynamics for
inflation are in circulation. The non-Gaussianity of primordial perturbations is one of
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the prediction of inflation th a t is possibly feasible for observation with current or near
future data. CMB anisotropy is the oldest signal from the big bang available so far and
it is believed th a t if any non-linear/non-Gaussian perturbation was produced during
inflation, it should have affected on the CMB anisotropy. The curvature perturbation
from inflation can be related to the anisotropy of CMB as
J-o
- vM x)
(5-35)
where T0 = 2.725K is the thermodynamic tem perature of CMB today, r/T is the
radiation transfer function and $ is the curvature perturbation. For the super-horizon
scale, the Sachs-Wolfe effect dominates and gives r)t = —1/3 for adiabatic fluctuation.
The primordial curvature perturbation $ emerged from quantum fluctuation itself
is not necessarily Gaussian. In addition to this, the transfer from $ to tem perature
anisotropy described in (5.35) can also cause non-Gaussian contribution. Taking these
effects into account, the curvature perturbation can be neatly prescribed as [76]
$ ( x ) = $ L(x) +
where
$ l(x )
/jv l[$ £ (x )
- ( $ l( x ) ) ] + 0 ( f NL)
is an auxiliary random Gaussian field with its mean
($ l(x ))
(5.36)
= 0 and the
variance ( $ | ( x ) ) . The non-Gaussianity param eter f NL collectively represents various
non-lincar cou p lin g effects.
Slow roll inflation generally produces tiny non-Gaussian perturbations,
| / jv l |
0 (1 ) while the detailed degree of non-Gaussianity depends on models [23, 20, 3,
39, 45, 80, 107, 108]. For a generic slow roll inflation model, the non-Gaussianity
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89
parameter can be expressed in terms of slow roll parameters defined in (5.20)
f NL ~ 3e - 2V.
(5.37)
While inflationary non-Gaussianity stays tiny f ^ latwn) ^ 0(e,t]), post-inflationary
processes such as preheating can greatly enhance non-Gaussianity [36],
0 (1)
,,
0(10).
,
i
~
This would make the inflationary non-Gaussianity sub-dominant
ri
■
Aobs)
Ainflation) , pfpost)
+ Jnl ■
m the observed non-Gaussianity f NL = j NL
Observations of J'ni . mostly have been performed using the CMB anisotropy data
and the progress of the results is line with the improvement of the quality of data.
Some representative results of the estimates on J nl are summarized in Table 5.2. The
Experiment
COBE DMR 4-year [74]
MAXIMA-1 [109]
WMAP lst-year [75]
WMAP 3 year [124]
Method
bispectrum
bispectrum
bispectrum
Minkowski Functionals
bispectrum
Minkowski Functionals
Limit on f NL
I/jvlI < 1500 (6 8 %CL)
\fNL\ < 944
(95%CL)
-5 8 < / n l < 137
(95%CL)
\fNL\ < 139
(95%CL)
-5 4 < f NL< 114
(95%CL)
—59 < f NL< 73
(95%CL)
Table 5.1: Limits on f nl from some representative observations.
minimum required magnitude of / ml for inflationary non-Gaussianity to be detected
in the past and ongoing experiments are reviewed in Table 5.2 [76]. In this estimate,
the effects from cosmic variance, instrumental noise and foreground are included.
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90
Experiment
COBE
WMAP
PLAN C K
ideal
f^ L (bispectrum)
600
20
5
3
f ^ L (skewness)
800
80
70
60
Table 5.2: The minimum necessary (primordial or inflationary) non-Gaussianity pa­
rameter to be detected for various experiments.
5.3
C alculations o f N on-G aussianity
5.3.1
M axim u m E n trop y M eth o d
We introduce a new technique th a t uses the maximum entropy theorem to probe
primordial non-Gaussianity in the CMB anisotropy data. The idea is extremely simple
and can be easily applied to numerical estimation of non-Gaussianity.
For a given continuous probability density function f (x), the entropy functional is
defined by
(5.38)
Maximum Entropy Theorem:
For a continuous probability density function f(x) defined on R with variance a 2,
S[f{x)] < ^ (ln27r<T2 + 1)
(5.39)
with equality if and only if f(x) is a Gaussian distribution with variance a 2, i.e., for
some mean p, f ( x ) = (l/y/frrfi2) exp [—(x —p )2/(2 a 2)].
