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Dark energy parameters from type Ia supernova number counts and the cosmic microwave background anisotropy

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Dark Energy Param eters from T ype la
Supernova N um ber Counts and the Cosm ic
Microwave Background A nisotropy
by
Ja m e s 0 . D u n n
A dissertation subm itted to the faculty of the University of North Carolina at
Chapel Hill in partial fulfillment of the requirements for the degree of Doctor of
Philosophy in the Department of Physics and Astronomy.
Chapel Hill
2003
Q
Approved:
\ CLia^
P.H. FVajmpton,'Advisor
uder
Rohm, Reader
D. Reichart, Reader
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UMI Number: 3086522
UMI
UMI Microform 3086522
Copyright 2003 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
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James
A B ST R A C T
O. D u n n : Dark Energy Parameters from Type la Supernova Number
Counts and the Cosmic Microwave Background Anisotropy
(Under the Direction of P.H. Frampton)
The discovery of the dark energy through type la supernova distance-magnitude
measurements is the most im portant cosmological discovery of recent years. The
most pressing problem for cosmologists since has been to understand the prop­
erties and the source of dark energy. The proliferation of increasingly precise
observations of the cosmic microwave background anisotropy, measurements of
large-scale structure and surveys of supernovae is rapidly constraining the values
of a myriad of cosmic parameters.
In this thesis, we suggest possible ways of extracting information from mi­
crowave background measurements and supernova counts. We address the effect
of dark energy on the position of the first acoustic peak in the CMB power spec­
trum. We derive an analytic formula for the first peak, £i, and with the aid of
the computer program CMBFAST compare our results to the current Microwave
Anisotropy Probe limits on Qm and Da- The accelerated expansion rate due
to the negative pressure of dark energy can significantly alter the value of i\
and CMB measurements should soon be precise enough to tightly constrain the
equation of state with this method.
We also suggest the possibility of applying the classical number count test
to type la supernova. We derive formulae for the rate of type la supernovae in
as a function of redshift, Qm ,
and the dark energy equation of state. Future
surveys should amass large catalogs of 1000’s of supernovae. Due to the extremely
high detection efficiency predicted for the SuperNova Acceleration Probe, it is
estimated th a t only a modest improvement in measurements of the cosmic star
ii
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formation rate and initial mass function should make this type of analysis at least
as useful for measuring cosmological observables as number count tests proposed
for galaxies and clusters.
iii
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ACKNOWLEDGMENTS
James 0 . Dunn
April 2 2003
I would like to thank my advisor Paul Frampton. He has been a superb role
model as a physicist through his hard work and devotion to the subject. The
example and expertise of Paul, Jack Ng and Ryan Rohm have been indispensable
to my education.
Many thanks to Dan Reichart for making many helpful suggestions.
I must also thank my family. The support of my wife, Christy, my parents
Oliver and Maria, and my siblings Lawrence and Teresa has been fundamental
throughout my academic career.
Finally, thanks to my hamster Maurice, author Terry Pratchett and the late
pianist Emil Gilels for keeping me sane.
iv
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D e d ic a te d to: M y wife, C h risty .
V
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CONTENTS
Page
LIST OF F I G U R E S ................................................................................................. viii
LIST OF T A B L E S ..................................................................................................
x
Chapter
I.
In tro d u c tio n ..................................................................................................
1
1.1
In tro d u c tio n ......................................................................................
1
1.2
Standard Model of C osm ology......................................................
3
1.2.1
Expansion D y n a m ic s ...........................................................
6
CosmicC h ro n o lo g y ............................................................................
7
Dark E n e rg y ..................................................................................................
13
2.1
D a t a ....................................................................................................
13
2.1.1
Large Scale Structure ( L S S ) ..............................................
13
2.1.2
Supernovae Type la (SNe la)
...........................................
13
2.1.3 Cosmic Microwave Background ( C M B ) ...........................
17
M o d e ls..................................................................................................
18
2.2.1 Cosmological C o n s ta n t........................................................
18
2.2.2 Q uintessence............................................................................
19
2.2.3 More Exotic Id e a s ..................................................................
21
Dark Energy and C M B ...............................................................................
23
3.1
CMB anisotropy.................................................................................
23
3.1.1 The Position of First Doppler P e a k ..................................
25
Dark Energy Parameterization (Introducing P ) .........................
26
3.2.1 Constraints on the Param eter P ........................................
27
1.3
II.
2.2
III.
3.2
vi
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P and the Acoustic P e a k .............................................................
31
3.3.1
P a ra m e te riz a tio n .................................................................
31
3.3.2
CMBFAST
...........................................................................
33
The Equation of State and the First Doppler P e a k ...................
34
Number C o u n ts ...........................................................................................
38
4.1
In tro d u c tio n .......................................................................................
38
4.2
Comoving Volume E lem en t.............................................................
39
4.2.1
General Formula
.................................................................
39
4.2.2
Dependence on Cosmological P a ra m e te rs .......................
41
Comoving R a te s .................................................................................
43
4.3.1
Core Collapse R a t e .............................................................
44
4.3.2
Type la R a t e .......................................................................
45
4.3.3
Model Dependence of Star Formation Rate (SFR) . . .
46
4.3.4
Core Collapse C alculations................................................
47
4.3.5
Type la C a lc u la tio n s ..........................................................
48
4.3.6
Corrections for Model Dependence
.................................
49
4.3.7
Combining Number Density and V o lu m e ........................
49
Simulations and F i t s .......................................................................
53
4.4.1
Estimation of Uncertainties
..............................................
54
4.4.2
Confidence Intervals
...........................................................
57
C o n c lu sio n s..................................................................................................
64
5.1
C M B .....................................................................................................
64
5.2
SNe l a .................................................................................................
65
5.3
Closing Remarks
..............................................................................
66
R E F E R E N C E S ...........................................................................................
68
3.3
3.4
IV.
4.3
4.4
V.
VI.
vii
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LIST OF FIGURES
2.1
SCP+High-Z SNe la D ata.............................................................................
17
3.1
CMBFAST Power S p e c tru m .......................................................................
24
3.2
Age of U n iv erse..............................................................................................
28
3.3
P = 0 C o n to u rs..................................................................................................
29
3.4
P=1 C o n to u rs.................................................................................................
31
3.5
P = 2 C o n to u rs..................................................................................................
32
3.6
Contours of Constant l \ Varying P ...........................................................
33
3.7 Contours of Constant i\ = 220 and Varying w ........................................
36
3.8 Contours of Constant i\ — 220 and Varying w ........................................
37
4.1 Comoving Volume Element vs. z for Different Dm .................................
42
4.2 Comoving Volume Element vs. z for Different w > —1
43
4.3 Comoving Volume Element vs. z for Different w < —1
44
4.4
Star Formation R a t e .....................................................................................
48
4.5
Type II SNe R a t e ..........................................................................................
49
4.6
Type la SNe R a te ..........................................................................................
50
4.7
Correction Factor for Star Formation Rates
..........................................
50
4.8
Corrected Star Formation R a t e .................................................................
51
4.9
Corrected Core-Collapse R a t e ....................................................................
51
4.10 Corrected SNe la R a t e .................................................................................
52
4.11 d N / d z vs. Redshift..........................................................................................
52
4.12 Transition Redshift vs. Equation of S tate..................................................
56
4.13 Ratio of Dark Energy and Dark M atter vs. Redshift
..........................
57
4.14 Confidence Intervals of
w and Da for 50% uncertainty d a ta .......
59
4.15 Confidence Intervals of
w and Dm for 50% uncertainty d ata......
59
4.16 Confidence Intervals of
Dm and Da for 50% uncertainty d ata .........60
4.17 Confidence Intervals of
w and Da for 20% uncertainty d ata............60
viii
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4.18 Confidence Intervals of w and Q,m for 20% uncertainty d a ta ................
61
4.19 Confidence Intervals of Qm and 0 A for 20% uncertainty d ata.................. 61
4.20 Confidence Intervals of w and 0 A for 10% uncertainty d a ta .................
62
4.21 Confidence Intervals of w and Qm for 10% uncertainty d ata................
63
4.22 Confidence Intervals of O m and fiA for 10% uncertainty d ata .................. 63
ix
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LIST OF TABLES
2.1
Best Fit Parameters from M A P ...............................................................
18
3.1
Comparison of CMBFAST with Analytic Formula
............................
34
4.1
SNe Counts for Various Redshift I n te r v a ls ............................................
53
4.2
SNe Counts for z=0.1 Redshift B in s .........................................................
54
x
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Chapter 1
Introduction
1.1
Introduction
W hether one dates the birth of cosmology from the musing of ancient philoso­
phers or Einstein’s application of his equations of general relativity to the uni­
verse, through most of its history it has been a difficult and primarily speculative
endeavor. Measurements of the redshifts of nearby galaxies and the detection of
the cosmic microwave background (CMB) radiation indicated an expanding uni­
verse with a hot dense past provided a general picture of the evolution of the
universe. But, the difficulty of making the precise astronomical measurements
needed to choose between the multitude of available cosmological scenarios made
progress in the field slow for most of the twentieth century.
Since the 1990’s the subject of cosmology has experienced a revolution due
to precision measurements of the cosmic microwave background anisotropy, the
large-scale structure of the universe, and the magnitude-redshift relationship of
distant type la supernovae. These measurements have constrained the values of
cosmic parameters to a degree not imaginable before. There is now the hope
th at this once speculative science will lead to the next leap in understanding of
fundamental physics.
The anisotropy of the cosmic microwave background radiation has provided
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information about fluctuations in the early universe and very strong evidence th at
the universe is flat, both supporting the predictions of inflation. The CMB also
verifies the big-bang nucleosynthesis prediction th at baryons make up only ~ 4%
of the critical density [1]. CMB polarization measurements may also eventually
be able to probe physics of the inflationary epoch[2].
Observations of the rotation curves of galaxies and the motion of galaxies in
clusters have long suggested th at much of the universe is composed of some sort
of dark m atter. Large-scale structure surveys have now determined th at cold
dark m atter comprises about 30% of the critical density [3, 4].
The most exciting revelation was from high redshift supernovae in the late
1990’s. Low redshift (z < .1) supernovae (SNe) had been used to measure the
present expansion rate, i.e. the Hubble constant. Higher redshift SNe can be
used to determine the deceleration parameter. It was expected th a t the expansion
rate was decreasing, but it was found th at the (perhaps inappropriately named)
deceleration param eter was negative [5, 6, 7, 8]. The combined supernova, largescale structure and CMB d ata tell us th at approximately 70% of the universe
consisted of a previously unknown homogenous energy with negative pressure
th at was driving the accelerated expansion of the universe.
