NOTE TO USERS T h is re productio n is the best copy available. ® UMI R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. U n iv e r s it y of C a l if o r n ia Los Angeles E ffects o f B uckyballs and C osm ic S trin gs on th e C osm ic M icrow ave B ackground A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Astronomy by A m y S h iu -M ei Lo 2005 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. UMI N um ber: 3202772 IN F O R M A T IO N TO U SER S T h e quality o f th is re productio n is d e p e n d e n t upon th e q uality o f th e copy subm itted. 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F urther reproduction prohibited w itho ut perm ission. © Copyright by Amy Shiu-Mei Lo 2005 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. The dissertation of Amy Shiu-Mei Lo is approved. /? w ' O Rene Ong Veerayalli Varadarajan Kastushi Arisaka Edward L. Wright, Committee Chair University of California, Los Angeles 2005 ii R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. T able of C ontents 1 In tr o d u c tio n .................................................................................................... 1 2 Buckyball R a d ia tio n .................................................................................... 6 2.1 Anomalous Free-Free E m is sio n s .......................................................... 6 2.2 Small dust g r a i n s .................................................................................... 11 2.3 The Monte Carlo S im ulation............................................................... . 14 2.4 3 2.3.1 Buckyball Initial Conditions . . ; .......................................... 15 2.3.2 Hydrogen Collision Process .................................................... 18 2.3.3 Photon Collision P ro c e s s .......................................................... 21 2.3.4 A Short Justification for Classical M echanics...................... 24 Simulation A nalysis................................................................................. 27 2.4.1 Typical Simulation N u m b ers.................................................... 29 2.4.2 Emission C h a racteristics.......................................................... 32 2.4.3 Anomalous Emission? ............................................................. 39 2.4.4 Possible O bservations................................................................. 47 Signatures o f C osm ic Strings in th e C osm ic M icrowave Back ground .................................................................................................................... 3.1 Topological Defects 50 ............................................................................. 50 3.1.1 Symmetries and G r o u p s .......................................................... 55 3.1.2 Fundamental Homotopy G r o u p ............................................. 59 3.1.3 Cosmic Strings and G U T s ....................................................... 64 iii R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 3.2 WMAP ..................................................................................................... 69 3.3 String Search: The E dgefinder.............................................................. 72 3.3.1 Simulated M a p s ........................................................................... 81 3.3.2 Edgefinder L im its ........................................................................ 83 3.3.3 Edgefinder Limits fromEdgeworth Coefficients...................... 86 3.3.4 Multiple String M a p s .....................................................................100 3.4 WMAP R e s u lts ........................................................................................... 102 3.4.1 WMAP 2nd Year Sim ulation........................................................103 3.5 A Cosmic String C an d id ate?.....................................................................104 3.5.1 String S e a r c h ..................................................................................105 3.6 Simulated PLANCK R e s u lts .....................................................................108 4 C onclusions and Future W o r k ................................................................... 113 5 B ibliography ................................................................................................... 117 iv R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. L is t 2.1 of F ig u r e s The correlated intensity fluctuations of the microwave and farIR emission of various experiments, The y-axis is the emissivity measured in units of optical depth at 100 microns. From Finkbeiner, 2003. The data points are from: Wilkinson Microwave Anisotropy Probe (WMAP) (Bennett et al, 2003), Green Bank (GB 140) (Finkbeiner et ah, 2002), Tenerife (de Oliveira-Costa et ah, 1999), Cottingham 19.2 GHz (de Oliveira-Costa, Tegmark, Page, & Boughn, 1998), OVRO (Leitch et ah, 1997) Saskatoon (de Oliveira-Costa, Tegmark, Page, & Boughn, 1998), and, Dif ferential Microwave Radiometer (DMR) (Kogut et ah, 1996). An explanation of all the lines are in the te x t......................... 2.2 8 A graph of the emissivity (ju) per Hydrogen atom (n# is hydro gen density). Graph taken from Draine & Lazarian (1999a). The data points with error bars are data from five experiments listed in the legend. Also plotted are dashed, dotted, and solid lines representing vibrational dust emissivity of four different dust tem perature and spectral index, /3T . The grey band is the dust corre lated free-free emission. A schematic of our buckyball radiation in the anomalous free-free emission frequencies plotted in the thick dotted line labeled Buckyball.................................................................... 2.3 A picture of a Buckminsterfullerene molecule, C 2 4 # 22- 10 The light grey spheres are the carbon orbitals, and dark grey spheres are the hydrogen orbitals. The two dehydrogenation sites are indicated by arrows............................................................................................................. v R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 16 2.4 A plot of the photon probability distribution used to generate the photon density in each simulation......................................................... 2.5 Plot of the emission frequency vs. 22 simulation time of the C2o buckyball in the CNM region. Solid lines indicate times during which the buckyball is radiating. The dashed lines correspond to a collision event.......................................................................................... 2.6 25 The Energy-frequency plot for C2o buckyball in the WNM region. The peak is taken as the peak emission frequency, and most graphs show a clear drop off in the energy emitted vs. frequency after the peak.............................................................................................................. 2.7 28 Spectral index vs. simulation Hydrogen density for each of the 4 tem peratures simulated. Crosses indicate hydrogen tem perature T = 25 K, Squares are T = 6000 K, Diamonds are T = 8000 K, Triangle are T = 100 K ............................................................................ 2.8 35 Spectral index vs. simulation tem perature for each of the 4 densi ties simulated. Crosses indicate hydrogen number density, n H, = 100000 cm-3 , Squares are nn = 0.4 cm-3, Diamonds are n # = 0.1 cm-3 , and triangle are nn = 30 cm ” 3 ................................................... 2.9 36 Peak emission frequency vs. simulation tem perature for each of the 4 densities simulated. Crosses indicate hydrogen number density, n Hl = 100000 cm” 1, Squares are n H = 0.4 cm” 1, Diamonds are Tin = 0.1 cm” 1, and triangle are n H = 30 cm ” 1 ................................. 37 2.10 Peak emission frequency vs. simulation Hydrogen density for each of the 4 temperatures simulated. Crosses indicate simulation tem perature, T = 25 K, Squares are T = 6000 K, Diamonds are T = 8000 K, Triangles are T = 100 K ........................................................... vi R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 38 2.11 Power vs. Frequency of the composite emission from all three buckyballs in equal proportions for the WIM simulation. The thin line is a nearest 8 neighbor sum to show a smooth spectrum. The thick dot dashed line is a flat (P oc zT) spectrum for comparison. . 42 2.12 Power vs. Frequency and a smoothed version of the composite emission from all three buckyballs in equal proportions for the CNM simulation......................................................................................... 43 2.13 Power vs. Frequency and a smoothed version of the composite emission from all three buckyballs in equal proportions for the CNM simulation........................................................................................ 44 2.14 Power vs. Frequency and a smoothed version of the composite emission from all three buckyballs in equal proportions for the CNM simulation.................................................... 3.1 45 Graph a): the double welled potential which leads to a p 4 -kink, with A = 1 and 77 = 1.7. Graph b): the solution to the potential depicted in a). The kink through the origin which interpolates between the two vacuum solutions is the defect called the </>4 -kink. 3.2 54 Figure A: / (the solid lines) and g (the dashed lines) are homotopic to each other, but not to c (the dotted lines). The grey filled circle is a hole in the manifold. Figure B: / and g are freely homotopic, because a path c can be constructed to wind around the defect separating them ......................................................................................... vii R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 60 3.3 A 3-D representation of the Edgefinder. The z-axis represent filter values, and the x and y axis are the pixel numbers. The right bottom inset is a plot of a slice along the y = 0 plane of the filter, with the axis in units of pixels................................................................ 3.4 Edgefinder values around a string horizon. The Edgefinder is ori ented North (up) to South (down). The light blue circles are masked pixels in the WMAP data due to foreground sources. 3.5 75 . . 77 The response function of the Edgefinder Filter. Note the peak at I ~ 200, around where the first Doppler peak of the CMB anisotropy angular power spectrum occurs. The different lines correspond to different a angles of the orientation of the filter with respect to the north-south Galactic axis..................................... 3.6 Plot of the input string tem perature Tf vs. the maximum EV of the set.......................................................................................................... 3.7 80 84 Detail of the plot of the input string temperature T f vs. the max imum EV. Also plotted are the multi-string set data as well as limits from the No-String sets. The stars are the Max EV for each simulated map containing a string of strength shown on the x-axis; the square indicate simulated maps containing multiple strings; the triangles indicate maps containing no inserted strings...................... 3.8 Plot of the input string temperature Tf vs. the 3rd Edgeworth coefficient.................................................................................................... 3.9 86 95 Plot of the input string temperature Tf vs. the 5th Edgeworth coefficient................................................................................................. . viii R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 96 3.10 Plot of the input string tem perature T f vs. the 7th Edgeworth coefficient..................................................................................................... 