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91
The maximum entropy theorem explicitly tells the unique property of a Gaussian
probability distribution that, among infinitely many continuous probability distri­
butions with a fixed variance, a Gaussian distribution gives the maximum entropy.
Thus, a distribution which deviates from a Gaussian one will have smaller entropy
than the entropy for the normal distribution given in (5.39). This idea can be readily
utilized to test a probability distribution function whether it departs from a Gaus­
sian one. The curvature perturbation $ by primordial seed during the inflation is
transferred to CMB anisotropy with the relation
(5.40)
where T0 = 2.725K , the thermodynamic tem perature of CMB today, and r)t is the
transfer parameter. For super-horizon scale, rjt = —1/3 for Sachs-Wolfe effects. It is
conventional to prescribe the non-liner coupling of the curvature perturbation as
(5.41)
where f NL is the non-Gaussianity param eter and <fig is an auxiliary random Gaussian
field with its variance ((I>g). The last term in (5.41) ensures th at the mean of $
becomes zero, (4>) = 0.
Then, the probability distribution function of the non-
Gaussian field 4> can be derived as
/
oo
fa ( % ) S D [ * - * » - f N L
- ( « 5 » ] < »„
•OO
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92
where $± are defined by
( 5
and
-4 3 )
has a Gaussian probability distribution function,
f a ($ s) =
.
1
exp
f
<E>92
(5.44)
V2^
From (5.40) and (5.41), we can find the relation between the variances of CMB signal
and th at of curvature perturbation as follows
=
(5.45)
where (TqMB = (ST^MB), the variance of CMB fluctuation of physical origin free from
noise. The variance of the Gaussian random field
vcmbi
can be expressed in terms of
r]t and f NL,
1
<*» “ 4 /fN L
(5.46)
We can use the probability distribution function in (5.42) to find an analytic cal­
culation of entropy. Figure 5.1 shows the entropy of non-Gaussian distribution as a
function of
J n l-
The variable $ is rescaled so th a t the variance is kept as 1.
A real CMB anisotropy d ata is composed of finite number if pixels and on each
pixel, considerable am ount o f instru m en tal noise is incoheren tly added to th e physical
anisotropy signal. A continuous probability distribution function of the tem perature
anisotropy is extrapolated from finite number of pixels (for WM AP W-band, the num­
ber of pixels at the highest resolution is 3,145,728). So, it is very im portant to choose
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93
1.41B95
1.41890
£
1.41885
1.41880
1.41875
-1 0 0
-50
0
50
10 0
Figure 5.1: Entropy for non-Gaussian distribution calculated from analytic expression
of a probability distribution function with mean 0 and variance 1 (red curve). Straight
line at the top represents the entropy for a Gaussian distribution with variance 1.
an appropriate bin size (or equivalently number of bins) with which the extrapolated
distribution function can represent the nature of the underlying distribution of the
data set best. W hen the size of bin is too large compared to the number of data
points, we would lose the details of the true distribution completely in the interval of
the bin size or smaller. The opposite case is equally bad because, for a too small bin
size, we would miss the overall perspective of the true distribution. Thus, it is mani­
fest th a t there exists an optimal bin size th a t is not too small or too large. Here, we
adopt a well-established procedure to select the optimal bin size for a given number
of data points [118],
For a data set {xi, i = 1,2, • • • N } with its range R, select a bin size A. Then,
the number of bin becomes n = R / A. Define variables k and v and a cost function
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94
N
102
103
104
10s
106
3 x 106
5 x 106
107
n *bin
~ 7
~ 20
~ 60
~ 230
~ 1000
~ 1700
~ 2300
~ 3200
dt*
~ 0.88
~ 0.36
~ 0.13
~ 0.037
-0 .0 1
— 0.006
- 0.0046
- 0.0033
Table 5.3: The optimal bin sizes for various sizes of data set. D ata sets used here are
randomly generated ones th a t have a Gaussian distribution f a (x) = (27r)-1^2 e- *2/2.