The existence of the ‘dark’ energy is now one of the most im portant problems
in cosmology. It is clear th a t dark energy exists, but its nature is not understood.
This mystery has encouraged particle physicists and even string theorists to begin
studying cosmology.
In the following chapters we will address the possibility of constraining dark
energy properties via the position of the first Doppler peak of the CMB anisotropy
and number counts of type la supernovae (SNe la).
We begin with a review of the standard Friedmann-Robertson-Walker cos­
mology. Then we give a brief summary of cosmic events from the big bang and
inflation to the emergence of a dark energy dominated universe.
In the next chapter we detail the evidence for dark energy. This comes from
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a combination of supernova data, surveys of large-scale structure and the CMB.
In chapter three we will discuss some of the models used to describe dark
energy and attem pts to motivate dark energy from string theory, branes or su­
pergravity. Next we use a parameterization of the dark energy to explore its
affect on the CMB anisotropy. We then address the issue of number counts of
SNe la to probe the comoving volume element. Finally we state our conclusions
on the feasibility of these methods for measuring dark energy properties. We
base our conclusions partially on the expected results of the supernova survey
SNAP and the CMB probe PLANCK.
1.2
Standard M odel of C osm ology
The standard model stems from the assumption of large-scale isotropy and
homogeneity of the universe about every point.
That is, the distribution of
m atter and energy throughout the universe is uniform. Additionally, regardless
of where one is the universe looks the same in every direction.
There are cosmologically relevant scales for which these assumptions fail.
The scale of galaxies is on the order of tens of kiloparsecs. Most galaxies exist in
clusters on the order of 100 Mpc and even these collect together in superclusters.
Fluctuations on these scales are im portant since we would not exist without
them, but if one is only concerned with the dynamics of the universe as a whole
they can initially be ignored and treated perturbatively later.
Deviations from homogeneity averaged over volumes on the order of the Hub­
ble volume
are very small. Since it is the m atter and energy distribution of the universe
on this scale th a t controls the expansion rate, the assumption of an isotropic
homogenous universe is not only convenient to assume, it is an excellent approx­
imation.
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The conditions of isotropy and homogeneity are enough to specify the appro­
priate space-time metric which is called the Friedmann-Robertson-Walker uni­
verse (FRW). The FRW metric can be w ritten as
ds2 = dt2 - R(t)2
^ kr2
r dO + r sin ddcj)'1
( 1 . 1)
where k is just a number th at may be -1, 0, +1 if the universe is respectively open,
flat or closed and R is the scale factor. An open universe is infinite with constant
negative curvature. A closed universe is a finite 3-dimensional hypersphere and
a flat universe has zero curvature, i.e. spatially Euclidean. The observational
evidence, as well as theoretical prejudice, strongly favors a flat universe so this
will usually be assumed in what follows.
The metric alone says nothing about dynamics. The additional information
needed is the m atter and energy content of the universe. We then use Einstein’s
equations to determine the evolution of the FRW space-time background. Ein­
stein’s equations with a cosmological constant are:
Ryu, -
= 8ttG n T^u + Ag ^ .
(1.2)
The Ricci tensor R ^ and the Ricci scalar 1Z are constructed from the metric
and the information on the m atter and energy distribution is contained in
the stress-energy tensor T ^ .
An isotropic, homogeneous matter-energy content is modelled by a perfect
cosmological fluid. The stress-energy tensor of such a fluid at rest in a locally
Minkowski frame is given by:
T£ = diag(p, - p , - p , - p ) ,
(1.3)
where p is the pressure of the fluid and p is the density . These are typically
related by the equation of state:
p = wp,
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(1.4)
where w is a param eter th at depends on the nature of the ‘fluid’. Ordinary
m atter
ispressureless since it isapproximately stationary in the cosmological
rest frame so w — 0. The equation of state for radiation is w
= 1/3. The
equation of state for vacuum energy is w = —1. In these cases the equation of
state is a constant parameter, but it may in principle depend on time as we shall
see when we consider inflation and dark energy.
The Bianchi identity:
=0,
(1.5)
implies the conservation of stress-energy :
T% = 0.
(1.6)
The ji — 0 component of this equation yields the first law of thermodynamics:
T-v
d(R3)
1 -d {,p r * ) + p A
_L
Rs
.R
P+ 3R P + 3R P
= 0.
(1.7)
At this point we can use the first law and the equation of state to say some­
thing about the behavior of radiation, m atter and vacuum energy as a function
of the scale factor. If we have an equation of state pw = p and apply the first
law of thermodynamics, then we see the energy density dependence on the scale
factor goes like:
P K 7j5(77H)'
(L8)
PR °c ^4>
(1-9)
Thus, for radiation (w = 1/3):
while for m atter dominated (w — 0):
Pm
oc— .
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(1.10)
Finally, for vacuum energy (w = -l):
Pa = constant.
From these relations it is clear th at at early times when the scale factor is
small, radiation will dominate. Then as the universe expansion m atter will come
to dominate. Since the vacuum energy density remains constant as the universe
expands, once it starts to dominate it will become more and more dominant.
The energy associated with the curvature of the universe would be propor­
tional to 1/i?2. This means we can neglect it at early times when radiation is
driving the expansion. Since we will take the standard view th at inflation leads
to a flat universe, we will ignore curvature throughout cosmic history.
1.2.1
E xpansion D ynam ics
Now we will put all of the elements of the preceding section together. Inserting
the above ingredients into Einstein’s equations, with k — 0,
Ruu
N^fjv
+
^-9fiui
(1.12)
we get from the 00 component of Einstein’s equations,the Friedmann equation,
2
R2
~ R?~
87tGNp A
3 3
and from the ii component of Einstein’s equations
87tG n P + A.
(1.14)
The difference of 00 and ii components is also useful since it gives us the accel­
eration:
(1.15)
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If we solve the Friedmann equation with radiation, we find th a t R oc t 1^2. In
the m atter dominated era we have R oc t 2/3. Generally, for
1
P^
fl3(l+ti»)
( 1 . 16)
we find
R oc t 3(1+u’) .
( 1 . 17 )
If the expansion rate is dominated by a cosmological constant or another energy
with w — —1, then R oc exp(Ht), where H = R / R .
1.3
Cosm ic Chronology
The accumulation of observational data has not yet brought us to the point
of a complete physical theory. However, we have enough information to describe
in some detail many of the im portant events in the evolution of the universe.
In this section we summarize the generally accepted version of cosmic history as
indicated by our measurements and the current state of physical theory.
B ig B an g and Planck Era
In the standard model of cosmology, m atter and radiation are created in a
singular event with infinite tem perature and density. During this epoch, at about
t ~ 1CT43 sec and T ~ 1019 GeV, quantum corrections to classical space-time
become significant and the theory of general relativity fails. The nature of the
universe at this time is highly speculative and waits for a consistent theory of
quantum gravity such as string or M theory to elucidate its structure. Observa­
tions are not yet sensitive enough to probe this era.
G U T Scale and B aryogenesis
As the universe cools, it goes through one of a number of phase transitions.
When the standard model coupling constants are evolved to high energies they
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meet at common point around T ~ 1016 GeV. At this point, the SU{3) x SU{2) x
U( 1) gauge theory of the standard model of particle physics may become a gauge
theory of a larger group which has the standard model group as a subgroup
known. Such theories are known as grand unified theories (GUTs).
A generic feature of GUTs is the of violation of baryon number. The unifica­
tion of the strong and electroweak interactions allow the interrelation of quarks
and leptons. This is expected to be the origin of baryon-antibaryon asymmetry
known as baryogenesis. It is also necessary have to C and CP violating interac­
tions to generate the slight excess of baryons over antibaryons. Since particles
and antiparticles have the same mass we need non-equilibrium conditions to ex­
ist as well, so the time scales of these interactions must be greater than the
expansion time scale.
Another consequence of the GUT phase transition is the creation of topolog­
ical defects. These include monopoles, cosmic strings and domain walls. These
defects can appreciably affect the dynamics and structure of the universe and
the tem perature fluctuations in the cosmic background radiation. The lack of
evidence for these provides a problem to such defect theories.
Inflation
The universe may go through a brief period of extremely accelerated expan­
sion called inflation at f ~ 1(U34 sec and T ~ 1014 GeV. Inflation is used to
address a number of problems th at cannot be solved within the Standard Big
Bang model.
In order to account for the large scale homogeneity and isotropy of the uni­
verse even though much of the universe lies outside of causal contact with the
rest, inflation produces an exponential de-Sitter expansion th at takes causally
connected regions before inflation into regions the size of or greater than the
observable universe. This expansion also produces a universe th at is flat, dilutes
the presence of unwanted monopoles and relic particles and is responsible for the
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seeds of structure formation.
Inflation is accomplished by a spatially homogeneous scalar field “rolling”
down a potential to its minimum. The equation of motion for this “inflaton”
field is
0+
+
a<p
= 0
(1.18)
which resembles the equation of motion of a ball rolling down a hill.
If the potential is sufficiently flat so th at the kinetic energy of the field is
dominated by the potential energy, the equation of state of the inflaton will be
w ~ —1 and the universe will expand exponentially, R oc exp(Ht). The inflaton
must roll down the potential long enough for this expansion (~ 60 e-folds) to
solve the problems mentioned above.
The inflaton field eventually reaches a point where the potential rapidly de­
creases. The inflaton falls into this minimum picking up kinetic energy. The field
oscillates around the minimum on a time scale much smaller than the expansion
rate, i f -1 , and decays on a time scale ~ T_1. The decay of the inflation field
reheats the universe and the expansion becomes radiation dominated.
Since inflation will also dilute any baryon asymmetry th at may have existed
before, baryogenesis must either occur through the decay of the inflaton or the
reheating caused by the decay must reach a tem perature comparable to the mass
of the so-called X boson of GUTs.
A particularly convenient feature of inflation is th at while solving the largescale homogeneity problem, it can also provide the small-scale inhomogeneities
needed to produce the structure of the universe.
Quantum perturbations of
<f) during the slow roll phase,< 8(j) > « H /2 n ,cross outside the horizon. The
fluctuations are “frozen in” as classical metric perturbations and re-enter the
horizon in proceeding epochs as density perturbations.
Since the expansion rate is constant during inflation and perturbation modes
on all scales have the same physical size (H ~1), the density perturbations are
nearly scale invariant.
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T h e E lectrow eak Scale
As the universe continues to expand and cool to T ~ 250 GeV, electroweak
symmetry is spontaneously broken via the Higgs mechanism. At about a GeV
we have the quark/hadron transition and quarks form baryons and mesons.