98 3.11 Detail of the plot of the input string tem perature Tf vs. the 7th Edgeworth coefficient. Also plotted are the multi-string set data as well as limits from the No-String sets.............................................. 3.12 Transfer function for the PLANCK simulation; the filter had R A D 99 = 0.5.....................................................................................................................109 3.13 Input string tem perature vs. the maximum EV of the EV set for simulated PLANCK d ata............................................................................ I l l 3.14 Blow up of input string tem perature vs. the 7th Edgeworth coef ficient of the EV set for simulated PLANCK d ata.................................112 ix R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. L is t of T ables 2.1 Simulation Parameters of the 3 Buckyballs ................................... 2 .2 Environmental Properties of Simulation E n v iro n m e n t........... 2.3 Buckyball Power Law Spectral Indices, /?, and their oy fitting co efficients 2.4 13 19 30 Peak Frequency, u p of buckyball emission and power at peak fre quency, P ( u p) 31 2.5 Photon vs. Hydrogen Impact 2.6 Simulation Rotational Energy of B u c k y b a lls ........................... 33 2.7 Thermal Rotational Frequency of Buckyballsa ........................ 40 3.1 Relevant WMAP Characteristics 3.2 Partially Exposed Strings 3.3 Edgefinder Gain C a lib ra tio n ........................................................ 3.4 Number and Strength of Input Strings for Multi-String Simulated ............................................................. 32 ....................................................... 71 .................................................................... 78 79 Maps......................................................................................................101 3.5 Select Properties of the WMAP composite QVW D ata.............103 3.6 Edgefinder Values for CSL-1 ................................................................. 106 x R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. A cknow ledgm ents Getting a PhD. is hard work, and the following people deserve mention: Ned Wright, for content, reading N drafts (N —>■oo as t —>■graduation), and always having funding. Steve Warwick, for support, a proposal, and the occasional smack. XI R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. V 1975 Born, Taipei, Taiwan 1997 B.S.Honors (physics), it a Brown University, RI 1999 M.S. (Astronomy and Astrophysics), UCLA, Los Angeles, California. 1998--2000 Cota Robles Fellowship 1999--2001 Teaching Assistant Physics and Astronomy Department, UCLA 1999--present Research Assistant Physics and Astronomy Dept, UCLA xii R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. A bstra ct of the D i s s e r t a t io n E ffects o f B u ck yb alls and C osm ic S trin gs on th e C osm ic M icrow ave B ackground by A m y S h iu -M ei Lo Doctor of Philosophy in Astronomy University of California, Los Angeles, 2005 Professor Edward L. Wright, Chair This thesis consists of two Cosmic Microwave Background related projects: a simulation of an anomalous foreground component, and a search for a distinct background signature. The Cosmic Microwave Background forms one of the three major pillars of support for the Big Bang theory of the origin of the universe, and is an important source of information about the early universe. The first of two components of this thesis proposes a possible explanation for an anomalous component of our Galaxy’s foreground contribution to the Cosmic Microwave Background. The second component of this thesis searches for signatures of early universe phase transition products called Cosmic strings. We propose that Fullerene molecules, or, buckyballs, may compose part of the interstellar medium. Their thermal rotational angular velocity is on the order of GHz, which lies within the range of the “anomalous free-free emission” that is correlated with interstellar dust found in Galactic foreground maps of the Cos mic Microwave Background. We have written a Monte Carlo code to simulate the radiation from spinning partially hydrogenated fullerene molecules. We quantify the emission, compare it to the Galactic foreground, and find that if C2o com- xiii R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. prises 0.5% of the Galactic carbon budget, then there are sufficient buckyballs to make it possible for fullerene molecules be responsible for the anomalous free-free emission. We also performed a search for signatures of cosmic strings in the the Cosmic Microwave Background data from the Wilkinson Microwave Anisotropy Probe. We used a digital filter designed to search for individual cosmic strings and found no evidence for them in the WMAP CMB anisotropies to a level of A T / T ^ 0.29 mK. This corresponds to an absence of cosmic strings with Gfi ^ 1.07 x 10" 5 for strings moving with velocity v = c / \ / 2. We have searched the WMAP data for evidence of a cosmic string recently reported as the CSL-1 object. We found that if the signatures at CSL- 1 were produced by cosmic strings, these strings would have to move with a velocity ^ 0.94c. We also present preliminary limits on the CMB data that will be returned by the PLANCK satellite for comparison. With the available information on the PLANCK satellite, we calculated that it would be twice as sensitive to cosmic strings as WMAP. xiv R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. CHAPTER 1 In tro d u ctio n The Cosmic Microwave Background (CMB) is a rich source of information about the early universe. Studies of the CMB carried out by satellite surveys can also give us information about the contents of cur own Galaxy. This thesis deals with both aspects of the CMB: it contains an investigation into a possible source for an anomalous component of the CMB foreground, as well as a search for signatures of topological defects in the CMB signal itself. The very early universe was an extremely hot place; electron and positron pairs coexisted in abundance. As the universe expanded, it cooled, and these “ pairs annihilated to create many photons. The photons interacted with charged particles in the early universe and remained in equilibrium, so th at the photons we see from the early universe traces the distribution of m atter at that epoch. As the universe cooled, the energy dropped below the electron-proton binding energy, and the universe became neutral when hydrogen atoms were formed. Photons could no longer interact in a significant way with the surrounding m atter, and they were able to free-stream with the expansion of the universe. This epoch is known as Last Scattering (LS), and it occurred approximately 300,000 years after the Big Bang. Structures of the universe were frozen into the CMB at LS. These photons have been redshifted since they decoupled from m atter, and they now fall into the microwave region of the electromagnetic spectrum. For this reason, these 1 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. photons are called the Cosmic Microwave Background. Due to the fact th at the CMB traces m atter distribution at LS, it has been the focus of numerous studies ranging from the extraction of the primordial power spectrum (which gives us information on the seeds of the current distribution of clusters and galaxies), to searches for the formation of the first stars (whose UV radiation would trigger an episode of reionization of the hydrogen in the universe). The second portion of this thesis contributes to our knowledge of particle physics by searching in the CMB for signatures of by-products of certain particle physics theories known as topological defects. In particular, we search the CMB signals returned by NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) for a form of topological defect known as cosmic strings. Launched in 2001, the satellite WMAP continues to conduct an all-sky survey of the CMB by observing the microwave universe. Any measurement it makes of the CMB is actually the cosmic CMB signal combined with non-cosmological noise. The strongest source of noise is microwave processes within our own Galaxy. An accurate measurements of the CMB must carefully remove the Galac tic foreground. Many maps of the Galactic foreground exist to serve this purpose. However, because the processes within our Galaxy are very complex, not every identified component can be adequately explained. It is, in fact, well known th at there is an anomalous component in the Galactic foreground in the frequencies observed by WMAP. The first portion of this thesis is a simulation of radiation from Buckminsterfullerene molecules motivated by the presence of this unidentified component of Galactic emission, frequently referred to as the anomalous free-free emission (Leitch et ah, 1997). This emission was first discovered in studies of the Galactic foreground (e.g. Leitch et al. 1997 and de Oliveira-Costa et al. 1999) as a component with a spectral index resembling 2 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. free free emission which could not be satisfactorily fitted by thermal dust emission or dust-correlated gas emission. The emission starts to deviate from fits to known sources below 60 GHz, and the mechanism responsible for this emission has been difficult to identify. One of the most popular explanations for this emission came from Draine & Lazarian (1998), who argued that its source is ultra small spinning dust grains. Recent debate over the origin of this emission was kindled by the WMAP mission itself. Bennett et al (2003) used a Maximum Entropy Method (MEM) analysis to fit known components of the Galactic foreground to the WMAP data, and they found th at their model showed that less than 5% of the anomalous emission came from spinning dust grains. Finkbeiner (2003), however, performed an alternate analysis on the same WMAP foreground data and concluded that spinning dust grains can still con stitute a major portion of the anomalous free-free emission. This claim does not challenge the authenticity of the CMB data produced by WMAP, as the chief concern of Bennett etal. was an accurate representation of the flux from the Galactic foreground, regardless of the underlying mechanism. W ith renewed de bate about the source of the anomalous emission, I followed Draine & Lazarian (1998) and put forward a specific candidate for this emission in the form of par tially hydrogenated Buckminsterfullerene molecules, which are called buckyballs for short. This portion of the thesis is detailed in Chapter 2. With the Galactic foreground removed, the WMAP science team has produced first year maps of the CMB. Chapter 3 of this thesis contains a search for cosmic strings in the CMB. Topological defects (TD) such as cosmic strings have been proposed as large scale structure (LSS) candidates but have fallen out of favor due to the lack of evidence for their existence. TDs form as results of phase transitions, 3 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. many of which occurred in the first year after the Big Bang. They are ideal LSS candidates for three reasons. First, defects reflect the energy scale of the phase transition, and, therefore, the earlier the phase transition, the higher the energy density of the topological defect, and the larger the gravitational potential well for structure formation. Second, these high energy phase transitions occurred a long time before observational evidence of the first stars and galaxies. This would give gravity enough time to coalesce m aterial to form structures. Last, cosmic strings, the most popular class of topological defects, are geometrically similar to the filamentous LSS first observed in deep redshift surveys (see early papers on LSS especially Vachaspati 1986). However, despite their convenience, results of the COsmic Background Ex plorer (COBE), an all-sky CMB satellite, showed th at cosmic strings