as follows
1 n
— 7 ki
n z—
■'
1=1
k
=
v
=
—V (ki — k)2
n 1= 1
C (A)
=
———
1
71
(mean occupation)
(5.47)
(variance of occupation)
(5.48)
(cost function)
(5.49)
where ki is the number of events th at falls in ith bin. Then, the bin size A* which
minimizes C (A) is the optimal bin size. A realistic data set has always finite number
of data points, so even if the best-fit probability distribution function for the data
set is a Gaussian one, it would deviate from the true normal distribution and thus its
entropy will be smaller than the expression (5.39). We performed computer simula­
tion s w ith p ixel num bers ap proxim ately equal to th o se o f W M A P d a ta (3 x 106) and
projected P L A N C K data (107). The results tell two remarkable features (see Figure
5.2
and Figure 5.2): first, the entropy peaks at the Gaussian distribution (J n l = 0)
and it decreases as the non-Gaussian feature is in effect ( / n l / 0), second, the dis­
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
95
persion gets narrower for the d ata set with larger number data points making smaller
non-Gaussian feature detectable.
First simulation is performed with the number
1.41870
1 .41865
^ 1 .4 1 8 6 0
►»
o0.
aV
Blue : 68%CL
Cyan : 95%CL
w
1 .4 1 8 5 5
1 .4 1 8 5 0
1 .4 1 8 4 5
-100
-5 0
0
fNL
50
100
Figure 5.2: Distribution of entropy for non-Gaussian distribution simulations with
3 x 106 pixels. Flat bands (red, orange) represent widths of entropy distribution with
6 8 % CL and 95% CL for Gaussian realizations. Curved bands (blue, cyan) show the
dispersion of entropy with 6 8 % CL and 95% CL for non-Gaussian realizations with
non-Gaussianity param eter /jv l as the operational variable. For each value of J n l ,
we performed 1,000 simulations to calculate the errors on entropy. From this plot,
the sensitivity for non-Gaussianity can be read out as I / j v l I — 50 at 95% CL and
\fNL\ ~ 40 at 6 8 % CL.
of pixels 3 x 106 which is approximately the same as th at of WM AP . The instru­
mental noise isnot added. The necessary magnitude of non-linearity param eter f ^ L
so th a t the non-Gaussian feature can be distinguished from Gaussian distribution is
estimated by
\fNL\ > 50, 95%CL
(5.50)
\/ n l \ > 40, 6 8 %CL.
(5.51)
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
1 .4 1 8 6 0 1
,
-1 0 0
,
I
-5 0
.
■
,
■
I
0
,____,____I___ i____i____i____i___
50
100
Figure 5.3: Distribution entropy for non-Gaussian distribution simulations with 107
pixels. Other conditions are the same as the simulations described in Figure 5.2. The
sensitivity for non-Gaussianity reads out as | / j v l | — 30at 95% CL and | / j v l | — 20 at
68% CL.
Second simulation is done with the pixel number 107 which is approximately the pro­
jected number of pixels of P LA N C K d ata to be released in the future. The necessary
magnitude of non-linearity parameter
/ ml
is estimated to be
\/n l \
> 30,
95%CL
(5.52)
\In l \
>
68%CL,
(5.53)
20,
the instrumental noise is not taken into account in this simulation. For the W M AP
W -b an d (94G H z) 3 year d ata, th e instru m en tal noise per m easurem ent per pixel
is given by a0 = 6.5112mK.
the sky map.
The number of measurements is not uniform over
The average number of measurements for W -band 3 year d ata is
roughly 4500 and, thus the white noise due to instrumental noise included per pixel
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97
is 6.5112/v/4500mK. When the map is smoothed in 0.9° scale by demoting 64 pixels
to suppress the sub-horizon scale features, the instrumental noise is further decreased
to 6.5112/V64 x 4500mK ~ 12/iK. Then, the variance of the signal becomes
S ig n a l =
°C M B
+
^
^ K ) 2 + (12*iK)2 = (66.1^K)2
(5.54)
where <Jc m b here stands for the variance for super-horizon scalefluctuation. The
simulation result shown in figure 5.4 is the dispersion of entropy with the white noise
included. Evidently, the instrumental noise blunts the sensitivity of the algorithm
requiring larger non-Gaussianity signal to be identified.
The estimated values of
necessary magnitude of f Ni to be detected for this simulation is given by
5.3.2
\fNL\ > 57,
95% CL
(5.55)
\fNL\ >
68%CL.