N u cleosyn th esis
Prom t ~ 10-2 sec to t ~ 102 sec and T ~ 10 MeV to T ~ 0.1 MeV is the era
of primordial nucleosynthesis. During this epoch the reaction rates of
n <— >p + e~ + i>e
n + ue <— >p + e~
n + e+
■»p + ue
are greater than the expansion rate of the universe and can be considered to be
in equilibrium. This determines the ratio of neutrons to protons. The neutrons
primarily end up in4He, giving us mass fraction
4He
Y = 7 7 T ^ - ~ 25%,
4He + p
but some 2H, 3H, 3He also remain and can form heavier elements in processes
such as
3He + 4 He — ►7Be + 7
4He + 3 H — ►
7Li + 7 .
The abundances of these elements and their isotopes and 77, the ratio of
baryons to photons, provide tight constraints on cosmological models.
During this epoch at a tem perature of about an MeV, reactions like
uu <— >e+e_
ue <— >■eu
are no longer in equilibrium since their reaction rates become greater than the
expansion rate. Hence the neutrinos become decoupled.
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R ecom bination
The energy densities of m atter and radiation scale differently with the ex­
pansion and eventually m atter comes to dominate the dynamics. M atter and
radiation equality occurs at a tem perature of order one eV and t ~ 10,000 yrs
after the big bang.
After a few hundred thousand years at a tem perature of about 3000°K, or a
few tenths of an eV, the universe goes through another transition. The photonbaryon plasma has been opaque to radiation up to this point since it was too hot
for electrons, protons and the nuclei formed during nucleosynthesis to form bound
states. M atter becomes neutral during this era, so the photons have nothing to
scatter them and they are free to propagate. These photons have cooled, as the
universe has expanded, to the 3° K CMB we measure today.
Since the photons were in thermal equilibrium with m atter up to this time,
their tem perature distribution and power spectrum can tell us much about the
state of the universe for higher redshifts and hence what happened before recom­
bination. We will discuss this in more detail in the following sections.
D ark M a tter and Dark Energy
W ith the amount of observational and theoretical progress th at has been made
in cosmology, it is frustrating th at we understand less than 5% of the constituents
of the universe, namely the baryons.
Most of the Dark M atter may consist
of so-called weakly-interacting massive particles, WIMPs. The only significant
interaction of WIMPs with ordinary baryonic m atter is gravitational.
There
are many possibilities to explain WIMPs, the most popular being the lightest
super symmetric partners (LSPs), or neutralinos, in supersymmetric extensions
of the standard model. All we know about WIMPs is through their gravitational
effect on visible m atter. Other candidates for dark m atter include MACHOs
(MAssive Compact Halo Objects).
11
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Until a redshift of z ~ 2 the universe is dominated by dark m atter. Now
the expansion rate is controlled by dark energy. This could be a cosmological
constant, a scalar field similar to the inflaton, or something completely different.
Its existence is the biggest problem in cosmology.
12
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Chapter 2
Dark Energy
In this chapter we review the data from supernovae, large-scale structure and
the cosmic microwave background radiation which supports the existence of dark
energy. We then discuss various models and parameterization of dark energy.
2.1
2.1.1
D ata
Large Scale Structure (LSS)
The LSS of the universe can be used to extract cosmological data. LSS arises
from gravitational instabilities of small fluctuations in the density of the early
universe. The evolution of these fluctuations depends on the nature and amount
of dark m atter.
Galaxy surveys such as 2dF [3, 4] have determined the present value of VLm
to be approximately 30 (±10)% of the critical density without any assumptions
about the existence of vacuum energy or the curvature of the universe.
2.1.2
Supernovae T ype la (SN e la)
The use of supernovae as cosmological distance indicators has been proposed
since the 1930’s [9] since they are the brightest stellar objects in the sky, ri­
valling even the brightness of their host galaxies. This possibility has a great
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advantage over the classes of variable stars th at are used as standard candles.
The periods of Cepheids, or the significantly fainter RR Lyrae, are correlated
with their luminosities, but are only useful for distance measurements within the
Local Group.
Supernovae may be classified by their spectra, see [10]. Type II supernovae
(SNe II) are those th a t exhibit hydrogen spectra and type I (SNe I) are those
th at are hydrogen deficient. Type I may be further subdivided into types la, lb
and Ic. SNe la have strong absorption near 6150 A due to Si II. SNe lb lack Si
II, but have strong He I lines. Sne lie have neither Si II nor He I.
SNe may also be classified by their progenitors.
SNe II are the result of
core collapse of very massive stars and this mechanism is well understood. This
is also thought to be the mechanism behind SNe Ib/c, their progenitor stars
having previously lost their hydrogen and perhaps helium layers. SNe la are
less well understood, but are most likely the result of rapid nuclear burning of
a carbon-oxygen white dwarf. The disruption in the burning of these stars is
caused through the accretion of mass from a binary companion. The possible
companions stars range from red giants, near main-sequence He-rich stars and
even another white dwarf.
SNe la occur in all types of galaxies. They are most common in spirals, but
do not show a preference for the spiral arms. The white dwarf progenitors of
SNe la are thought to be of masses about 4 - 7 M0 and about .1 - .5 billion years
old.
Type la supernovae are of interest to cosmologists as standard candles due
to their extreme brightness and the homogeneity of their spectra [11, 12]. They
are not, however, perfect standard candles. This does not negate the utility of
SNe la since most may be regarded as approximately standard and it has been
found th at the shape of SNe la light curves are strongly correlated with their
luminosity [13, 14, 15]. W ith a sufficient number of observations, this correlation
may be used to calibrate measurements and reliably determine the luminosity of
14
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SNe la events.
The luminosity distance of an object is defined by
/
*
-
C \ 1/2
f e
)
-
(21)
where C is the luminosity of the object and T is the measured flux. In a m atter
dominated FRW universe with a cosmological constant we find this to be
(i+£)
dL = ----- ;----- T /? sin n
^k \'1/2
H0 \nr
| ^ | 1/2 [ dx ((1 + x ) 2(l + VtMx) - x(2 + x)Qa)
Jo
-
1/2
( 2 .2 )
where “sinn” means sinh for f i x > 0, sin for VLx < 0. If k=0, then
dr,
=
(1 + z) [f Z dx f((1
t i + x ) 2(l + flMx) - x(2 + x) ttA) 1/2 .
Ho
lo Jo
(2.3)
The measured magnitudes of the SNe are fit to the distance modulus
= 5 log d;, + 25
(2.4)
and likelihood fits may be used to determine the parameters Ho, Qm and 12aOne can also examine the relation between the luminosity distance and redshift:
Hodi = 2 + ^(1 —qo)z2 + • ••
_ R(tp)
_
0 _ fl(to)
90 ~
R(tp)
. .
(2.5)
R
.
.
RH«
*>(t0)
where R is the scale factor and to is the present time. H0 is the Hubble constant
which is the present rate of expansion and qo is the deceleration parameter which
measures the rate at which the expansion is slowing. In the flat cosmology above
we can calculate the deceleration parameter
Qo = g
~
•
If we use LSS to give us CIm , then this way we can find ClA.
15
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(2.7)
At redshifts z < . 1, SNe la observations have been used to determine the value
of the Hubble constant, H q « 65 km s-1 Mpc-1 [16]. Precision measurements
of high redshift SNe la, .2 < z < 1 have shown them to be an average of 0.25
magnitude dimmer than local SNe la.
If a flat universe was assumed as evidenced by the CMB, Qm + Ha = 1, it
was found th a t the d ata was consistent with a m atter density of the universe
Qm ~ -3. This implies th a t the vacuum energy density, Ha ~ -7, comprises most
of the energy density of the universe. The value of the deceleration parameter
is found to be
~ ~ -6 which tells us the expansion rate of the universe is
accelerating.
These observations were first made by the Supernova Cosmology Project
(SCP)[5] and the High-Z Supernova Search [6, 7, 8] from a combined total of
nearly 100 SNe la. The proposed SNAP satellite project [17] will have the ca­
pability to identify ~ 2000 SNe la per year up to redshifts of z < 1.7. This will
provide improved accuracy and precision in the values of fi^f and HaOne must examine alternate explanations for the supernova data. The age
and metallicity of the progenitor system as well as the morphology and redshift
of its host galaxy [18] can affect the brightness of SNe la. This may diminish
the utility of SNe la as standard candles In spiral galaxies, SNe la occur at a
greater rate with higher luminosities. It has been suggested th at the main source
of variability in SNe la brightness is difference in the C /O ratio [19]. This ratio
depends on the main-sequence progenitor of the white dwarf and its metallicity.
Older, metal poor environments seem to yield dimmer SNe la. The metallicity
of spiral galaxies in particular are significantly lower at redshifts of z > 1.
Another explanation for the apparent dimming of SNe la is extinction due
to the presence of interstellar dust [20] [21]. While it is quite certain th at there
are particles in space th a t can potentially affect observed luminosites of objects
[22] it is not clear if the extinction is great enough to explain the SNe data. The
re-radiation of this light by the dust into the far-infrared may eventually provide
16
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SCP+High-Z SNe la, Magnitude vs. Redshift
45
ai
40
cn
3
r—I
£
35
<u
u
(0
4J 30
m
■H
tf
25
0
0.2
0.4
0.6
0.8
1
redshift z
Figure 2.1: SCP d ata are triangles and High-Z points are squares. The solid line
represents the theoretical relationship for Qm — 0.3,
— 0.7, the dotted line
is Qm = 0.2, Oa = 0.8. and the dashed line is Qm = 0 .2 ,0 a = 0. All assume
w — —1.
an independent test on the presence and effect of this dust [23].
2.1.3
Cosm ic M icrowave Background (C M B)
During the first few hundred thousand years of the universe, m atter and
radiation achieved thermodynamic equilibrium. When radiation decoupled from
m atter, it maintained a blackbody spectrum with a tem perature of ~ 3000 K.
As the universe expanded, the radiation cooled to its present value of ~ 3 K as
first measured by Penzias and Wilson in 1965 [24].
The COBE experiment [25] confirmed the large scale tem perature isotropy of
the universe but also showed th at there existed very small tem perature fluctua­
tions on the order of A T /T ~ 10-5 . The tem perature anisotropy was consistent
with the scale-invariant primordial density perturbations th at may be produced
during inflation.