were not responsible for large scale structure formation, because the CMB anisotropies were consistent with a Gaussian signature, while numerical simulations of all topological defects show th at they would leave distinct non-Gaussian signatures in the CMB (see, e.g., a review by Allen et ah 1997). The COBE results were interpreted to mean th at the observed LSS’s are products of perturbations with Gaussian seeds, and hence cosmic strings fell out of favor as a possible mechanism for structure formation. To date, there does not exist any conclusive observational evidence of the existence of cosmic strings or any other TDs such as magnetic monopoles. On the other hand, COBE data do not preclude their existence in small enough numbers such th at they do not appreciably affect the CMB angular power spectrum; there could be on the order of a few cosmic strings in the visible universe. Furthermore, TDs are necessary products of certain types of phase transitions in the early universe, and they are useful as possible explanations for variety of phenomenon 4 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. such as Gamma Ray Bursts (e.g. Berezinsky et al. 2001), and ultra high energy cosmic rays (see e.g. review article Torres & Anchordoqui 2004 and references therein). This persistent interest in topological defects motivated our production of their observational limits. The cosmic strings th at do exist need to be few in number. W ith th at in mind, I performed a search for individual cosmic strings based on all-sky CMB survey data. The data returned from WMAP have an angular resolution of 13 arc-minutes which enabled searches for individual cosmic strings, because I expect to start to be able to detect cosmic strings on angular scales of approximately 2 degree scale. While maps with greater resolution have been produced by balioonhorne experiments (e.g. Bouchet, Peter, Riazuelo, & Sakellariadou 2002), these experiments only observe a very small area of the sky. WMAP covers the entire 4n steradian of sky and is therefore uniquely suited for searches of individual cosmic strings. Our search involves a pixel by pixel filtering method, as opposed to the wavelet analysis th at some others have performed on the small scale CMB experiments (e.g. Barreiro & Hobson 2001), although the underlying principles of searching for non-Gaussian signatures are similar. This thesis is divided into two sections. Chapter 2 details our buckyball simulation to model an anomalous component of the CMB foreground. Chapter 3 is a search for signatures of cosmic strings in the CMB data. I present conclusions to both projects in Chapter 4. o R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. CHAPTER 2 B u ck y b a ll R a d ia tio n 2.1 A n o m a lo u s F ree-F ree E m issio n s The CMB foreground contains more power at mid GHz frequencies than can be explained by conventional theories. The term “anomalous emission” was first used by Leitch et al. (1997) to describe a component of Galactic foreground that could not be fit by thermal dust emission. In fits to the 14.5 GHz Galactic foreground data, Leitch et al. (1997) discovered a component with a temperature index of 3t ~ 2 (where T oc v~ 0T, v is the frequency) th at was in excess of what was expected. This tem perature index corresponds to the signatures of free-free emission, which has a power spectral index of {3 = 0 , where f3 is defined as, P(v) oc iA (2.1) where P is the power. This component has now acquired the name “anomalous free-free emission” , although “free-free” may be misleading as we now know that free-free emission constitutes only a fraction of the total anomalous emission. Due to the fact that its signature was discovered via experiments designed to quantify the Galactic foreground, this anomalous free-free emission is sometimes called “Foreground X” (de Oliveira-Costa, Tegmark, Page, & Boughn 1998). In general, the Galactic foreground at GHz frequencies is composed of three parts: synchrotron, free-free, and dust emission. As the source of the anomalous 6 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. free-free emission is unclear, we discuss evidence for and against each mecha nism to motivate our work in explaining this anomalous emission with a new mechanism. Galactic synchrotron emission comes from the interaction of magnetic fields and accelerated electrons. Bennett et al (2003) found th at the spectrum of syn chrotron emission at GHz frequencies typically has a tem perature index of fix ~ 2.5 to 3, while the anomalous emission is best fit with pT ~ 2. In addition, the data taken by Reich & Reich (1988) indicated th at the anomalous emission does not correlate with the observed synchrotron emission at the lower frequencies of 408 MHz and 1.42 GHz. We rule out synchrotron processes as the source of the anomalous emission. The anomalous emission is spatially associated with Galactic free-free emis sion. Finkbeiner (2003) recently mapped the free-free “haze” permeating the Galaxy; this haze accounts for a portion of the anomalous emission, as shown in Figure 2.1, taken from Figure 2.1 of Finkbeiner (2004). The free-free haze is the solid line labeled as “free-free” in the plot. In addition to the free-free haze, there is a component due to thermal emission from vibrational dust, extrapolated from higher frequency data, which also constitutes a small part of the anomalous emission. W hat is the source of the free-free haze? Galactic free-free radiation arises from plasma interactions in the interstellar media. Free electrons come from ionized gas, usually as a result of UV photodissociation. This is an equilibrium process, which necessarily means there should be recombination radiation, so a search for free-free emission involves looking at Galactic H a regions. McCullough (1997) showed th at the H a gas that is spatial correlated with the anomalous freefree emission did not account for all of the anomalous emission. Typically, the 7 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. - - - - - - - - - ,- - - - - - - - ,- - - - - - - J- - - - - - - - - - (— 1- ,— j- - - - - - - - - - - - - - - - - - - - - - - - - - - ,- - - - - - - - - - - - - - - - ,- - - - - - - - - - - !- - - - - - - - - 1- - - - - - - 1- - - - - - 1- - - - - 1- - - - 1 1000.0 j- _ • WMAP high latitude "spin"+thermal : X GB 140ft - Lynds 1622 & LPH 201.663+1.643 : .4 Tenerife (b>20) ' □ Cottingham (t» 3 0 ) ; ■ O VRO O Saskatoon 100.0 w —3 10.0 ooi 1.0 WNM WIM ■ 0.1 10 100 v (GHz) Figure 2.1 The correlated intensity fluctuations of the microwave and far-IR emis sion of various experiments, The y-axis is the emissivity measured in units of optical depth at 100 microns. From Finkbeiner, 2003. The data points are from: Wilkinson Microwave Anisotropy Probe (WMAP) (Bennett et al, 2003), Green Bank (GB 140) (Finkbeiner et ah, 2002), Tenerife (de Oliveira-Costa et ah, 1999), Cottingham 19.2 GHz (de Oliveira-Costa, Tegmark, Page, & Boughn, 1998), OVRO (Leitch et al., 1997) Saskatoon (de Oliveira-Costa, Tegmark, Page, & Boughn, 1998), and, Differential Microwave Radiometer (DMR) (Kogut et al., 1996). An explanation of all the lines are in the text. 8 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. amount of dust correlated Ho emission was a factor of 3 to 10 times less than the anomalous emission. This can be seen in Figure 2.2, from Draine & Lazarian (1999), where the dust-correlated free-free emission is plotted as a band of grey dots, under-representing the measured emission (the data points) by up to an order of magnitude. Finally, gas above 104 K may emit free-free radiation without significant contributions of recombination radiation. These gas molecules emit mostly in the X-ray band, and searches for spatial correlations between X-ray gas with the anomalous free-free emission have come up negative (Finkbeiner et al. 1999). Known mechanisms are insufficient to explain all the anomalous emission. The main interstellar dust grain emission mechanism is thermal emission; for the majority of the cool dust in the Galaxy, the peak of this emission at frequen cies greater than 100 GHz. According to Draine & Lazarian (1999a), at GHz frequencies there are two other major dust radiation mechanisms: electric dipole and magnetic dipole radiation. Simulations dene by Draine & Lazarian (1999) showed th at unless the magnetic dust grains are specifically tailored, they con tribute too little rotational emission to fully account for the anomalous emission. This leaves electric dipole radiation, which is radiation coming from rotating dust grains with electric dipoles. Only very small dust grains can rotate fast enough to emit in the frequency range of the anomalous emission. Small dust grains is therefore a possible explanation for the anomalous emission. Finkbeiner (2004) investigated Draine & Lazarian (1999) dust emission for various interstellar regions. These regions include Cold Neutral Medium (CNM), Warm Neutral Medium (WNM), and Warm Ionized Medium (WIM). These dust models span a large range in emissivity, which reflects the large uncertainty in the dust grain models. This work hopes to limit some of these uncertainties by modeling radiation from a specific dust candidate. We hope to shed some light 9 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. — I— 10 -3 9 I— I— I— I--------------------------- 1-------------------1-------- A de O l i v e i r a - C o s t a e t a l 1998 ♦ de O l i v e i r a - C o s t a e t a l 1997 ■ L eitch e t al 1997 ▼ Lim e t al 1996 .♦* O K o g u t e t al 1996b 1 0 -40 \ I N X S-, m w1 0 -41 iao Sh <D K U d u st-c o rre la te d free-free / /cP/qr /<o 10 - 4 2 A'A A / ' // / 10 -4 3 10 A »V V/ /V A m 20 30 50 F r e q u e n c y (GHz) 100 200 Figure 2.2 A graph of the emissivity ( j / per Hydrogen atom (n# is hydrogen density). Graph taken from Draine Sz Lazarian (1999a). The data points with error bars are data from five experiments listed in the legend. Also plotted are dashed, dotted, and solid lines representing vibrational dust emissivity of four different dust temperature and spectral index, 0t - The grey band is the dust correlated free-free emission. A schematic of our buckyball radiation in the anomalous free-free emission frequencies plotted in the thick dotted line labeled Buckyball. 10 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. on the process of spinning dust grains as the source of the anomalous free-free emission. 