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

код для вставкиСкачатьDark 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. D e d ic a te d to: M y wife, C h risty . V R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. 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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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 2 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 3 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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, 4 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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— . R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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) 6 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 7 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 8 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 9 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 10 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 17 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 18 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 . 20 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 22 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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) R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 ‘ 25 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 26 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. ( 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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) 28 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. , (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. = /> 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. (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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 53 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 54 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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% 55 R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 56 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 57 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission . 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- 58 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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). 59 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 60 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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). 61 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 62 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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). 63 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. 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. 67 R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission. REFERENCES [1] C. L. Bennett et al., arXiv:astro-ph/0302207. [2] J. 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