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Value
Uncertainty
1.02
0.02
W
< -0.78
95% CL
flA
0.73
0.04
VLm
0.27
0.04
0.044
0.04
Parameter
Table 2.1: Best Fit MAP Parameters with Hubble parameter h = .7llo!o3The CMB anisotropy can be used as a probe of certain cosmological param­
eters. Specifically, the two point tem perature autocorrelation function may be
used to extract the angular power spectrum of the CMB which reflects the acous­
tic oscillations of the baryon-photon plasma at the time of recombination. The
location and heights of peaks in the power spectrum depend on numerous cosmic
observables.
Since COBE there have been a number of ground based and balloon experi­
ments yielding progressively tighter constraints on numerous parameters [26, 27,
28, 29, 30]. The most recent and best data is from the Microwave Anisotropy
Probe(M AP)[l]. This experiment confirms a universe with Qtot ~ 1 with UA ~
.73 and Q,m — -27 with approximately 15% of the m atter density Ctb — 0.04
consisting of baryonic m atter.
2.2
M odels
2.2.1
C osm ological C onstant
The simplest solution to the problem of dark energy is to introduce a cosmo­
logical constant into Einstein’s equations.
R^v ~
= 8nGNT ^ + A
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(2.8)
The energy density of this type of energy is p oc Agoo and the pressure us
p oc ga so clearly the equation of state is w = —1 which we know results in a
scale factor th a t grows exponentially with time.
The most naive estimation of the size of the cosmological constant is made
by identifying it with the vacuum energy of quantum field theory. This yields
the naive prediction th at
A
8*G ~
( 2 ' 9 )
which is about 122 orders of magnitude greater than the measured value! Since
there is no fundamental reason to reject a non-zero A, explaining this discrepancy
is a very serious fine-tuning problem. One might invoke some symmetry to make
A identically zero, but there is no known symmetry th at will make A small and
positive [31].
Adjusting model parameters by hand to satisfy the observational constraint
requires fine-tuning to an unacceptable degree. A possible resolution to this is
to find a dynamical way to force the cosmological constant to zero [32, 33, 34].
2.2.2
Q uintessence
An alternative to a simple cosmological constant is quintessence.
In this
approach to dark energy, a spatial homogenous scalar field in a rolling potential
is introduced. This is similar to inflation.
Even before the evidence for the dark energy was observed, the cosmological
effects of a rolling scalar field in the present epoch were studied [35, 36]. The
mass density of such a field was found to behave as a time dependent cosmolog­
ical constant. These models were studied to reconcile low dynamical estimates
of the mean mass density with the negligible spatial curvature predicted by in­
flation. This was possible if the energy density of this scalar field happened
to dominate in the present epoch. Once the accelerating expansion rate of the
universe was observed, this idea rapidly became a popular way of modelling the
19
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dark energy [37].
These are sometimes realized in the context of Brans-Dicke theory [38], the
dilaton or other moduli fields from string and M theory [39, 40], supergravity
models [41] or in Randall-Sundrum brane world scenarios [42] [43] since scalar
fields naturally arise in these theories. One may also parameterize quintessence
as a time dependent cosmological term in Einstein’s equations [44, 45].
Quintessence can only be allowed to interact gravitationally with ordinary
m atter. This is to prevent observable long range interactions and time depen­
dence of physical constants. Ordinarily these couplings are simply taken to be
small without explanation; however, it is possible to invoke an approximately
conserved global symmetry to achieve this [31]. A quintessence field th at inter­
acts with m atter other than gravitationally violates the Equivalence Principle,
so care must be taken not to violate limits from Eotvos-type experiments [46].
The quintessence field is minimally coupled via the Lagrangian
C = d^d^-V {(t> ).
( 2 . 10)
The stress-energy tensor for this field is
(2 . 11)
The equation of state of a scalar field is
( 2 . 12)
Quintessence is assumed to be spatially homogenous, V 0 = 0, so
w = — = ---------------
(2.13)
The equation of motion for this field is
0 + 3Hj> + — = 0 .
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(2.14)
A variety of potentials such as power-law, exponential and cosine have been used
to describe quintessence.
The quintessence equation of state can be constant, but is in general depen­
dent on the scale factor so it evolves in time. However this dependence is difficult
in principle to measure since cosmological observables depend on w through var­
ious integrals and information on time dependence is lost [47, 48, 49, 50, 51].
There are two other problems th at must be addressed in constructing models.
The m atter density and the dark energy density have comparable values in the
present era, but they evolve at different rates. The initial conditions must be
carefully chosen for this to occur. This is the “coincidence problem” and may
be addressed by so-called tracking solutions [52, 53] which are solutions of the
quintessence equations of motion th at are attracted to a common solution.
Fine-tuning must also be addressed in quintessence models since the dark
energy density is much smaller than typical particle physics scales. This problem
may also be addressed through tracking solutions.
So far the d ata is consistent with quintessence models as well as a cosmological
constant [1, 54].
2.2.3
M ore E xotic Ideas
Possible solutions to the dark energy problem are not limited to the above. An
alternative proposed in [55] in order to avoid the fine-tuning problem is k-essence.
K-essence is a scalar field model of dark energy with a non-canonical kinetic
energy. During the radiation dominated epoch, k-essence mimics the equation of
state of radiation. As m atter begins to dominate this energy density decreases to
some small fixed value. As the universe continues to expand k-essence eventually
becomes dominant over m atter. Such tracking models in quintessence must be
tuned to produce accelerated expansion during the right epoch which is not a
problem for k-essence.
Another possibility from string theory is a tachyonic condensate [56, 57, 58,
21
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59] th a t can be described by a scalar field with an effective Lagrangian
£=-V(<f>)[l
(2.15)
Such theories can produce cosmic accelerations with the right choice of potential
and have been proposed as the dark energy [60, 61, 62, 63].
An interesting feature of theories with nonlinear kinetic energies is th at their
equation of state may become less than negative one, impossible for quintessence.
Such theories necessarily violate the Null Energy Condition of general relativity.
Equations of state less than negative one cannot be excluded by the data so far
and may even be required in some theories [64, 65, 66, 67]. Despite the theoretical
difficulties these models may present they must still be considered. .
Another interesting possible source of dark energy is some as yet unknown
physics from the trans-Planckian regime as suggested in [68, 69]. A theory of
quantum gravity such as strings may produce modified dispersion relations on
very short distance scales. Ultra-low frequency and high momentum modes pro­
duced during the trans-Planckian era are frozen out by the expansion of the
universe. These so-called “tail” modes cannot decay while the expansion rate is
larger than the frequency of the modes. Using a particular class of functions to
describe the dispersion relations, the frozen tail modes have an energy density
consistent with th a t of dark energy without any fine-tuning.
Despite the vast improvement in the quality of data, not enough is known
of dark energy parameters to confidently accept or reject any of the physical
models. In the next chapter we address the possibility of measuring properties
of the dark energy by using precision cosmic background radiation data.
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Chapter 3
Dark Energy and CMB
In this chapter we examine the CMB anisotropy and the possibility of mea­
suring the dark energy with CMB measurements. We begin by describing the
angular power spectrum of the CMB which arises from analysis of the tem pera­
ture auto-correlation function. We derive an analytic formula for the first Doppler
peak [70]. We then define a parameterization of the dark energy and derive the
expansion dynamics of the universe for this model [45]. Then we attem pt to place
constraints on this model in light of certain cosmological measurements [5, 6, 7].
A new formula is then derived for the position of the first acoustic peak with
dark energy. This result [45] and the material in Chapter 4 [71] are the original
contributions made in this thesis.
3.1
CM B anisotropy
The tem perature of the cosmic microwave background radiation is almost
completely uniform across the entire sky. The only deviation from this isotropy
are tiny tem perature fluctuations on the order of A T / T ~ 10—5[25, 72].
We study the CMB fluctuations using the tem perature autocorrelation func­
tion:
(3.1)
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8x10
’V
u
+
'V
6x10
CN
4x10
2x10
-10
-10
-10
-10
0
200
400
600
800
1000 1200 1400
Figure 3.1: Plot of the Angular Power Spectrum of the CMB produced by CMBFAST.
where
AT(n)
T
is the fractional tem perature perturbation as a function of direction n.
(3.2)
The
perturbation is first expanded in spherical harmonics :
AT(n) =
T
(3-3)
lm
Using the assumptions of statistical isotropy and homogeneity and the orthonor­
mality of the spherical harmonics, it is found th at the coefficient satisfy:
(3.4)
and a plot of the Ci s versus I, figure 3.1, characterizes the angular power spec­
trum.
24
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3.1.1
T he P osition o f First D oppler Peak
The position of the first Doppler peak is given by
(3 '5>
where Ad is the angle subtended by the horizon at the time of last scattering. We
address the cases of open, closed and flat universes in FRW cosmology separately.
Consider the metric in the open case:
ds2 = dt2 - R 2 [d^2 + sinh2 ^dO2 + sinh2 $ sin2 Odcj)2} .
(3.6)
Geodesics satisfy ds2 = 0, which implies
^ _I
dt
(3.7)
R
The chain rule gives us:
d'F _ d& .
d^ _
~ d t~ d R
1
dR~~RR'
”
Now we need Einstein’s equations, specifically the Friedmann equation:
•
\
R\
Rj
2
_ 87t G n p m A
3H q
3
1
R2
.
1
.
}
The value of ^ at t = 0 is zero, so at some time t
f dR
.
.
( 3 ' 1 0 )
If we make the the definitions:
8-k G n P m o
O
“
Qk =
3HS
0
{n
a
" A “ 3H$
—k
1
H$R% = f l p g
,
(3' U )
^°p6n universe)
and the substitutions R = R 0r and x = 1 /r, then
Rp
^t=
^
l
+
q k Xt ^
nx ‘
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(3-12)
The angle subtended by the horizon is
1
Ad
(3.13)
H R sm b 'll,’
where H — R / R is the expansion rate in the FRW universe. The position of the
first acoustic peak is
7T
i\ = — = n H R s m h ^ t
A0
( 3 . 14 )
1 /2
it
/ R
\Ro,
-TCQ
x sinh
v VLk
11
dx
\/Q, m %3 + Qr X2 +
Modifying this result for the closed and flat universes is simple. In the case
of a closed universe (k = 1) we have instead
1/2
k
=
( j l)
Q“(f)3+^ (f)2+^
(3.15)
dx
x sin
\/Q m x 3 + FIk x 2 + Oa
In the flat case (A; = 0):
1 /2
) +fiA
3.2
a
da:
I
(3.16)
\/VLm x 3 +
Dark Energy Param eterization (Introduc­
ing P)
In [45] it was proposed th at the dark energy could be usefully parameterized
as a time dependent cosmological term in Einstein’s equations. This was done
by making the substitution
9tivR
9[ivR
^
-
p
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( 3 .1 7 )
where P is some fixed power. The equation of state of this energy is still th at of
a cosmological constant, w = —1.