2.2 S m all d u st grain s In order to chose a candidate, we narrow the list by examining what we already know about the anomalous free-free emission. Since the grains radiate by rota tion, we can do a first estimate on the grain size based on the necessary rotational frequency. A “normal” dust grain of radius 1000 A contains roughly 2 x car 108 bon atoms. Typical thermal angular speed, cu, of such a dust grain at tem perature T is given by, ( 2 .2) where I is the moment of inertia for the dust grain, and k the Boltzmann constant. The factor 3 reflects the 3 degrees of rotational freedom for a sphere. At T = 50 K, the angular velocity of this 1000 A spherical dust grain is ^ 3.6 x 105 rad s e c '1, or 57 kHz. Compared to the anomalous emission at ~ 30 GHz, this normal dust grain spins too slowly. For these large dust grains, suprathermal rotation may be caused by mechanisms such as multiple recombination events with interstellar hydrogen (Purcell, 1979). However, even these mechanisms only have a maximum rotational speed of uj 109 rad s 1 (W hittet, 1992), causing MHz emission: still too low. On the other hand, an ultra-small dust grain can achieve a much higher rotational frequency. At 50 K, a 3 A dust grain containing 30 carbon atoms will have a thermal rotational velocity of cu ~ 3.1 x 10u rad s e c '1, or, 50 GHz, which is in the range of the anomalous emission. There is one further advantage to ultra small dust grains: according to Mathis et al. (1977), the number density, dN, of 11 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. interstellar dust obeys the following relationship, d N oc a ~ 3 Sda (2-3) where a is the radius of the dust grain. There are many more small dust grains than large grains, but most of the mass of dust is in large grains. The major drawback of small dust grains is that they are fragile. Barely larger than a molecule, ordinary ultra-small dust grains form either by adhesion of small molecules, or by fragmentation of larger particles. Many of these dust grains are easily destroyed by UY photons. The mean Galactic UV field is ~ 102 photons cm - 2 s - 1 sr-1 ; our ultra-small dust grains candidate must be stable against this UV field. G uhathakurta & Draine (1989) have calculated the survival time of dust grains against UV photons as a function of their size. Graphite dust grains with 20 carbon atoms would have a mean lifetime of 1011 seconds, or about 3000 years. Silicate dust grains with 30 atoms will have a mean lifetime of a little more than 1010 seconds. Such dust grains are not the ideal candidates because they would need a constant replenishing source. These considerations led us to propose that Buckminsterfullerenes, or buckyballs, are better spinning dust grain candidates than “normal” silicate or graphite dust grains. Aside from C&o, the famous geodesic dome-shaped molecule, there is a whole family of buckyballs ranging from 20 to 300 carbon atoms in ellipsoidal configurations composed of various combinations of benzene and penta-carbon rings. By the virtue of these benzene rings, buckyballs belong to the class of polycyclic aromatic hydrocarbons (PAHs). Specifically, we consider partially hy drogenated buckyballs, e.g. C 6 oHeo_n where n is the number of lost hydrogens via the dehydrogenation process (where a hydrogen is dissociated from the rest of the molecule). In interstellar conditions, is usually the result of an impact with a photon of specified energy. We discuss dehydrogenation later in more detail. 12 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.1. B uckyball Simulation Parameters of the -3 Buckyballs R ad iu s T ypical D ipole M om ent ^initial (p h o to n rate) (M OI) (A) (Debye) (GHz) ( s ' 1) g cm2 C 20 3.41 2.0612 38 4.50 x 10- 7 l . l x l 0 “ 37 c24 3.57 4.9937 29 4.71 x 10- 7 1.9x 1 0 ~ 37 C60 4.96 8.1410 13 6.55 x l O -7 1 1 .4 x l0 " 37 Buckyballs make compelling ultra small dust grain candidates. First, they are small; three species of buckyballs and their characteristics are listed in Table 2.1. CqqH^q has a radius of ~ 5A, and C 2 0 H 20 has a radius of ~ 3.5 A. The compact structure of buckyballs means that buckyballs have the smallest moment of inertia for a given mass of carbon. Larger dust grains are more stable than smaller dust grains, but their moments of inertia are too large to spin in the GHz range. At the same time, buckyballs are not large enough so that their collision cross section to UV photons and hydrogen atoms cannot be neglected. This means that they will encounter enough photons to significantly alter their emission characteristics. In this way, their emission spectra will be very non-thermal, and the spectra will probably (as we indeed will show) obey a power law. The second reason to consider buckyballs is that they are very strong. Unlike most dust grains whose formation relies on adhesion, buckyballs are chemically bonded molecules. Their hollow spherical structure allows deformations under collision to absorb impact energy which prevents fragmentation. Buckyballs can also withstand high temperatures. Experiments have shown that buckyballs, specifically, Cgo, remain stable up to 1000 K. The smaller PAHs can be excited to temperatures as high as 3000 K without breaking apart (Omont, 1986). The general class of PAHs has been proposed to be a component of interstellar dust 13 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. by Dorm as early as 1968. Aside from physical reasons to consider buckyballs, they are appealing for simulations. These molecules are nearly spherical which enables much simpli fication when considering their angular momenta and inertia. Their particular dehydrogenation process meant that their dipoles moments can be easily calcu lated. We note th at there are at the present, no unambiguous interstellar detections of signatures of buckyballs. There have been extensive observations of general PAH spectral features (starting with, e.g. Duley 1973, to e.g. Rapacioli et al. 2005), and benzene has been discovered in many circumstellar regions (Cernicharo et ah, 2001). More and more PAH emission lines are being discovered, (see e.g. Peeters et al. (2004), and a discussion of their relation to the anomalous free-free emission (Iglesias-Groth, 2004)). To date, there are only two unambiguous extra terrestrial signatures of buckyballs. The first is in the vaporized product of the Allende meteorite (Becker et al., 1994). The second was found in craters formed on the Long Duration Exposure Facility (di Brozzolo et ah, 1994). Terrestrial buckyballs are formed by vaporizing graphite. We speculate that similar circum stances around carbon stars allow the formation of buckyballs as condensates. 2.3 T h e M o n te C arlo S im u la tio n This portion of the thesis describes a Monte Carlo simulation of the electric dipole radiation coming from buckyballs experiencing collisions with interstellar hydrogen atoms and Galactic photons. My code tracks the angular momentum of an individual buckyball as it suffers collisions that alternately spin up or spin down the molecule. From the changes in the angular momentum, we deduce 14 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. the energy output and obtain a spectrum of the buckyball’s radiation from the energy-time analysis. The code was written in C + + . 2.3.1 B uckyball Initial C onditions Buckyballs are composed of a core shell of carbon atoms, with the hydrogens radially extended from the carbons. We simulate three different buckyballs: C2 0 H 2 0 -T1 , C 2 i H 2 i ~n, and CeoHeo^n, where n is the number of hydrogens lost during the dehydrogenation process. Throughout the simulation, I trace the an gular momentum of the buckyball and generated from th at the energy, rotational frequency, and radiated power. This requires a knowledge of the moment of in ertia of the buckyball. As part of the initial conditions, I explicitly calculate the moment of inertia tensor, I, for each buckyball, in the buckyball center of mass frame, = E m *(c2 - 4 ) (2-4) i Ikj = - ' 5 2 m irijrik, (2.5) i where j and k runs from 1 to 3, representing the three Cartesian coordinates, + is the distance from the center of the buckyball to the ith particle in the system, r^ is the j th coordinate distance from the center of the buckyball to the ith particle, and rrii is the mass of the ith particle. The number and position of the carbon atoms is fixed and their positions are static relative to each other. With the interstellar medium (ISM) hydrogen to carbon ratio of 103'5, bucky balls are likely to have formed fully or nearly fully hydrogenated. A fully hydro genated buckyball is neutral and symmetric with no dipole moment. However, the bond between the hydrogen and the carbon is relatively weak and can be dissociated. The leftover electron is easily lost so the whole molecule stays neu- 15 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Dehydrogenation Figure 2.3 A picture of a Buckminsterfullerene molecule, CzaH-2 2 - The light grey spheres are the carbon orbitals, and dark grey spheres are the hydrogen orbitals. The two dehydrogenation sites are indicated by arrows. tral. However, the carbon atoms surrounding the missing hydrogen site will incur an excess local charge, so a resonance structure forms in the surrounding carbon bonds, creating an electric dipole. Once one hydrogen is detached, the next hy drogen detachment will likely be from a carbon adjacent to the first detachment, since this allows the two carbons which have lost their hydrogens to form a car bon double bond, which is a more stable structure than two single bonds (Rubin, 2000). A picture of a partially dehydrogenated buckyball is shown in Figure 2.3. The light grey spheres indicate carbon orbitals and the dark grey spheres are the hydrogen orbitals. This is a C 2 4 H 2 2 molecule, with the two dehydrogenation sites indicated by the arrows. 16 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. The value of the dipole moment can be calculated by standard chemistry programs such as Spartan Pro, which semi-empirically solves the Schrodinger equation for a given molecule. These calculations show th at one missing hydrogen generates a dipole moment of 0.8 debye. W ith two missing hydrogens, the carbon double bond creates an electric dipole moment of 1.4 debye, a little less than two times th at for the single dehydrogenation case. The number of dehydrogenations that takes place in interstellar space is un certain. We have therefore followed the suggestion of Omont, (1986): for our C 2 0 H 2 0 -T1 buckyball, the median number of dehydrogenations is set at n = 3. I scale the number of dehydrogenations to the number of carbon atoms for the larger molecules. There is an upper limit to the number of dehydrogenations, because, with the loss of two hydrogen atoms, the excess local negative charge of the surrounding carbons maintains tighter bonds with the remaining hydrogen atoms. In my code, I set the number of dehydrogenations as part of the initial con ditions of the simulation. Once n is determined for an individual buckyball, it is static for the rest of the simulation. I acknowledge that in real situations, the buckyball may lose a few hydrogen atoms to UV photons, and this will affect the strength of the dipole moment. However, the magnitude of the dipole moment only affects the overall scale of the emitted power, but it does not change the shape of the power spectrum. The random orientation of the dipoles results in a dilution of the strength of individual dipoles, as two dipoles pointing in opposite directions cancels each other out. Unless the dipoles are aligned serendipitously on top of each other, different reasonable distributions give ~ ± 5% change from the mean value of the dipole moment. I then generate the locations of the de hydrogenation in pairs. For example, with n — 9, we generate four sites in the 17 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. buckyball for double bonded carbons and one site for a resonance structure. The value of the total electric dipole moment is a superposition of the five individual electric dipoles moment. Once I determine the position of the dehydrogenations, I can calculate the moment of inertia tensor I and the strength and orientation of the electric dipole of the molecule. I obtain the rotation axis by diagonalizing the moment of inertia tensor. I then switch to the principle axis coordinates of the buckyball, denoted by primes, where, I z>zi > Ix>xi , I yryi. In these coordinates, calculations of the properties of the buckyball becomes simpler. The angular momentum vector L, becomes, hi/ y \ iji j i u j j . y ( 2 .6 ) The oniy non-zero component of rotational velocity is cjz/ by definition, so that the angular momentum is simply, L\ = Jj/z/wz/. The rotational kinetic energy of the molecule is therefore (2.7) The buckyball and impacting particles are allowed full three dimensional degrees of freedom. I now let photons and interstellar hydrogen atoms collide with the buckyball while monitoring the buckyball angular momentum. 2.3.2 H ydrogen C ollision P rocess In my simulation, I consider buckyball collisions with interstellar hydrogen atoms and photons. We first describe the simpler hydrogen atom collisions. Upon collision with a hydrogen atom, I make the approximation that all of the angular momentum of the hydrogen atom is transferred to the buckyball. 