(5.56)
48,
M inkow ski F un ction als
The Minkowski functionals provide a tool for estimating morphological character­
istics for a data set which form a field in a configuration space. For a smoothly varying
scalar field defined on a ro-dimensional configuration space, th ere exist n + 1 M inkowski
functionals th a t describe the morphology of the field. A CMB anisotropy map is a
tem perature fluctuation field defined on a spherical shell with unit radius, thus there
are three Minkowski functionals. Minkowski functionals for two-dimensional sky map
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98
1 .4 1 8 7 0
1 .4 1 8 6 5
1 .4 1 8 6 0
w
1 .4 1 8 5 5
Blue (d a r k )
Cyan (lig h t)
: 68% CL
: 95% CL
1
1 .4 1 8 5 0
1 .4 1 8 4 5
-1 0 0
-5 0
0
50
100
[NL
Figure 5.4: Distribution of entropy for non-Gaussian distribution simulations with
3 x 106 pixels. In this simulation, the instrumental noise added is unoise =
6.5112/V64 x 4500mK.
are area, contour and genus respectively. Consider a smooth scalar field u ( x ) defined
on a spherical shell §2, then the three Minkowski functionals are defined as follows
[115]:
where u :i represents the covariant derivative with respect to x t defined on §2, Qv is
the excursion set over a threshold v.
Qv = {x e §2 1u(x) > i/}
R eproduced with perm ission of the copyright owner. Further reproduction prohibited without permission.
(5.60)
99
I
1
(■
•a
o
-i
0
-2
•2
a
Figure 5.5: Average Minkowski functionals for a Gaussian random field. Left: VqG\ u)
(area), middle: V^G\ v ) (contour), right:
(genus).
and dQ u is the smooth boundary of Qu.
k
=
k
is the geodesic curvature defined by
2 u :1U:2U:12 ~ u \ u :22 - U%U,n
(5.61)
( A + » a 3/2
The analytic forms of the average Minkowski functionals for Gaussian random fields
are given by [128]
r„(GV)
1
2 1“
(5.62)
T1/ 2
r fV )
where r =
r
(5.63)
( v
-i/exp I ——
(2 tt)3/ 2
V
2
(5.64)
and the scalar field u (x) is rescaled and shifted so th a t it can
have mean and variance 0 and 1 respectively without loss of generality. The expres­
sions in (5.62)-(5.64) are the surface densities of Minkowski functionals, which are
obtained by dividing actual Minkowski functionals by the surface area. Deviations of
Minkowski functionals of real d ata from those of Gaussian field will be the signal of
non-Gaussianity.
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100
I
I
I
X
-4
-8
0
8
x
-8
0
8
-8
o
8
Figure 5.6: Minkowski functionals left: Vo(r) (area), middle: Vi(r) (contour),
right: V2 (r) (genus) calculated from super-horizon scale fluctuation sky map. CMB
anisotropy signal used here is W M AP 3 year W -band with ST = J^i = 2 aimXim-
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101
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118
A p p en d ix A
Friedm ann-R obertson-W alker
U niverse
A .l
Sign C onventions and N otation s in G eneral
R elativity
There are several equivalent but different sign conventions and notations in the
literatures of general relativity. Depending on the choice of ”space-like” or ’’time­
like” notation or on the way of contraction of Riemann tensor, signs of terms in the
Einstein field equation flip. In this section, we list the signs and notations adopted
in this thesis.
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119
Q uantity
Symbol
Representation
Signs of Metric Tensors
S V
(+
Christoffel Symbols
rA * V<J
4
2
Riemann Tensor
L vpa
Ricci Tensor
R /J .I
Stress Energy Tensora
Tfxu
Einstein Field Equations
/
G tw = + 8 itG T n U
-----------------------
)
i,9vX,a + 9o\,v
vo,p
9vo,\)
up,a - i1- o.pr “vo
—
a
or avp
g afjRo.pdv = R a„av
(.P + p)up,uu - p g
Rpv — \ 9pvR — + 8 irG T fW
aFor a perfect fluid.
Table A .l: Basic mathematical quantities in General Relativity and their represen­
tations employed here.
A. 2
Friedm ann-R obert son-W alker (FRW ) M etric
The Friedmann-Robertson-Walker metric describes a model of the Universe which
has the constraining symmetries of homogeneity and isotropy.