It is now possible to solve for the expansion rate in a FRW universe with the
usual cold dark m atter component.
2
(3.18)
= H q Qm
where
(3.19)
and Qm and Qa are the ordinary cold dark m atter and cosmological constant
things. The standard cosmological model is recovered for P —►0.
3.2.1
C onstraints on th e Param eter P
We first seek to restrict P through the consistency of the expansion dynamics
of the universe. Let us first examine the flat case VLk = 0, which Nature seems
to favor. The expansion rate becomes
(3.20)
where x = (R0/R). The root, x, of the right-hand side is
(3.21)
If 0 < Qm < 1) consistency requires th at P <30, m This is apparent when we calculate the age of a flat universe in this model.
The age of the universe in a FRW universe is given by the formula:
(3.22)
Changing variables to R! —> R!/ R 0 makes this
1/2
27
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(3.23)
0
0.2
0 .4
0.6
0.8
P
Figure 3.2: Age of universe with Qm = 0.3 and
of P.
= 0.7 in Gyrs as a function
dx
We can ignore the radiation dominated phase of the universe in this calculation.
We set z = 1100, the redshift at the surface of last scattering and choose fIm = -3
and Oa = .7. Now we plot this as a function of the parameter P.
The age of the universe diverges as P —> .9 = 3Qm - If P > 3Gm then there
exists a x > 0 such th at R —> 0 and changes sign. This arises from the condition
for energy conservation. In this case, there is a bounce in the past if x > 1,
and this contradicts the evidence for monotonic expansion from nucleosynthesis
to the present epoch.
If we rewrite the equation for x in terms of flA:
(3.24)
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P=0
3
2.5
2
1.5
<
a
-i
1
1 -Q
0.5
0
-0.5
0.5
1
1.5
Qm
2
2.5
3
Figure 3.3: Past and future bounces for P=0.
then we see th at for P < 3 th at any Vt\ < 0 gives a solution with 0 < x < 1
corresponding to an allowed bounce in the future. If P > 3, the condition for a
reversal of the expansion in the future, a ’bounce’, is Qa <
This behavior is present in the more general case with Qk ^ 0 as well. We
illustrate this by first considering the case P — 0, the standard FRW cosmology.
The VLm ~
plane may be divided into three regions, see figure 3.3. The
boundaries of these regions are where the expansion rate goes to zero and changes
29
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sign. The upper shaded region indicates values of Dm and Da th at yield such
a ‘bounce’ in the past. This region is disallowed since it is inconsistent with
evidence for a Big Bang. The lower shaded region indicates the existence of a
bounce in the future, i.e. the evolution of the universe back to a ‘big crunch’.
The unshaded area contains the values of Dm and Da with no bounces in the
past or future.
We can create similar figures for non-zero values of P with different allowed
values of Dm and Da, figures 3.4 and 3.5
An additional constraint on this model, from big bang nucleosynthesis, is th at
the expansion rate for very large x must be close to th at th at of the standard
model. To this end, it is sufficient to study the ratio:
(k/R fp
3a „ - p
(R/R)P
2 =0
(3 - P)( lM
(3'25)
(R/Rfp
4Qr - P
\ ' !p = -—
(R/R)P
2=0 (4 - P ) n R
,
(3. 26)
y
’
in a m atter dominated universe and
lim
in a radiation dominated universe.
This ratio at the BBN era is therefore
(R /R )j
X" ~ ( R / R f P=„
3Hm — P
4 n g “- - P
(3 - P)Slu
(4 - P ) n g “”
{
>
where the superscript “trans” refers to the transition from radiation domination
to m atter domination. If we put VIm = 0.3 and flp ans = 0.5 and require th at the
expansion rate of this model be within 15%, the equivalent of the contribution to
the expansion rate at BBN of one chiral neutrino flavor, of the standard model,
then P < 0.2.
30
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P=1
3
.5
2
.5
1
(1-Qm -Q a )2-3(2Q m -Qa )<0
.5
0
-0
0
0.5
1
2
1.5
Q■M
2.5
3
Figure 3.4: Past and future bounces for P = l.
3.3
P and the A coustic Peak
3.3.1
Param eterization
We can now expand the derivation of position of the first acoustic peak [45]
to include the time dependent cosmological term considered above. For £1% — 0:
h
=
7
r
R
S M ^rl
~K
Rq
~
\ P
+ f!A
Ro \
~RJ
x/2*
V
«£L
f R
h
d'r
y/& m x 3 + &h.xp
31
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(3.28)
P = 2
1
1-Q,
0.8
0.6
0.4
<
G
0.2
0
0.2
W -vr^M~Tr
0.4
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Figure 3.5: Past and future bounces for P=2.
for VLk < 0:
1/2
7T
h =
R
x
Ro
(3.29)
dx
sin
yj $Im %z + &a x p + £Ik %2
for Qk > 0:
7T
h =
R
V&K Ro
1/2
^(1) +Mt) +n-(l)
Ro
(3.30)
dx
R
sinh I \J VLk
'i
X
v & m x 3 + & a x p + Q,kx2/
We now consider how the restriction on P, 0 < P < 0.2, could affect the ex­
traction of cosmological parameters CIm and ft a - If the value of the first acoustic
32
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0 . 66
0.65
Qa 0 . 6 4
0.63
0. 62
0.26
0.27
0.28
0.29
0.3
0.31
0.32
Qm
Figure 3.6: Constant
in the Q \ — Qm plane
= 197 contours for varying P = 0, .05, 0.10, 0.15, 0.20
peak is chosen to be l\ = 197 and we let P take the values P = 0, 0.05, 0.10,
0.15, 0.20 we find th a t Q m and Qa can vary as much as 3%.
In the figure, we have added the line for the acceleration parameter qo = —0.5
as suggested by SNe la measurements.
3.3.2
C M BFA ST
The formulae derived above measure the angle subtended by the photon hori­
zon at recombination. But since the universe up to this point is a baryon-photon
plasma we must consider whether we should be examining the acoustic horizon
instead. This introduces a factor of \/3 in our formula for £\ from replacing the
speed of light, 1, with the sound speed l/y/3.
The solution to this problem lies somewhere in between the two horizons.
33
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Da
CMBFAST £y h [45, 70]
0.5
284
233
0.6
254
208
0.7
222
182
0.8
191
155
Table 3.1: For P = 0, Dm—0.3 and h — 0.65 we find the above values for the
position of the first Doppler peak with CMBFAST and our analytic formula.
The decoupling transition does not happen instantaneously so fluctuation are
able to grow during this time. If we compare our formula to the public code
CMBFAST [73], which numerically computes the power spectrum of the CMB
anisotropy, we find th at the results are related by a constant factor of ~ 1.22.
This normalization is intermediate between the acoustic (\/3) and photon horizon
(I)-
Let us now use this normalization in our formula and use the MAP data that
indicates Dm = -27, Da = .73 and pick the equation of state to be w = —1, a
cosmological constant (P = 0). The formula for the Doppler peak yields precisely
£i = 220.2, which is very consistent with the current best value of 220.1 ± 0.8.
3.4
The Equation of State and the First Doppler
Peak
Another useful parameterization of the dark energy is through the equation
of state. In the above calculation this was fixed at w = —1, the equation of state
for vacuum energy. It is a simple m atter now to recalculate the position of the
Doppler peak for general values of w.
The expansion rate in a universe with cold dark m atter and a dark energy
34
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component with equation of state, w, is
Rn
\R
R
R )
3(iu+l)
Ro
(3.31)
This means th a t the acoustic peak is:
1 /2
dx
R
h = n--„ ClM
sio
I
y / C l M X z + O aX3^
1) ’
(3.32)
for Qk < 0:
h =
7r
R
+ Cl a
y/—Qk R o
■RoN
r
3(w +l)
)
/
jj \
W i)
2 1/2
X
(3.33)
dx
sm
y/Qhix3 + Oa^3^ +1^ +
for CIk > 0:
71
n —
R
y/Clx Ro
ci M
Ro
R
+
Ro
~R
a
* ”- ( f
X
(3.34)
dx
sinh
n
1/2
3(w +l)
a/CI m x 3 +
Qao;3^ 1) + CIk x 2t
In figure 3.7 we see contours of equal l \ = 220 in the ClA — CIm plane for a
range of equations of state. In figure 3.8 we have also plotted in the same plane
with equations of state less than negative one. Observations are consistent with
this possibility, though this may require some exotic physics.
If we assume a perfectly flat universe, the effect of this parameterization of
the dark energy on l \ can be relatively large. A choice of w picks out precisely
the m atter and energy content of the universe over a significant range of values.
The MAP d ata indicates th at the total energy density is Cltot — 1.02 ± 0.02.
An uncertainty of this magnitude allows an even greater degree of freedom in
picking a consistent equation of state using this information alone.
It has also been pointed out to the author [74] th at it is possible to fit the com­
plete MAP power spectrum with Cltot = 1.04 and w = —4, although supernovae
35
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f 1=220 C o n t o u r s
0.8
0 ' 60 . 1 5
0.2
0.25
0.3
0.35
0.4
Figure 3.7: Constant i\ = 220 contours for varying w = -1, -.9, -.8 -.7 (top to
bottom) in the
—Q,m plane. The thicker gray line represents VLm +
= 1data discussed in Chapter 4 will partially reconcile this degeneracy. This type
of param eter degeneracy should perhaps be taken seriously. A positively curved
universe limits the number of e-foldings of inflation th a t could have taken place,
but there are models consistent with these limits [75]. These models still solve
the usual cosmological problems and are consistent with structure formation.
So even with the greatly improved precision of the MAP results there is a lot
of room for different dark energy models and we must look to additional sources
of d ata to break the degeneracy.
In this chapter we have studied how the CMB anisotropy can be used to
measure dark energy properties. In the next chapter we examine how supernova
number counts may provide an additional source of cosmological information.
36
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Al=220 C o n t o u r s
0.95
0.85
0.75
0.15
0 . 25
0.2
0.3
Figure 3.8: Constant t\ = 220 contours for varying w = -1, -1.5, -2, -2.5, -3
(bottom to top)in the Cla ~
plane. The thicker gray line represents Om +Ha =
1.
37
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Chapter 4
Number Counts
4.1
Introduction
In order to constrain successfully cosmic parameters it is necessary to gather
data from multiple complementary sources. In this chapter we suggest a new
method of measuring cosmic observables and the dark energy. This new strategy
involves probing the comoving volume element of the universe by measuring the
distribution of type la supernovae. We first derive and study the formula for the
comoving volume element of a CDM cosmology where the dark energy component
has an arbitrary equation of state. We then discuss the details of computing the
comoving number density of type I and II supernovae using the star formation
rate (SFR) and assumptions about the progenitors of the supernovae. Finally,
using simulated data of the quality expected from the SNAP mission [17], we
study the feasibility of extracting cosmological information using this method.