18 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.2. Region Environmental Properties of Simulation Environment H D ensity H T em p eratu re T im e Betw een Collision H Velocity (# cm - 1 ) (K) (s) (cm s _ 1 ) W IM 0.1 8000 2.10938 x 109 813395 W NM 0.4 6000 6.08926 x 10s 704421 CNM 30 100 6.28897 x 107 90940 VCM 100000 25 1.88669 x 104 78742 The hydrogen atom was immediately ejected from the buckyball with a negligible percentage of its incoming kinetic energy. This approximation is justified because the buckyball is at least 200 times more massive than the hydrogen atom so the re-emission process does not appreciably affect the buckyball’s internal energy. The time scale between hydrogen collisions and the energy of the incom ing hydrogen atoms depends on the characteristics of the type of interstellar media I model. I follow the Draine and Lazarian division of the interstellar re gion into three types: Warm Neutral Medium (WNM), Cold Neutral Medium (CNM), Warm Ionized Medium (WIM), and we added a new region, the Very Cold Medium (VCM) to expand the range of density and temperatures. The temperature and hydrogen density characteristics of the four regions are summa rized in Table 2.2. The temperature range simulated went from 25 K to 8000 K. The hydrogen-buckyball collision has a characteristic time scale r = (n Ha v ), where nn is the number density of the hydrogen atoms, o is the cross-sectional area (collision cross section) of the buckyball, which I take to be its physical size, 7rr 2 (ignoring Coulomb interactions since the buckyball is neutral), and v is the velocity of the hydrogen atoms. The probability density function for the arrival 19 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. time,f, or the time between hydrogen collisions, is proportional to exp (—t / r ) . The hydrogen atoms are assumed to be in thermal equilibrium and therefore their speeds obeyed a Maxwellian distribution p W = 4Tr( ^ r r ^ ’”’!/2tT (2.8) where m is the mass of a hydrogen atom, k the Boltzmann constant, and v is the velocity of the hydrogen atom. I obtained average cross section, < av >, by computing, A A g A J p(v)dt! (2.9) in our case the cross section, a, does not depend on velocity. The hydrogen atom collision site on the buckyball is chosen by randomly generating the x and y coordinates, [px,Px\, of a point, p, in the unit circle per pendicular to the incoming velocity, v, of the hydrogen, where x and y are unit vectors in the plane of the unit circle, perpendicular to the unit vector, v, of the incoming hydrogen atom. The buckyball has a rotational velocity of vrot. At the start of the simulation, this is set to the thermal rotational velocity. As the simulation progresses, it changes according to the angular momentum of the buckyball. The angular momentum imparted to the buckyball, AL,from the hydrogen atom is given by, AL = where A v = \ rot + v, and b m A v x b (2-10) is the impact parameter, given by b = r ■[px ■i + py ■y - ^/l - p2 x - p 2yw] . (2 . 1 1 ) Since the buckyball is a rigid rotator, its rotational velocity is vTOt = no. This alters the rotational kinetic energy, which is simply L 2/21 and produces a new angular velocity, given by uo' = y 2 E / I Z 20 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Between collisions, the buckyball radiates away its energy according to the classical Larmor formula 2 u 4d2 P = sin 3 c3 ( 2 . 12 ) 0 where 9 was the angle between the electric dipole moment, d, and the rotation axis, z', and c is the speed of light. Since to is proportional to y/E, then d E /d t oc —E 2, so th at the final energy, E f , at some time t seconds after the collision, is, 1 e , 1 8 d 2 sin 2 9 e, 3 c > /y , (2.13) where jE* is the energy of the buckyball right after the collision. 2.3.3 P h o to n C ollision P rocess The buckyball is also bombarded by Galactic background photons. I calculate the distribution and energy of the photons according to the M athis et al. (1983) estimates of the energy density of ambient radiation in the solar neighborhood. The energy density of this radiation is given by, zl 7r = u ux v + E 4-7T W i — B x (Ti) c + — B X(2.725K) c where u\ is in units of erg cm 2 pni-1 , (W2, W 3, IT4 ) = (1 (2.14) x 10~1 4 ,10~13, 4 x 10~13), with (T2, T3, T4) = (7500,4000, 3000)K. 4wJ\ has units of erg cm 2 pm -1. B \ is the blackbody function. The UV components of the Galactic background photons are given, in units of flux, F\ = 4:irJ\ , and relates to energy density by, i n J x / c = u\. The UV components are, 47Tj \ 0 0 —>0.0912 fim 38.57A3-4172 0.0912 —>• 0.110 n m 2.04510-2 0.110 —> 0.134 pm 7.115 x 10- 4 A^ 6678 0.134 —>■0.246 pm 21 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (2.15) _Q _ Q ~0 0.05 0.00 10 6 10 5 10 -4 Log Wavelength of Incident Photon ( c m ) Figure 2.4 A plot of the photon probability distribution used to generate the photon density in each simulation. The probability density function of ambient solar neighborhood photons is plotted in Figure 2.4. I use this photon distribution in my simulation. Bombardment by a photon raises the internal energy and therefore the tem perature of the buckyball. The buckyball then emits photons to radiate away the energy. The actual number of photons emitted by the buckyball is a stochas tic process, but the average number is dependent on the tem perature to which the buckyball was excited. The tem perature is calculated from the enthalpy, Un (T) (given in ergs) of the buckyball; I use the G uhathakurta & Draine (1989) approximation, 4 11 x 10_ 22 T 3 -3 Un (T) = (1 - 2 / N ) N 1 + g 5l x l 0 __3 T + L 5 x l Q _ 6 T 2 + g 3 x 1 0 _ 7 r 2 .3 (2-16) where N is the number of carbon atoms in the buckyball, and T is the temperature 22 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. of the buckyball in Kelvin. If we assume th at all the energy of the absorbed photon went into heating the buckyball, then equating U^(T) to hv, the energy of the photon, yields the temperature increase of the buckyball after photon absorption. The number of photons emitted by a buckyball with energy E suffering a collision with a photon of energy hv is, E+hv r N A N = / -rdE E E (2.17) ^ where, N = I E = I QuB v(T)dv (2.18) Qv is the absorption cross section for micron-sized graphite dust grains calculated by Draine and Lazarian (1999), scaled for the buckyballs, and B u is the Planck function. The dotted variables means the time derivative, for example, E is the rate of change of the energy. The total number of photons emitted is therefore, » iV S hQ ,B „(T,)dv SQ.B „ (T ,) S S Q M T i) ’ ’ where T; is the tem perature of the buckyball with energy E, T f is the temperature of the buckyball with energy E + hi/, both obtained by solving Equation 2.16. I constrained the number of photons emitted to be a positive integer. The minimum number of photons emitted is 0 and the maximum, given by a collision with the highest energy Galactic photon, is 27. The emitted photon has both orbital angular momentum and spin, but the orbital angular momentum of a photon about the buckyball is much less than h = h /2 tt. Therefore, I only need to consider the spin of the photon. Each photon emitted will change the angular momentum of the buckyball by ±h. The sign of the change in angular 23 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. momentum due to photon emission is determined randomly. I set up two random numbers to separately generate a) the time between successive photon impacts, and b) the number of photons emitted by the buckyball according to Equation 2.19. Between photon impacts, the buckyball radiates classically according to the Larmor formula as it did between hydrogen atom impacts. The photon and hydrogen collision processes drive up the variance in the angular momentum, as the buckyball angular momentum executes a random walk due to the random directions of the collision particles. The competing radiation process reduces the angular momentum of the buckyball. An equilibrium between these two processes produces the spectrum for my buckyball radiation. Each simulation was set for 10 years, with as little as 1000 events in the rarefied region, and as many as 100,000 events in the densest region. The ini tial conditions of the simulation is wiped out in as little as five events, so th at running the simulation for 10 years produces a good representation of the dis tribution of buckyball radiation. Plotted in Figure 2.5 is the emission frequency vs. simulation time for a C 20 buckyball in the CNM region. The distribution in the Figure is representative of the direct output of the simulation from which I derive parameters such as spectral index and emissivity. 2.3.4 A Short Ju stification for C lassical M echanics The fact th at buckyballs are very small may raise concerns about using classical mechanics in our simulation. I show that the rotational quantum number of C 2 0 H 2 0 is large enough for a classical analysis (for references, see, e.g. Thompson (1994)). Consider a rigid body rotator. The Hamiltonian of this system is exactly the 24 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. cr 20 520 54 0 560 58 0 600 S im u la tio n T im e (M s ) Figure 2.5 Plot of the emission frequency vs. simulation time of the C 20 buckyball in the CNM region. Solid lines indicate times during which the buckyball is radiating. The dashed lines correspond to a collision event. 25 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. kinetic energy of the system in the absence of outside forces (e.g. gravity) H h2 t 2 p °xi _|_ y V _|_ 2 I'Xl lyl j 2 zi ( 2 .2 0 ) Izl where J is the angular momentum operator, and I x,T y, andlz are the principle moment of inertia for each axis. The system is characterized by three angular momentum quantum numbers: J for the rotation of the rotator, M for its projec tion onto the z axis, and K the projection on the z’ axis defined previously. The energy eigenstates are given by (J M K \ H | J M K ) . Buckyballs are very symmetric molecules (dehydrogenations introduce small deviations which 1 took pains to include in the simulation, but contribute very little moments of inertia), so we can easily find a coordinate system with I x — Iy which makes all off-diagonal elements of I approximately zero. In fact, this is exactly the Principal Axis Coordinates (the primed coordinate in our simulation). The energy eigenvalues are, E t = h2 j(j + 1) / 1 \Iz 2/ x 1 \ k2^ IxJ ( 2 .2 1 ) 2 j where j is the quantum number of state J, and k is the quantum number for state K. Compare this to the energy level of the diatomic molecular vibrational bands, Ej = — ( 2. 22) Different J ’s represent different bandheads, with K ’s being the bands within a bandhead. For simplicity, consider K = 0, the case where the rotation is perpendicular to the z' axis; the energy difference between adjacent states is, A E j = E J+i —E j ~ J + 1. The rotational kinetic energy of a C 2 0 H 2 0 molecule at 50 K is 3.5 xlO - 1 5 ergs, and the moment of inertia is 2 x l 0 ~ 37 g cm2. This gives J = 35. The energy difference between the two states is about 6 %, or, about 10- 1 6 ergs. The fact that the energy bands are so close together leads us to believe treating the rotational energy level as a continuum was justified. 