In spherical polar
coordinates, it can be written as
k > 0,
ds2 = dt2 —a2(t)
dr2
+ r 2d92 + r 2 sin2 Od(j)2
1 —k r2
\ k = 0,
A; < 0,
closed
flat
(A-1)
open.
where a (t) is th e d im en sion less scale factor and k d eterm ines th e geom etric n atu re o f
space-time. Since a (t) is a dimensionless parameter, the co-moving radial coordinate
r must carry the unit of length and thus k has the unit of (length)-2 . The scale
factor is scaled to a(to) = 1 where to is present time. We read out the metric tensor
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120
according to the sign convention given in Table A .l,
#00
—
1
)
#11
—
~ a \t)
, #22 = ~ a 2(t)r2,
1 —k r2 ’
#33
= - a 2(t)r2 sin2 9.
(A.2)
The metric tensor (A.2) has the following non-zero components of Christoffel symbols
a
9ik
a
r°
^ ik —
(A.3)
“ i
F*
1 0k —
— a 9k
(A.4)
^r*ik
(A.5)
9H9u,k
—
F*
— —j19 a 9kk,i
1 kk —
(i j- k)
(A.6)
here no summation is implied on the repeated indices i or k. The non-zero components
of the Ricci tensor and the Ricci scalar are
(A.7)
R ij
(A.8)
R
A .3
a
a
a „ a 2 2k
~ + 2 - j H— 2 9ij
a
az
or
a
a2
k
- 6 --- 1--- 2 4 2
a
az a*
Roo — - 3
—
=
—
(A.9)
Friedm ann Equation
The sources of the gravitational field in cosmology can be regarded as perfect
fluids as a first approximation. This assumption simplifies the stress-energy tensor in
the field equation given in Table A .l to
TyU — (# T p)UfiUv
PfJf.iv
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(A.10)
121
where u[L = d x ^ /d r and t is the proper time in the rest frame of the fluids. Using
the equations (A.2) - (A. 10), the field equations become
H2+
— —^ - p ,
a2
3
a
A/
2— I- H 2 -\— - = —8nGp,
a
a2
(time —time component)
(A .ll)
(space —space components)
(A.12)
where H = a /a is the Hubble parameter and (A .ll) is called the Friedmann equation.
Since the metric (A .l) with A; = 0 describes a flat space-time, we define the energy
density p = pc at which the curvature parameter k vanishes as the critical density,
3 H2
(A.I3)
We can combine (A .ll) and (A. 12) to derive an equation for the acceleration status
of the Universe using the equation of state pi = WiPi
a
47tG ,
_ .
4nG
\
- = ---- — (p + 3p) = --- — > p i(l + 3Wi),
a
3
3
I
.
.
<■
i = species of energy.
(A.14)
It is convenient to define the energy density fraction of species i to the critical density
SU = ^
(A.15)
Pc
and then, the curvature parameter k can be expressed in terms of 12 and it reveals
the straightforward relation between the amount of total energy and the geometry of
the Universe by comparing to (A .l),
/
a2H 2
Q,tot > 1,
closed
ntot =
flat
1,
Qtot < 1,
open
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(A-16)
122
where Cltot =
is the total energy density fraction. It is custom to rescale the
co-moving radial coordinate r in the metric (A .l) so th at k can have simple discrete
values +1, 0 or -1 for closed, flat or open universe respectively. The param eter k has
physical unit of L 2.
Under the assumption th a t the Universe has undergone the adiabatic expansion (dS =
0) from the earliest moment of radiation dominant era up to today which is best
described with the ACDM model, the first law of thermodynamics can determine the
aspects of the energy density evolution over the expansion history of the Universe.
W ith the aid of the equation of state,
'
wm = 0,
Pi — wiPi,
m atter
< wr = 1/3,
w \ ~ —1,
radiation
(A-17)
vacuum energy,
we can write the first law of thermodynamics, dE = T d S —pdV, for each species of
energy as
d(pia3) = - w{pid(a3)
(A. 18)
and the standard form of energy density as functions of the scale factor a can be
derived from (A. 18),
Pi(t) — Piido)
a(t)
— 3 ( l + i u i )
(A.19)
a(to).
W ith the aid of (A. 13) - (A. 19) and the normalization of the scale factor a(to) = 1
where t0 = present time, we rewrite the Friedmann equation (A .ll) in a convenient
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123
form
H 2 = IU ^m,0O ^ + ^r,0a 4 + ^A,0
^tot,0
1
(A.20)
where lh,o = ^h(tn) i.e, the cosmological param eter Q{ defined in (A. 15) at present
time.
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