The distribution of galaxies and clusters of galaxies in space and in redshift
depends on the cosmological volume element and may be used to determine cos­
mological parameters. This was first used by Loh and Spillar [76] who considered
the redshifts of ~ 400 galaxies over a redshift range 0.1 < z < .9. In this man­
ner they found the ratio of the critical density to the total energy density of
the universe to be flo = 0.9to!s- However, their measurements depended on the
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assumptions th a t the comoving number density of galaxies and the luminosity
function of the galaxies did not change over th at redshift interval. It is not clear
whether this is valid since a complete theory of galaxy evolution is lacking. This
shortcoming highlights one of the major difficulties of using number counts.
Newman and Davis in [77] suggested using counts of galaxies with the same
circular velocity at different redshifts as an improvement. Via the semi-analytic
method of [78], one may calculate the probability th at a dark m atter halo with
a particular mass will have a formation time such th at it will virialize with a
given circular velocity: (v% = G M /r ) . Using the analytic calculation of galaxy
abundance as a function of mass predicted by [79], which fits very well with dark
m atter simulations, the total abundance of galaxies with a given circular velocity
in a particular era can be found. This method of determining the evolution
conveniently does not depend strongly on the cosmological model.
The abundance of clusters above a certain mass as a function of redshift is a
sensitive probe of VLm and Da [80, 81, 82]. The comoving number density of such
clusters may be determined from the mass function of dark m atter halos which
results from numerical simulations [83].
4.2
Com oving Volum e Elem ent
4.2.1
G eneral Formula
The number of objects within a certain redshift interval and solid angle di­
rectly probes the geometry of universe. Let d V be a comoving volume element
containing d N sources.
If we let n(t) be the number density of sources per
comoving volume element, then we have:
d N = n(t)dV = n (t) — V
. drdtt.
(1 —hr2) '
Note th a t both n(t) and dV are dimensionless.
39
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(4.1)
If r\ is the coordinate of a source at red shift z, then we have the relation:
r\
dr
_
(1 - kr2)1/2 ~
I
r t0 _dt_ _
Jh
R(t) ~
f Ro
JRl
dR(t)
R(t)R(t)
3(l+ui)
dR
1 /2
-
r
=
J r , IP IIo
3(l+iu)
-
1/2
R qH q Jn
'*■ (A )
Now change coordinates to x — R / R q:
rV [ 1
^ [S2„oT3 + n K x ~ 2 + nAi - 3<1+u’1]
0 # 0 7 (1+ ^ - ! X2
1
= -p „
R qH
q
/
J' f( 11 +*)
+Z)-1
-
1/2
J
dx
[fi-MX + Q k x 2
+
J
We now use the fact that:
sin 1ri
/*n
Jo
dr
(1 - &x2)1/2
( ri
k=l
(4.2)
k=0
sinh~1 ri
k = -l
So the comoving coordinate ri is:
n =
1
sm JUL
R qH q
k=l
dd
UoUo
sinh JR qiH±q
k=0
(4.3)
k = -l
where
r( I + 2 )
I{z) =
J
dx [£Im x 3 + Qk x 2 + Oax3^1+^ ]
-
1 /2
(4.4)
Prom the above calculation we can see that
dr\
(1
dz
- k r f f 2 - J t j r a [(l + z)2nK + 0 + z f n M+ (l + z)
3(l+u>)
-
1 /2
0/
( 4 .5 )
Using this and our formula for r\ above we have for k = 1:
40
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dN
dzdQ,
n(z) sin2 [I (z) / {RqH q)]
(4.6)
1/2 1
(.R0H 0) (1 + z f t t K + (1 + z f
+ (1 + *)3(1+w)
for k = 0:
dN
dzdfl
n(z)I(zf
1/2
(4.7)
’
(Ro Ho f (1 + z f n M + (1 + z f 1+w) Qa
and for k = -1:
dN
dzdQ
______________ n(z) sinh2 [/ (z) /{R qH q)]______________
i/2 •
(r 0h 0) (i + z f n K + (i + z f n M + (i + ^)3(1+" }a*
(4.8)
Since we are concerned with the rate of supernovae, we must account for time
dilation and insert a factor of 1/(1 + z).
4.2.2
D ependence on C osm ological Param eters
Now let us examine the behavior of the volume element as we vary different
parameters. We will restrict ourselves to the case of a flat universe for convenience
and as suggested by the CMB data. The volume element is then:
dV _
dzd£l
I(zf
1/2
(R qH q)
(4.9)
•
(1 + z) Qm + (1 + z f (l+w^
Let us first consider the case of a universe with only cold dark m atter and
a cosmological constant. In figure 4.1 we plot dV/dzdQ in arbitrary units as a
function of redshift for various value of the density of cold dark m atter.
In figure 4.2 we fix the values of density of dark m atter and dark energy and
vary the equation of state. Since the current data allows the possibility, we plot
in figure 4.3 the volume element for several values with w < —1 as well.
If the number of objects remains constant over a range of redshift then:
dN
dzdn =
dV
udzd n ’
41
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,
(4 -10)
1—I
w
Q)
o 0.15
>
tn
"rH 0 . 1
>
o
| 0.05
u
^
0
1
2
3
4
5
R ed sh ift
Figure 4.1: The comoving volume element dV/dzdQ in arbitrary units vs. red­
shift for flj 4 =.25, .27, .30, .33 (top to bottom).
where no is a constant equal to the comoving density of the objects within th at
redshift range. This is an adequate approximation if one is considering a redshift
span small enough so the sources in question do not evolve appreciably over this
time. If none of a particular type of object are being created or destroyed then
the comoving number density is constant.
This obviously is not the case for galaxies or clusters of galaxies if we consider
a large enough redshift range. The problem becomes even trickier when the
number of sources evolves even for relatively small redshift intervals.
Since supernovae, especially type la, are extremely bright and hence visible
over cosmological distances they are good candidates for tests of number count
versus redshift. The number of supernovae is tied to the cosmological star for­
mation rate (SFR). The core collapse SNe II rate is expected to be proportional
to the overall rate, but since a detailed understanding of type la SNe is still
42
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I—I
w
<D
o 0.15
>
di
ej
0.1
■rH
>
o
| 0.05
u
0
0.5
1
1.5
2
2.5
3
R ed sh ift
Figure 4.2: The comoving volume element in arbitrary units vs. redshift for
varying equations of state greater than negative one. From top to bottom w =
-1, -0.9, -0.8, -0.7.
lacking the precise relation of SNe la rate to the SFR is less clear. Fortunately
the various scenarios for SNela events may be parameterized in a simple manner.
The situation is complicated, however, since the star formation rate varies sig­
nificantly over the redshift range we are interested in and measurements of the
SFR are model dependent.
4.3
Com oving R ates
The primary difficulty in using number counts to probe cosmological param­
eters is disentangling the comoving volume element from the comoving number
density. In this section we will address the problem of determining this quantity
for supernovae.
43
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r—H
w
Q)
I—I
o
>
b )
d
°
>
-
1 5
0.1
o
e
° 0.05
0
0.5
1
1.5
2
2 .5
3
R ed sh ift
Figure 4.3: The comoving volume element in arbitrary unit vs. redshift for
varying equations of state less than negative one. From top to bottom w =-2,
-1.5, -1.3, -1.
4.3.1
Core C ollapse R ate
The rate of core collapse SNe and the type la SNe rate must be related to
the SFR since they are stars, but of course not all stars become SNe. We first
address the simple case of the core collapse SNe rate as an illustration.
The rate of core collapse supernovae SNRCC is approximately proportional to
the SFR for stars with masses greater than 8 M0 . Such massive stars become core
collapse SNe and their lifetimes are about 50 Myr or less. Since this time scale
is much smaller than cosmological time scales, we can make the approximation
that as soon as a star greater than 8 M0 is formed it becomes a supernovae. If
we have an initial mass function (f>(m), where m is mass in units of solar masses,
then the fraction of stars formed at a given epoch th at become core collapse
supernovae is
44
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= /> 8 dm(j)(m)
f dm(p(m)
( 4 .1 1 )
Hence the core collapse rate at some time t is
f „ dm&im)
S N R cc(t) = S F R ( t ) ^ A -—
= kSF R (t).
J dmcpym)
4.3.2
(4.12)
T ype la R ate
The SNRCC is simple because we can ignore the lifetime of the progenitors.
This approximation is not possible for the type la rate (SNR/a). Although it
relatively certain the SNela are the result of the explosion of a C -0 white dwarf
due to accretion from a binary companion, the nature of the companion can
drastically affect the time it takes the white dwarf to become a SNela. Low mass
stars also spend more time on the main sequence.
The delay between the formation of the white dwarf and its eventual explo­
sion, if it explodes at all, could be on a variety of cosmologically relevant scales.
Hence it is necessary to incorporate this delay time into the SNR/a. We will
follow the method used by Madau et al. in [84].
We assume th a t possible progenitors have an initial mass of greater than 3
M© and less than 8 M0 . A star with an initial mass greater than 8 M© will
become a core collapse supernovae. If the initial mass is less than 3 M© (final
mass > 0.72 M©), then the white dwarf will not be able to become a Type la
supernova for any value of the companion mass.
Now we define m c = rnax[rnmm, rn(t')\ where m(i') = (10 G y r/t')0-4. The
quantity m(t') is the minimum mass of a star th at reaches the WD phase at
some time t'. The quantity m c will be the lower limit of integration over the
initial mass function. We also set tm = 10 G yr/m 2 5, which is the lifetime of a
star with mass m in units of solar masses.
If we let r) represent the fraction of W D ’s th at eventually become SNela and
45
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r the characteristic delay time for a WD to explode, then the type la supernova
rate at some time t can be written:
S N R Ia(t) =
fldt'SFRtt') f 8 dm exp
1 >Jm\ , J \
t J dmq>[m)
<f>(m)
— •
(4-13)
This rate Eq.(4.13) is proportional to the SFR (unlike the core-collapse rate).
If we considers stars born with a particular mass m at a given time, the number
of white dwarfs th a t become supernova peaks at a time tm + r after the stars are
born.
We can alternatively use the parameterization of Dahlen and Fransson [85]:
S N R i a(t)=r ] f dt!SFR(t') f dm5(t —t' — tm — r)<f)(m),
J tjr
J
(4-14)
3
where tp is the time th a t corresponds to the redshift of the formation of the first
stars.