26 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 2.4 S im u la tio n A n a ly sis At each time-step in the simulation, we recorded the energy and rotational fre quency of the buckyball. Our main goal was to extract the spectral index (f3) of the three buckyball emissions as well as the emissivity. Using the energy-time data, we made a series of energy bins and tabulated the amount of time the buckyball spent in a particular energy bin. Specifically, we divided the energy range into 500 uniform bins so that later, when we wanted a composite spectrum of the three buckyballs, we could just sum the power in each bin. From this in formation, we obtained an e n e rg y -fre q u e n c y relationship: how much energy the buckyball radiated at what frequency. The peak of this graph we defined to be the peak emission frequency of the buckyball. We have plotted an example of this in Figure 2.6. The peak emission, lup and the power at peak emission, P ( ojp), are listed in Table 2.4, for the three buckyballs and four regions. The units of the lop are given in GHz, and the units of P(cop) are given in 1CT31 erg s_ 1 Hz-1. For simplicity, in this section, we use C 2 o, C'24, C 6 o, to denote 6 2 0 /^ 20 - 71, CftoHfto—n, where n is the number of dehydrogenation. We divided the total amount of energy in each energy bm by the total time a given buckyball spent in the bin to get the power radiated at each frequency. Dividing this by the central frequency of the bin, gave us the power per Hz emitted by the buckyball in each bin. The slope of this graph is the spectral index of the radiation, /?, given by Equation 2 .1 where P( uj) is the power of the buckyball, in units of erg Hz- 1 s-1, and to is the frequency of the buckyball, given in units of Hz. As a goodness-of-fit parameter, we cite the residual, ar, of the least squares fit to the spectral index, ar = m (togio-PmM - f i t ) 2 /m , 27 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (2.23) peak emission 107 1C8 109 1 0 10 1 0 11 LOG Emission Frequency (Hz) Figure 2.6 The Energy-frequency plot for C2o buckyball in the WNM region. The peak is taken as the peak emission frequency, and most graphs show a clear drop off in the energy emitted vs. frequency after the peak. 28 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. where Pm(a>) is the data, f i t is the best fit to the logarithm of the data, and m is the number of data points. In the cases where the spectral index was better fit by splitting into two power laws, we cite the spectral index for the higher frequency power law, labeled t for top, at ar ~ 0.07, so th at the assessment was the same for each simulation. The lower frequency power law is labeled b for bottom. Exception to the oy ~ 0.07 rule was made for some C qq cases because the scatter in these data was much larger than that for the other two buckyballs. In order to have oy ~ 0.07 for these cases, I could only use less than 10% of the data points for fitting the top power law which was not an accurate representation of the distribution. Therefore, for the Ceo data, we typically cited the slope that fitted a reasonable number of data points while maintaining a small oy. Where this actually occurred was judged on a case by case basis, but typically, oy was 0.3 for data with larger scatter. The spectral indices of all buckyball emission and their oy fitting coefficients are in Table 2.3. For a measure of the relative effects of photon vs. hydrogen impacts, we have included, for each simulation environment, the percentage of the angular momentum transferred from photons and hydrogen atoms in Table 2.5. The numbers cited are the number of hydrogen or photon impacts in the given region for each of these buckyballs; the numbers in the parenthesis are the percentage of the total angular momentum transferred to the buckyball by those collisions. In the more rarefied regions (WIM), there were no hydrogen impacts, and in the denser regions (VCM), hydrogen impacts dominated. 2.4.1 Typical Sim ulation N um bers As a first check, we can compare the buckyball rotational velocity, vrot, to the mean impacting hydrogen velocity, v h , to verify our hydrogen absorption assump- 29 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.3. Buckyball Power Law Spectral Indices, ft, and their oy fitting coefficients. . . Region W IM W NM CNM VCM P ro p e rty C 24 C-20 C60 ta bb t b t b 3 2.691 0.871 2.436 0.973 1.165 0.934 Tr C 0.0755 0.419 0.0666 0.324 0.172 0.297 0 2.360 0.935 2.498, 0.912 1.470 0.917 (7x 0.105 0.293 0.0772 0.352 0.193 0.393 3 2.603 0.883 2.580 0.940 1.867 0.962 CTr 0.0761 0.460 0.0721 0.278 0.171. 0.267 3 0.981 0.885 1.016 0.862 1.009 0.894 (Tx 0.0527 0.299 0.0540 0.298 0.0419 0.287 aT op (high frequency) p o rtio n of the em ission b B o tto m (low frequency) p o rtio n of th e em ission cuy for linear fits to th e sp ectral index 30 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.4. Peak Frequency, u p of buckyball emission and power at peak frequency, P ( u p). Region W IM W NM CNM VCM C haracteristic C 20 C 24 Ceo LOpa 37.25 23.03 9.13 P ( u p )b 1.688 1.760 1.324 UJp 22.91 13.11 7.63 P(udp) 0.6333 0.6573 1.639 UJp 16.84 20.75 5.41 P(uip) 0.9861 1.639 1.553 UJp 31.23 18.29 12.24 P(UJp) 19.82 16.32 145.8 ain units of GHz. bin u nits of 10" 31 erg s _1 H z- 1 . tions. Taking the largest value of the C 2 o peak frequency, u p from Table 2.4, and the C*2 o radius, we obtain the C*2 u rotational velocity to be vrot = ujpeak x radius, 1260 cm s - 1. Comparing this to the mean vh from Table 2 .2 , for the coolest region (VCM), showed that on average the impacting hydrogen atom velocity was almost 100 times faster than the rotational velocity of the buckyball In the buckyball reference frame, the much larger speed of the impacting hydrogen atoms meant that our assumption of the buckyball completely absorbing the hy drogen atom ’s angular momentum is feasible. For the other buckyballs, vrot is similarly small compared to the hydrogen atom speed: C 2 a- 821 cm s” 1, Ceo: 607 cm s_1. I present some characteristic results of the simulation. For a given simulation, the rms amount of angular momentum from a single photon absorption and sub sequent emission process is ~ 2.5h, or about 10% of the total angular momentum 31 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.5. Photon vs. Hydrogen Impact Region P ro p e rty W IM # P h o to n Im p act (%} 729 (100%) 792 (100%) 731 (100%) # H ydrogen Im pact(% ) 0 (0%) 0 (0%) 0 (0%) W NM # P h o to n Im pact (%) 731 (100%) 746 (89.48%) 1036 (65.01%) # H ydrogen Im p act (%) 0 (0%) 2 (10.52%) 2 (34.99%) CNM # P h o to n Im p act (%) 667 (95.38% ) 735 (97.65%) 1018 (94.3%) # H ydrogen Im p act (%) 20 (4.62% ) 17 (2.35%) 49 (5.7%) # P h o to n Im p act (%) 668 (6.02%) 760 (5.78%) 1062 ( 0.1% ) # H ydrogen Im p act (%) 42150 (93.98%) 46154 (94.22%) 88621 (99.9%) VCM C 20 C 24 C60 aPercentage of to ta l angular m o m e n tu m tran sferre d of the buckyball. Approximately 30% of all photon collisions im parts an angular momentum of one or two units of h to the buckyball. The average change in the rotational energy of a VCM region t72o buckyball, given in units of kT, is 8 E / k T = 0.0055. A typical hydrogen atom collision imparts ~ l / 1 0 th of the angular momentum of a photon impact, which means that hydrogen impacts are irrelevant when the hydrogen atom density is low compared to the photons. I list in Table 2.6 the mean rotational energy of the buckyball from the simulation, in temperature units kT, as well as the 97th percentile value for the rotational energy in temperature units. The mean rotational energy of the buckyball is subthermal in all cases, a fact reflected in the peak rotational frequency discussed in the next section. 2.4.2 Em ission C haracteristics Here I describe the simulation results, and I discuss our analysis of the processes which led to these results. In general, the b (lower frequency) power law for 32 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.6. Simulation Rotational Energy of Buckyballs C 2 0 #20-71 C 60 # 60—n C 2 4 #24 —71 Region < E >a E (97t/l %-tile) W IM 0.00314 0.00560 0.00191 0.00301 0.00188 0.00378 W NM 0.00188 0.00396 0.00104 0.00241 0.00225 0.00386 CNM 0.0719 0.131 0.125 0.2.34 0.0943 0.196 VCM 0.658 1.17 0.356 0.949 0.766 1.14 <E> E (9 7 fA % -tile) <E> E (97t '1 % -tile) ain te m p e ra tu re u n its of k T all buckyballs in all regions has a power spectral index of /3 ~ 1, with average (3 = 0.91. The t spectra (at higher frequencies), are steeper and obeys a f3 ~ 2 power law, with the average spectral index (3 = 1.9. The breaking of the power law into t and b is most pronounced in the CNM region, and it is least pronounced in the VCM region, where breaking the fit into two power laws produced a less than 10% improvement in oy. For the other regions, breaking the power law produces a 25% to 50% improvement in oy of the fits. The break in the power law is also less pronounced in the C qo buckyball. It is noteworthy that the t spectra contains significantly less scatter than the b spectra. Table 2.4 gives the peak frequency, cjp, and the power emitted by the buckyball at the peak frequency, P ( ujp). I define peak frequency as the frequency bin which contain the highest total emitted energy. I plotted a sample graph of the total energy emitted vs. the frequency in Figure 2.6. In every case, the buckyball emission showed a clear drop off past the peak emission frequency, so I am confident th at the buckyball emission range was limited. The value of u p from all buckyballs and simulation regions spans 5.4 to 37.3 GHz. The peak frequency of emission by the C$o buckyball is lower in every region than the 33 R eproduced with perm ission o f the copyright owner. Further reproduction prohibited w ith o u t perm ission. smaller buckyballs. Figures 2.7 to Figure 2.10 are plots comparing simulation region quantities against each other. Figure 2.7 compares the effect of hydrogen density on the spectral index; Figure 2.8 plots the effect of hydrogen tem perature on spectral index. Figure 2.9 plots the effect of hydrogen temperature on peak emission fre quency, and Figure 2.10 plots the effect of hydrogen density on peak emission frequency. In Figures 2.7 and 2.8, the spectral indices of the different tem per atures simulated are offset by the labeled amount for clarity. For example, the dot-dashed diamond lines (indicating the 8000K simulation) is offset by +1.0, which means 1.0 was added to all spectral index values. This is done so the spec tral behavior of each simulation is distinguishable from the other. The trends shown in these Figures are discussed together in the rest of the subsection. Two major effects governed buckyball dipole emission. 1) Buckyball size dic tates frequency response. The smaller buckyballs responded faster to changes in energy, largely because they could spin faster, and therefore emitted their absorbed energy quicker. This resulted in the presence of the hyper and quies cent states. The hyper states were emissions right after an angular momentum increasing impact which spun up the buckyball. The quiescent states were lulls between collisions. The large buckyball had more inertia and therefore took more energy to be spun up to a hyper state. This explained the fact th at it generally had a lower spectral index, as well as a lower peak frequency, as shown in Table 2.4. 2) The hydrogen density of the emission region drove the variance in the buckyball angular momentum. Even with the higher temperatures and therefore harder impacts, the rarefied regions has substantially subthermal emission. In Figure 2.10, the high temperature and high hydrogen density produces drastically 34 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 4 : offset = +1 C20 offset = +0.5 Q. "A , x 0 \ "O _c "cC o offset = -0.5 -t—' 0 Q. + A □ O CD 1-4 25K 1 00 K 600 0K 8 00 0 K 10 -2 " Hydrogen Density (g/cm 0 10 ur f Figure 2.7 Spectral index vs. simulation Hydrogen density for each of the 4 tem peratures simulated. Crosses indicate hydrogen tem perature T = 25 K, Squares are T = 6000 K, Diamonds are T = 8000 K, Triangle are T = 100 K. 35 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 4 3 offset = +0.5 x ■a 0 _c Io 2x>ffset = -0.5 0 Q. co 1 0 10° 101 102 10 3 10 4 IQ5 Hydrogen Temperature (K) Figure 2.8 Spectral index vs. simulation temperature for each of the 4 densi ties simulated. Crosses indicate hydrogen number density, n Hl = 100000 cm-3, Squares are uh = 0.4 cm-3 , Diamonds are n # = 0.1 cm-3 , and triangle are nn = 30 cm~3. 