For both of these formulae it is necessary to insert by hand the efficiency of
WD to SNela transition. This is done by fitting 77 to the local SNela rate. This
assumes th a t the mechanism th at causes SNela does not depend on redshift. If
one makes particular assumptions about the SNela progenitors it is also possible
to derive the fraction of WD th at become SNe as is shown in [86].
4.3.3
M odel D ependence of Star Form ation R ate (SFR )
If we assume a Salpeter [87] initial mass function, then all th at remains is to
insert the star formation rate. This quantity is measured only indirectly by as­
tronomers, since the relation of the SFR to observations depends on the assumed
cosmological model. This dependence appears in the formula for the distance
modulus fjb(z), the difference between the apparent and absolute magnitude of
the source,:
n(z) = m — M = 51ogdi(z) —5,
46
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(4-15)
where (Ll is the luminosity distance. In the case of a flat universe with a cosmo­
logical constant this is
< « * ) = ^ ( 1 + z)
-Wo
(4.16)
/
(4-17)
and
dx \flM%Z + O j 1 2 •
A star formation rate which is asserted based on observations must be con­
verted from whatever cosmological model was assumed in the derivation to the
cosmological model of interest.
Observational results are typically given under the assumption of
=
1
Einstein-de Sitter cosmology. If we are assuming for instance a ACDM cosmology
we must scale the luminosity using the formula [88]:
t
f L acdm\
logl ^
n1
f d L CDM(z ) \
) =21ogh F ( ^ ) -
(A
i c A
(418)
Since the comoving volume element will obviously change as well, luminosity
densities must be scaled as
,
Now we
(P
l
CDM\
nl
f d $ CDM( z ) \
g l p f ‘s ) 2k* (
proceed to actual
J
,
(V m dm \
(419)
calculations of star formation and supernova rates
based on the these considerations.
4.3.4
Core Collapse C alculations
Let us first consider the core collapse rate. We argued above th at this rate
should simply be the rate of formation of stars above 8 solar masses. If we assume
a Salpeter initial mass function:
(j) ocm 2'35,
47
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(4.20)
1
ro
I
u
1
£
1
1
!>i
i ~
0)
O
J
1
-2
CO
2
0
2
3
Re dshift
1
4
5
Figure 4.4: Fit to the star formation rate vs. redshift [84] [89] [90] [91].
where m is mass in solar units, the rate of type II SNe becomes:
-1 2 5
S N R U t ) = S FR( t ) &
d™m
(4.21)
We use a fit for the star formation history [84] [89] [90] [91]:
S F R ( t) = [0.336e~t/1,6 + 0.0074(1 - e~t/om) + 0.0197t5e- t/o-64] Me y r^ M p c -3 ,
(4.22)
where t = 13(1 + £)~3/ 2 is the Hubble time in Gyr. In figure 4.4 we plot the star
formation rate as function of redshift. In figure 4.5 we plot the corresponding
core-collapse rate.
4.3.5
T ype la C alculations
Now we plot rates using the formulae for SNela from [84] and [85] for the
delay times r = .1, 1, 1.5 Gyrs. The explosion efficiencies are fit to a local rate
48
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m
l
o
4
£ -4.1
Sh
>1 -4.2
-4.3
tn
o -4.4
J
cu -4.5
4->
rti -4.6
H
H
-4.7
(U
a
CO
0
1
2
6
5
3
4
Redshift
7
Figure 4.5: Core-Collapse rate vs. redshift.
of 1.3 ± 0.6 x 10-5 SNe yr_1 Mpc-3 [84] and the star formation rate used is the
same as above
4.3.6
C orrections for M odel D ependence
Since the determination of the star formation rate is model dependent, we
must rescale our rates to match the particular model we are interested in. We do
this now for the ACDM model with Qm = 0.3 and
= 0.7. The star formation
rate becomes:
2
SFR{Z) —
Y±c
S FR ( z )
d m
\
. VEdS J
4.3.7
(4.23)
C om bining N um ber D en sity and Volum e
We are finally able to calculate the number of type la supernova per redshift
per solid angle. We do this for each of the three delay times used above.
49
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m
i
U
&
- 3
4
Sh
>1
-4 . 2
D) -4 .4
o
w -4 . 6
a)
-U
(d -4
(d
H
a;
£
0.5
1
2
1.5
R e dShift
2.5
3
Figure 4.6: SNe la rate vs. redshift for three delay times.
1
.8
£
O
-H
-U
O 0 .6
aj
u
o
u 0 .4
U1
0 .2
1
2
3
R e dshift
4
5
Figure 4.7: The correction factor we must scale measured star formation rates
to convert to ACDM model with (1^ = 0.3 and Qa = 0.7.
50
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ro
I
o
1
£
1
u
>1
1
-2
D)
o
J
2
fa
C/2
2
2
3
Red shi ft
1
0
4
5
Figure 4.8: The corrected star formation rate for the VLm = 0.3 and Cla = 0.7
cosmology.
m
I
o
§
-4 .3
u
>1 "4 .4
-4 .5
-4 . 6
(U
.u
(d _A .7
fa
^
H
H
d)
£
U1
-4 . 8
0
1
2
3
4
Redshift
5
6
Figure 4.9: The corrected core-collapse rate for the fIm = 0.3 and
cosmology.
51
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7
= 0.7
0)
co
0
0.5
1
1.5
Re dshift
2
2.5
3
Figure 4.10: The corrected SNe la rate for the VLm = 0.3 and Qa = 0.7 cosmology.
175
(D
S
150
tj>
a)
'd 125
a)
!o 100
d
D1
w
U
ft
75
50
0.5
1
1.5
Red shift
2
2.5
3
Figure 4.11: Plot of d N / d z per square degree for three delay times. A flat
universe with fIm = -3 and an ordinary cosmological constant is assumed.
52
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z
t
- 1 Gyr
t
—1 Gyr
r =1.5 Gyr
0-.1
.01
.01
.01
.1-.5
1
1
2
.5-1
7
10
10
1-1.5
20
30
35
1.5-2
35
45
50
0-2
65
90
100
Table 4.1: The approximate numbers of type la supernova per square degree
within various redshift intervals for the 3 delay times .1, 1 and 1.5 Gyrs. The
Hubble param eter is set to h=0.7, 1Q,m — 0.3, Oa = 0.7 and the dark energy
equation of state is set to w = —1.
Integrating this we can determine the expected number of supernova over a
given redshift interval.
It is clear from the graphs and tables th at the delay time can significantly
affect the number of SNe la, especially in high redshift intervals. Measurements
with 15 — 20% uncertainty can distinguish r = . 1 Gyrs from the others and a
5 —10% uncertainty can distinguish r = 1 Gyr from r = 1.5 Gyr.
4.4
Sim ulations and Fits
Now we study the feasibility of number counts with particular reference to
the SuperNova Acceleration Probe (SNAP). First we make estimates of the un­
certainties of the comoving number density and in the number of SNe we expect
to see per redshift. We will then use these uncertainties to simulate measure­
ments of the comoving volume element over some range of redshift. Then we use
maximum likelihood fits to determine various cosmological parameters.
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z
r =.1
r =1
r =1.5
z
r =.1
r =1
r = 1.5
0-.1
.01
.01
.01
1-1.1
3.1
4.6
5.6
.1-.2
.1
.1
.1
1.1-1.2
3.8
5.5
6.6
.2-.3
.1
.1
.1
1.2-1.3
4.5
6.4
7.4
.3-.4
.2
.3
.3
1.3-1.4
5.2
7
8
.4-.5
.4
.5
.6
1.4-1.5
5.8
7.7
8.7
.5-.6
.6
.8
1
1.5-1.6
6.3
8.3
9.3
.6-.7
.9
1.3
1.7
1.6-1.7
6.8
8.7
9.8
.7-.8
1.3
2
2.5
1.7-1.8
7.2
9
10.3
.8-.9
1.8
2.8
3.5
1.8-1.9
7.5
9.4
10.8
.9-1
2.4
3.7
4.5
1.9-2
7.7
9.7
11.3
Table 4.2: The number of type la supernova per square degree within redshift
bins of width .1 for the 3 delay times .1, 1 and 1.5 Gyrs. The Hubble parameter
is set to h=0.7, Qm — 0.3, Qa = 0.7 and the dark energy equation of state is set
to w = —1.
4.4.1
E stim ation o f U ncertainties
We consider measurement of the comoving volume element. This is given by
dV _
1
dN
dzdO,
nsNe{z) dzdQ,'
so we must estimate the uncertainty of the comoving rate density as compared
to the actual number of SNe observed in a given redshift interval.
The random and systematic uncertainties of SNAP project data regarding
the measurement of the luminosity and redshift of each SNe la will be negligible
compared to the uncertainties we wish to consider here. Incidentally, the detail
with which each SNe in the SNAP mission will be measured is greater than any
existing SNe measurements.
A systematic error is the problem of identifying all SNe la th at occur during
the survey. The observed number will be only a lower bound on this number.
Detection efficiency depends mostly on the magnitude of the SNe. They must be
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distinguished from the host galaxy. In [92] the comoving SNe rate is measured
using a sample of 38 SNe from the Supernova Cosmology Project data set [5]
between the redshifts 0.25-0.85 which cover ~ 12 squared degrees of sky. The
detection efficiency was estimated using simulated data over these fields searching
for the SNe within the synthetic images. Detection efficiencies were typically
greater than 85% for any any object with R magnitude over 23.5 and is largely
independent of the position of the star relative to the core of the host galaxy.
We must also determine how many supernovae need to be observed to mini­
mize the random error in measuring dN/dz . Using Monte Carlo simulated data
we determined the statistical uncertainty of the number of SNe la per redshift
interval is reduced to < 10% for ~ 2000 observations. This is reduced to < 5%
for ~ 3000 —4000 SNe la. These numbers are possible with a 1 —3 year survey
over 20 degrees of sky.
Besides this issue, the primary source of uncertainty is in the calculation of the
comoving number density. Errors appear in this context in different ways: first,
we must know the star formation rate; next, we need an initial mass function;
third, we combine this with the SFR and incorporate a delay time, which we
need to fit according to the data; finally, the efficiency of WD explosions must
be fit to the local rate.
Since the SNAP mission will identify thousand of supernovae every year with
only tiny uncertainties in their redshifts, we estimate th at we will be able to de­
termine the delay time parameter within ~ 10 —20%. This assumes no evolution
in the mechanism th a t controls the delay time.
Comparison of predictions of the luminosity density and observation [93] show
th at the d ata can be well modelled by fits of the SFR [84] and a Salpeter IMF up
to redshifts of z « 2. Above this redshift, uncertainties in the SFR become quite
large since the understanding of evolution of luminous m atter in early epochs is
complicated by light being absorbed by dust and reradiated in the far infrared.
The present combined uncertainty from the SFR and IMF are ~ 20%
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1.75
N
0 .75
0 .25
1.8
1.6
1.4
1.2
-1
0.8
W
Figure 4.12: Transition Redshift vs. w for models with Q,m = 0.3 and f^A = 0.7,
O.M = 0.2 and Oa = 0.7,0m = 0.3 and
= 0.8. (bottom to top)
Fortunately, this is sufficient for our needs since we are primarily interested in
the era during which dark energy begins to dominate. Consider the acceleration
of the expansion of the universe:
R
H qR
= Hl
3ru + 1
f Ro
~ ^ ~ aA\ T
{ ¥
3 ( w + l )
(4.24)
When R / (H qR ) = 0 the universe makes the transition to accelerated expan­
sion due to the presence of dark energy. If we calculate the redshift at which this
occurs for various values of the dark energy and dark m atter energy densities
and for varying equation of state, it is easy to see we are safe to consider only
redshifts in this restricted range.
Furthermore, once we go out to redshifts ~ 2 the ratio of dark energy to dark
m atter becomes small enough so th at dark energy is not dynamically important.
This is true for a variety of equations of state and values of Q,m and fhv •
The largest uncertainty at present is in the local SNe rate. Current values
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0.25
0.5
0.75
1
1.25
Re dshift
1.5
1.75
2
Figure 4.13: Ratio of dark energy and dark m atter vs. redshift for models
with flM = 0.3 and ftA = 0.7,
= 0.2 and QA — 0 .7 ,0 ^ = 0.2 and flA =
0.8.(bottom to top)
are only known within ~ 30 — 50% [92] [94] [95] [96] [97]. The poor quality of
these measurement reflect the lack of large scale SNe surveys to date. The very
large number of expected SNe from the SNAP project will greatly improve the
situation.
4.4.2
Confidence Intervals
All of the uncertainties mentioned above will become smaller over time, but,
until SNAP or some comparable survey is performed, many of the errors involved,
especially in determining the comoving rate, are difficult to quantify with preci­
sion.
Let us assume the uncertainty in our knowledge of the comoving rate dom­
inates systematic problems in detection efficiency. This will indeed be the case
with the SNAP mission which expects to be able to identify every supernova
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up to a redshift of 1.7 within 20 square degrees. We also assume there are a
sufficient number of observations we can ignore the statistical error in measuring
dN/dz. The combination of errors from fitting the delay time, the SFR/IM F
and the local rate amount to an approximately 50% uncertainty in measuring
the comoving volume element.
We will determine the likelihood of cosmic parameters from our simulated
SNe d ata from a y 2 statistic,
(-N
theory ( ^ i j & M j ^
A >
W
0
A^ j,s ( ^ j ) ) )
(4.25)
We will take the Hubble parameter h = 0.7 as a prior for convenience, but we
will not assume flatness.
Assuming the observed SNe numbers are normally
distributed our likelihood function will be
L cc e x p ( - y ).
(4.26)
The following figures are the la, 2 a, and 3 a confidence intervals for simulated
measurements of the comoving volume element between redshifts of z = .2 and
z = 2. The d ata is binned into A z = .1 intervals. Our current uncertainties
would give us confidence regions like figures 4.14, 4.15 and 4.16.
We can expect the SNAP survey to improve the quality of our knowledge of
the delay time and the measurement of the local comoving rate. However, we are
still left with a large uncertainty from the IMF and SFR. Let us optimistically
assume th at by the time a SNAP-like mission is undertaken astronomical obser­
vations will improve the uncertainties related to the SFR and IMF to ~ 10 —15%.
Since we will have a factor of 10-100 more supernova per A z = 0.1 this new data
will also give us a comparable improvement of uncertainty due to the local den­
sity of SNe and r , so the volume element will be measurable in each redshift bin
A z = 0.1 to about 20%. The projected confidence contours in this circumstance
will look like figures 4.19, 4.17, and 4.18.
Finally, it is interesting to know how well under control we would need our
random and systematic uncertainties to constrain Qm ,
and w to a level com-
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2 .5
2
1.5
1
0.5
0
2.5
-2
-1
1. 5
0.5
0
w
Figure 4.14: la, 2 a and 3 a confidence intervals for w and
uncertainty).
(50 percent
0.5
0.4
0.3
0.2
0.1
2.5
-2
1.5
-1
0.5
0
w
Figure 4.15: la, 2 a and 3 a confidence intervals for VtM and w (50 percent
uncertainty).
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1 .5
c?
1
0.5
0.1
0.2
0.3
0.4
0.5
Figure 4.16: lcr, 2 a and 3 a confidence intervals for Qm and
uncertainty).
(50 percent
1.4
1.2
1
cf
0 .8
0.6
0.4
0. 2
0
-2
-1
1.5
0.5
Figure 4.17: la , 2 a and 3 a confidence intervals for w and
uncertainty).
0
(20 percent
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0.5
0.4
0.3
0.2
0.1
2.5
1
1. 5
-2
0.5
0
w
Figure 4.18: la , 2 a and 3 a confidence intervals for Q,m and w (20 percent
uncertainty).
l
0.8
0.6
0.4
0.1
0.2
0.3
0.4
0.5
£2m
Figure 4.19: la , 2 a and 3 a confidence intervals for Q,m and Qa (20 percent
uncertainty).
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1
0.9
0.8
0.7
0. 6
0.5
0.4
1.6
1.4
-
-1
1.2
0.8
0.6
w
Figure 4.20: lcr, 2 a and 3 a confidence intervals for w and Qa (10 percent
uncertainty).
parable to the limits expected from the distance-magnitude relationship and
CMB anisotropy. Our present results show th at an uncertainty of < 10% will
satisfy this condition.
It is therefore guaranteed th at the observation of thousands of SNe la per
year by the SNAP project will very significantly increase our knowledge of the
dark energy component including our understanding of the equation of state w.
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df 0 . 2 5
0.15
-2
1
1.5
0.5
0
w
Figure 4.21: la, 2 a and 3 a confidence intervals for £Im and w (10 percent
uncertainty).
0.9
0.8
0.7
0.6
0.5
0.26
0.28
0.3
0.32
0.34
0.36
0.38
Qm
Figure 4.22: la, 2 a and 3 a confidence intervals for f1m and 11a (10 percent
uncertainty).
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Chapter 5
Conclusions
In this thesis we have analyzed two different methods of measuring cosmic
parameters and constraining the properties of the dark energy. These are (1) the
use of the Cosmic Microwave Background (CMB) and (2) the use of Supernovae
Type la (SNe la).
5.1
CM B
First, we studied the position of the first acoustic peak in the CMB power
spectrum.
We parameterized the dark energy as an energy density with an arbitrary
power law dependence (with parameter P ) on the scale factor. The position of the
Doppler peak is related to the angle subtended by the horizon at recombination.
The value of l \ depends primarily on the total energy density Dm + Da- The
overall shape of the power spectrum is fixed at decoupling, but l\ depends on
the geometry of the universe between then and now. The choice of P we use
for the dark energy alters t\. We are constrained to small values of this power,
P < 0.2, by the age of the universe and big bang nucleosynthesis. Varying P in
this way we can change Dm and Da by ~ 3%. The MAP constraints on Dm and
Da are not strong enough to fix P within the currently allowed range. However,
continual improvement in CMB data is to be expected and precision constraints
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will soon be available on all of these parameters.
This analytic computation of the first peak must necessarily be supplemented
by a constant determined numerically by the computer program CMFAST. The
actual scale being measured by
lies somewhere in between the acoustic horizon
and optical horizon. In the future a better understanding of the recombination
transition may enable us to determine this constant of proportionality.
5.2
SN e la
The second method we proposed as a probe of dark energy is number counts of
type la supernovae. The primary focus of future supernova surveys is to measure
the distance-luminosity relationship which utilizes both the extreme brightness
and homogeneity of this subclass of SNe. Since these supernovae are visible on
cosmologically relevant distance scales, this data will be an opportunity to apply
the number count versus redshift relationship.
The number of SNe in some solid angle at a given redshift depends on the
comoving volume element and comoving number density of the SNe. The dif­
ficulty of this test lies in extracting this number density from observations. In
fact, the extremely high expected efficiency of future SNe surveys will make the
feasibility of this approach almost entirely dependent on our ability to calculate
the comoving number density.
Although our understanding of the type la progenitors is incomplete, we can
use measurements of the cosmic star formation rate and initial mass function
and parameterize our ignorance by the delay time between formation of the
white dwarf and its evolution into a SNe la.
The delay time and local rate measurements will improve once SNAP and
similar surveys are undertaken. This means th at the primary problem in the
future is in the initial mass function and star formation rate. If these uncertain­
ties can be lowered to ~ 10 — 15%, about half to three quarters of the current
65
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uncertainty, then SNe number counts will provide constraints at least as strong
as proposed number count tests using galaxies and clusters of galaxies [98].
It should also be emphasized th at information from the comoving volume
element is complementary to distance-magnitude data. The confidence contours
in the Qm — a plane from number counts will be oriented orthogonally to
those derived from distance-magnitude relationship. Thus it may be possible to
tightly constrain cosmic parameters with SNe la alone, without the aid of CMB
measurements.
In the future this method could be improved not only by better measurements,
but by a calculation of the number density directly from the physics of SNe la
rather than using fits to data. Once the delay time is well constrained it will
help fix a particular model for the white dwarf explosion which will let us derive
a result from more fundamental physics.
It is also likely th at there is not only one correct delay time, but a distribution
of r ’s depending on the details of the progenitor system. This distribution may
depend on the mass of the companion star and the difference between the mass
of the white dwarf and the Chandrasekhar mass. If the spread of delay times is
comparable to the average value of r , then it is necessary to account for this in
our SNe la rate calculation.
We should also consider the prospect of using this procedure in reverse. Us­
ing cosmological parameters derived through other observations, we can work
backwards from the number counts to place constraints on the astrophysics of
SNe la and the star formation rate.
5.3
Closing Remarks
In this thesis we have explored two techniques th at have demonstrated their
value in cosmology. Analysis of the first acoustic peak in the cosmic microwave
background has allowed us to explore the dark m atter and energy th at shape
66
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the universe we find ourselves in today. The last days of many stars are also
precious to cosmologists, as number counts of supernovae can be used as probes
of dark energy. These two tools will continue to inform our understanding of the
universe.
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