36 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 140 + A □ O N I o > 120 o c a; 100 cr 80 3 <D C o w 'E C20 10 0 0 0 0 / c m 3 30/cm 3 0.4/cm 3 0.1 / c m 3 / 60 'c/5 40 LU CC a> 20 CL 0 10 100 10 00 1 0 000 Hydrogen Temperature (K) Figure 2.9 Peak emission frequency vs. simulation tem perature for each of the 4 densities simulated. Crosses indicate hydrogen number density, n H, = 100000 cm-1 , Squares are n H = 0.4 cm-1, Diamonds are n H = 0.1 cm-1, and triangle are n # = 30 cm-1 . 37 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 120 N X 0 o | 80 + A □ O 25K 1 00K 60 0 0K 80 0 0 K C20 C7 £ ^ 60 c . o 'w •| 40 LU ® o 20 CL 4 2 Hydrogen Density (g/cm 6 f Figure 2.10 Peak emission frequency vs. simulation Hydrogen density for each of the 4 temperatures simulated. Crosses indicate simulation temperature, T = 25 K, Squares are T = 6000 K, Diamonds are T — 8000 K, Triangles are T = 100 K. 38 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. higher peak frequencies, compared to the lower temperatures. However, this is not a tem perature effect, because at high temperature but low density, only the high density simulation shows a jum p in peak emission frequency, as can be seen in Figure 2.9. The three other simulations have low peak densities even with higher temperatures. This behavior is due to the fact th at im pact rate is more im portant than impact strength. The ratio of the RMS hydrogen atom velocity between the WIM region and the VCM region is ~ 18, but the ratio of the mean free time between hydrogen atom arrival is 1/50000. The random walk executed by the buckyball’s angular momentum has larger variance with more steps, as opposed to larger steps. As a result, the highest density region has the highest emission frequency. Larger variance is also responsible for a flatter spectrum, as the averaged effect is to distribute the energy more evenly among frequency bins. Figures 2 7 and 2.8 illustrates this. Figure 2.8 shows that the spectral index does not vary as a function of the temperature, whereas Figure 2.7 shows that at all temperatures, the spectral index drops with respect to hydrogen density. In effect, the frequent and random collisions provides a cooling mechanism to the buckyballs. The rarefied regions are dominated by rare but powerful impacts which imparts large jumps (up or down) in the buckyball angular momentum, which in turn contributes to a higher spectral index. 2.4.3 A nom alous Em ission? Are buckyballs candidates for anomalous free-free emission? The anomalous freefree emission was measured to have a temperature index of fix ~ 2 , or a power spectral index of (5 ~ 0. This is different than the buckyball power spectral index that we measure, typically around 2 for t spectra and 1 for b spectra. However, 39 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 2.7. Thermal Rotational Frequency of Buckyballs3, Region C 20#20—n C 2 4 R 24 —n C qqH qq—n W IM 681.1 547.3 223.4 W NM 589.9 474.0 193.5 CNM 76.15 61.19 24.98 VCM 38.07 30.60 12.49 ain u n its of GHz. we are encouraged by two facts: of the anomalous emission, and 1) 2) the peak emission frequencies fits in the range buckyball dipole radiation drops off sharply at frequencies higher than the peak emission frequency. These two facts suggest that individual buckyballs do not produce the flat free-free power spectrum, but a composite of spectra from a population of buckyballs having many peaks in this region may contribute to the observed spectrum. In general, we expect the spectrum of the composite buckyball emission to resemble the C qq simulation result, because terrestrial graphite condensates pro duce more C§q than any other buckyball. Figures 2.11 to 2.14 are plots showing a composite spectrum of the three species of buckyballs for the WIM region. The power from each species is given equal weight, so the spectra contain equal proportion of C2o to C2± to Cgo- The scatter plot in each figure is the sum of the power for each frequency bin of all three buckyballs, fitted with a best fit straight line. I include a nearest 8 neighbor sum of the above scatter plot, offset by a factor of 103 below the scatter plot in each Figure for clarity. The equal proportion probably overestimates the expected number density of C2o and C 2 4 , however, it illustrates the point that, as expected, the composite spectra of all four regions strongly resemble the C e0 spectrum, with minor modifications at the 40 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. turn-off due to the higher peak frequencies of the smaller buckyball. The best fit line of the composite spectra gives the composite spectral index for each region: for VCM, (3 = 0.873; WNM, (3 = 1.12; WIM, /? = 1.01; CNM, /T = 1.15. In all cases, the composite spectral index was very similar to the spectral index for the region listed in Table 2.3. The smaller buckyballs also emitted with less power than 6 * 60, as shown in Table 2.4. We conclude based on the composite spectra that, in general, buckyball emission is dominated by C6o emission. However, in the higher frequency regime (above ~20GHz), the emission from most regions was dominated by the C2 o buckyball because the C 6o emission was restricted to lower frequencies due to its slower rotation rate. Therefore, we use the emission characteristic of C2o to fit the observed anomalous freefree emission between 35-45 GHz in order to estimate the number of buckyballs required to produce the anomalous free-free emission. Both the WIM and VCM regions contain emissions in this range. Draine & Lazarian (1999a) calculated the emissivity per H nucleon of the observed microwave emissions by cross-correlating the sky brightness with a FarIR sky map. Their data are shown in Figure 2.2. Also plotted on their graph is the emissivity of various populations of dust with at different temperatures and a spread of tem perature indices as references. We have plotted a schematic of our WIM composite data on top of their graph for a comparison. Our buckyball simulations produced more than enough power to account for the excess emission. The schematic for the emission from our VCM region would be another order of magnitude higher. In the WIM region, the buckyballs spent 3.6 % of simulation time in the 35-45 GHz range; the number for VCM is 3.1%. We calculate the power emitted per Hertz from each buckyball in this range by multiplying Equation 2.12, the Larmor formula, by the percentage of time the 41 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 5 o a. P = 0 reference line -3 6 10 ,-38 best fit nea rest 8 neighbor s m o ot hi n g 0 -40 i i I I i I 10 100 Frequency (GHz) Figure 2.11 Power vs. Frequency of the composite emission from all three buck yballs in equal proportions for the WIM simulation. The thin line is a nearest 8 neighbor sum to show a smooth spectrum. The thick dot dashed line is a flat (.P oc i/°) spectrum for comparison. 42 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. I 7/5 03 10 34 03 o Q_ 10 36 offset=10 10 38 best fit nadrest 8 neighbor s m o o t h in g 10 40 10 5 10 6 10 7 10 9 10 10 11 10 1 LOG Emission Frequency (Hz) Figure 2.12 Power vs. Frequency and a smoothed version of the composite emis sion from all three buckyballs in equal proportions for the CNM simulation. 43 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. o offset=10 Q_ 0 10 -36 o 10' -38 best fit nearest 8 n e ig hb or sm o o t h in g 10' -40 10 ' 10" . 10 10 10 10 10 1 ? LOG Emission Frequency (Hz) Figure 2.13 Power vs. Frequency and a smoothed version of the composite emis sion from all three buckyballs in equal proportions for the CNM simulation. 44 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 10 1 0 -28 ' I I I I I ] --------------1------- 1------1— I I T T T l-------------- 1--------- 1---- 1— I I I I I -30 ^ i °-3 2 IsT x o5 0 oHO $ -34 <^o 0 5 o ^ 0 o_ J n'~ 36 ^o offset=10v 1 10 ' 38 best fit nearest 8 nei ghbor sm oo th in g 10 -40 I 1.1__________i 1 0 ' -.9 10" 10w 10 i i i 10 i i m 0 I 11 LOG Emission Frequency (Hz) Figure 2.14 Power vs. Frequency and a smoothed version of the composite emis sion from all three buckyballs in equal proportions for the CNM simulation. 45 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. buckyball spends in the frequency band, assuming an isotropic distribution in the emitted radiation: P (C 20,35 - 45Gbfz) = 3 % ~ l ~ ~ s m 2d 4-/r 3 cs (2.24) = 3 % i | (247r 3 (2.25, 40x3 la9),d2. Given a typical value of the dipole moment of 1.3 debye, the power per molecule per Hz is, P(G 2 o, 35 - 45GHz) = 4.54 x 1 0 “ 3 4 er^ (2.26) Comparing this to the data, the free-free emissivity at 40 GHz from Figure 2.2 is, eo//(4 0 GHz) = 4 x 10" 4 1 er^ s~l s r ' 1 H z ~ l H ~ ' . (2.27) We obtain the number of buckyballs needed to radiate with this emissivity by equating the following, taff = P( C 2 o, 35 - 45GHz) x 109H z x 1^ 1 ^ — [Oj n H (2.28) where 109 Hz is the width of the band, [G2 o]/[G] is the concentration of C2o to the amount of carbon in the Galaxy, n c is the carbon density of the Galaxy, and n # is the hydrogen density of the Galaxy. The necessary C 20 to carbon ratio is ~0.5% for buckyballs to constitute all of the anomalous free-free emission. This is a large but not unreasonable amount of carbon to be in these ultra small molecules (compare to 0.9% obtained by Foing & Ehrenfreund 1997). If we assume spherical dust grains, the grain mass, M, scales with grain radius, a, as, M oc a05. The mass contained in 3 A grains (size of the smallest buckyball) is of the mass in 1000 ^3/1000 0.06 A grains, which is comparable to what we need to explain the galactic emission. 46 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 2.4.4 P ossib le O bservations We expect buckyballs, like many other PAHs, to form in a variety of interstel lar conditions. Buckyball searches have been conducted near carbon rich stars, specifically around R Coronae Boreal (RCorB) is stars (Clayton et al., 1995). RCorB itself is thought to have a fossil dust shell (Gillett et ah, 1986), extending from 1.3’ to 9’. Conservative estimates of carbon dust content of this fossil shell is around 0.013 M sun. Given our predicted buckyball abundance, we expect to see radio emission of approximately 0.75 mJy at 40GHz from RCorB, a value th at is very small and difficult to disentangle from other emissions processes at this wavelength. In our opinion, it would be virtually impossible to directly observe buckyball dipole emission around stars. High latitude (\b\ > 15) cirrus clouds (HLC), on the other hand, present ideal conditions for the detection of buckyball dipole emission. These clouds contain large numbers of carbon atoms, but have relatively simple chemistry so that the observation of a single process is possible. HLCs are by definition found at high Galactic latitudes where there are less objects interfering in the line of sight. They span a range of hydrogen densities from 100 cm - 3 in rarefied clouds to 105 cm - 3 in dense knots and filaments (Ingalls et ah, 1997). HLCs also have low kinetic temperatures, around 10 K. These factors make our VCM region very representative of conditions expected in HLCs. We present buckyball emission estimates for two HLCs, one in the Northern hemisphere and one in the Southern hemisphere. Meyerdierks et al. (1990) presented a study of LVC (Low Velocity Cloud) 127, whose center is located at (l,b) = (127°, 20°), part of the North Celestial Loop. They found this cloud to be a clumpy distribution of knots; the most prominent knot is centered at (l,b) = (128.2,20.8), 7’ (0.2 pc) in diameter. From 47 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. CO measurements, they have derived a carbon nuclei abundance of N(C) = 5.4 x 1016 cm-2 . This corresponds to a C /H ratio of 3 x 10-4, similar to the mean Galactic ratio. If we assume a uniform distribution, the total number of carbon atoms in the clump is therefore Ctot = N{C) - 7r(r ) 2 = 5.4 x 1016 ■tt(0.1 • 3 x 1018) 2 = 4.86 x 1051atoms (2.29) (2.30) Again, assuming 0.5% in C 2 0 , we get, N c 20 =-■ = x 4.86 x. 1051 (2.31) 1.2 x 1048molecules (2.32) The emission is therefore 0.61 mJy from this knot. An instrument like the Very Small Array (VSA) may be able to detect this level of emission. The VSA (see website: http://w w w .m rao.cam .ac.uk/telescopes/vsa/) is located at Tenerife and can observe the sky between DEC of -7 to 63. It is designed to obser ve between 26 and 36 GHz, with a tem perature sensitivity of 7 fiK. Our buckyball emission translates to a radio telescope main beam tem perature of 14 /j,K at 40 GHz. The intensity of a source given in terms of the main beam temperature of a radio telescope is given by, = 2 6 5 T MB{ K ) e H a r c , m n ) Xl {cm) where 9 is the source radius, and A is the wavelength. A sample of HLCs was measured in the Southern hemisphere by Ingalls et al. (1997). Of these, the cloud G225.3-66.3 (Keto & Myers, 1986) is a good southern hemisphere candidate. Images of this cloud show that it is a thick arc, spanning 25’ in length, and the thickest region is 5’-10’ in width. The cloud is located at (RA, DEC) = (02h 36m 45.6s, -19h 49m 01s), and we assume this cloud to 48 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. have an approximately triangular structure with a length of 25’ and a width of 5’. Ingalls et al. (1997) derived a carbon column density of 2.5 x 1016 cm - 2 for this cloud. The total number of carbon in this clump is, Nc = pc x A = 2.5 x 1016 • 0.5(0.7pe x 0.15pc) = 1.18 x 1052 atoms, (2.34) (2.35) and therefore 2.95 x 1048 buckyball molecules. Given our buckyball luminosity, we expect this cloud to have a buckyball radiation of 1.5 x 10~2&erg s~l cm~~2 Hz'~l , or 1.5 mJy. In the Southern hemisphere, the Cosmic Background Imager (CBI, see website http://w w w .astro.caltech.edu/ tjp /C B I/), located in the Andes, has a sensitivity of 13 to 41 p K . Our buckyball emission has a main beam tem perature of 14 p K across this source, if we assume an equivalent source beam of 10’, and should be detectable by CBI. 49 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. CHAPTER 3 S ign atu res o f C osm ic S trin g s in th e C osm ic M icrow ave B ack grou n d W ith the Galactic foreground removed, the intrinsic CMB signal contains a wealth of information. Almost any process in the early universe will leave its trace on the CMB. The photons permeate the universe and is a background to every process th at has taken place since the era of Last Scattering, which occurred about 300,000 years after the Big Bang. In this chapter, we describe how we constructed and calibrated a digital filter to set detection limits for a class of topological defects known as cosmic strings. Topological defects are predictions of Grand Unification Theories which seek to unite all the forces we observe as one. These theories predict physics behaviors at such high energies that it is currently impossible for us to perform experiments to verify the theories. This lack of experimental constraints has led to a proliferation of different Grand Unified Theories, and we hope by producing limits on a necessary product of these theories, we can start to differentiate between viable theories. 3.1 T o p o lo g ica l D e fe c ts As we go back in time, the universe was smaller and hotter. The extremely hot and dense conditions of the very early universe represent the perfect lab- 50 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. oratory to examine fundamental particles and interactions th at the subject of particle physics seek to explain. The Standard Model of particle physics gives us a good description of the physics of the universe up to the electroweak unifica tion, ~ 10- 1 0 seconds after the Big Bang. The interplay between particle physics and cosmology has intensified in the recent years as cosmological observations gained enough sophistication to produce evidence in support of particle physics theories. For example, the observed deuterium abundance of the universe reflects the weak interaction cross section which governs free neutron decay. The fraction of neutrons which had not decayed became incorporated into atoms when deu terium formation became energetically favorable. Similarly, topological defects are a possible observable consequence of certain types of particle physics theories. This chapter is a description of theories that produce cosmic strings. More conventional sources for topological defects can be found in condensed m atter systems and liquid-crystal systems. Nematic liquid crystals, widely used in digital displays, provide a good analogue for the formation of cosmological topological defects. In fact, the correspondence is so good th at laboratory ex periments with nematic crystals are used to predict properties of cosmic string networks (Ray & Srivastava, 2004). Liquid crystals are long organic compounds which flow like liquids, but their flow contains directional information akin to crystal lattice orientations in a solid. Nematic liquid crystals consists of rods of crystals whose flow has the crystals’ long sides oriented parallel to each other. The solid-liquid transition tem perature for these types of crystals range between 10 to 200 ° C. As their phase goes from liquid to solid, the crystals lose their ability to flow and the orientation of the crystal alignment in the original liquid flow is locked into the solid. Different domains with different orientations may develop in the 51 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. absence of an external aligning electric field. Boundaries of the different domains form defects, where the parallel orientation of the crystals is disrupted, resulting in a band of crystals with random orientations. These defects come from crystals maintaining continuity through the different values of the degenerate ground state. Hence, they are a consequence of the topology of the ground state. These topological defects are stable because in order to bring either sides of the defect into alignment, it is necessary to flip the orientation of all the crystals on one side of the defect to match the other, which is impossible without an external energy source. The formation of cosmic topological defects rests on the same principle of degenerate ground states for phase transitions. We begin with a simple model; consider a general Lagrangian density L = y 2 ( 9 ^ ') ( 9 ^ ) - V(<f). (3.1) where the </>represents a scalar field in one time and one spatial dimension, x, and V{(f)) is the field potential. For now, we are not concerned about the details of the transformation of the Lagrangian or with choosing any particular gauge. Let us consider the simplest ground state degeneracy: a discrete, two-valued solution, where the potential is V 'M = - x2)2; (3-2) A and r/ are some positive constants. This is the familiar double welled potential. The Lagrangian with a potential given by Equation 3.2 leads to a field equation solution th at has the form, (x) = ?7 tanh This solution is known as the c/>4 -kink, one of the simplest topological defects. 52 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. The potential, V(4>), and the solution, </>(x), are both plotted in Figure 3.1, where A = 1 and 77 = 1.7. The <J)4-kink interpolates between the two ground state values, ±r). The field, (f)(x), can be thought of as a mapping which maps x, the space dimension, to the ground state, or, vacuum values ±rj. The kink around the origin is the result of the field rising to the top of the central hump of the potential in order to preserve continuity between the two vacuum values. The </>4-kink is stable because, in order to remove the kink, we need to lift one of the sides of the solution (e.g. —oo to 0 ) over the potential hump. This process would cost an infinite amount of energy. In 4 dimensions, </>4-kinks are 2-dimensional sheet-like defects known as domain walls, which would occur at boundaries of regions with different vacuum values of a discrete symmetry. However, the existence of domain walls is ruled out by the observed homogeneity of the universe, because the large energy density of domain walls cannot be dissipated - they would create large discontinuities in the energy density of the universe. This simple model illustrates two important ideas critical to understanding topological defects. First, the mapping 4>(x), which maps the real spatial values (—oo < x < oo) to the vacuum values (±?7 ), is what determines the characteristics of the defect. Second, the stability of the defect is a consequence of the topology of the mapping, and while transient defects exist (a cosmic string formed with one side bounded to a monopole is possible, though unstable), topological defects are generally stable objects with measurable energy density and size. At energies much higher than the central hump, the potential of Equation 3.2 has only one effective solution at zero, as the field is energetic enough to traverse the potential barrier of the central hump. However, as the energy in the field drops, for example, due to an adiabatic expansion, the field falls into either of the 53 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 3.0 5 2.0 5 0 5 0.0 3 2 0 2 3 Figure 3.1 Graph a): the double welled potential which leads to a </>4 -kink, with A = 1 and 77 = 1.7. Graph b): the solution to the potential depicted in a). The kink through the origin which interpolates between the two vacuum solutions is the defect called the 0 4 -kink. 54 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. two solutions, and it is less energetic than the central hump. The symmetry of the held has been spontaneously broken. Even though the Lagrangian in Equation 3.1 is symmetric, the physical states of the vacuum given by the potential in Equation 3.2 are not. 3.1.1 Sym m etries and Groups The symmetries of a system can be characterized by the set of operations which leaves the system invariant. Noether’s theorem states that this symmetries is the result of the conservation of some quantity. For example, a Lagrangian which has rotational symmetry is invariant under rotational operators; this symmetry implies the conservation of angular momentum. We can perform rotations on a vector x through an angle 0, in 2-dimensional Cartesian coordinates where x —>■x ' , and x' =- Rgx. The matrix R.q is cos 9 Rd sin 9 (3.4) - sin 9 cos 9 In 3 dimensions, rotations in any arbitrary direction can be written as a composite of rotations about the three Euler angles: a , j3, 7 , corresponding to rotation about the Euclidean x-, y-, and z-axes, with / RJa) = 1 0 0 0 cos a sin a 0 \ ^ cos (5 0 —sin f3 ^ Ry(P) = —sin a cos a 0 ^ sin (3 0 I R z{j) 0 = V cos 7 sin 7 —sin 7 cos 7 0 0 0 1 0 (3.5) cos /? ^ \ These rotations have the following properties: two successive rotations is a rota tion (closure); with a = /3 = 7 = 0, the operation produces no rotation (identity); 55 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. three rotations, R a) R c, produce the same result, whether they are grouped (RaRb)Rc, or R a(Rt)Rc) (associativity); for any rotation operation R a, there ex ists an operation O ” 1 such that R ^ R a brings the system back to the original orientation (inverse). These properties makes the set of rotational operations a mathematical group. The matrices in Equation 3.5 are one representation of the rotation group. Therefore, the symmetries of a system or a theory can be described using groups. Mathematically, the group of rotations in 3-D space is given by 5 0 (3 ), which are represented by the group of 3x3 matrices which have determinant 1 and are orthogonal (an orthogonal matrix A and its transpose, A T, produce the identity, A TA = I). The 5 0 (3 ) group is closely related to the 5 0 (2 ) group, which governs spin and associated spinor transforms. One representation of 5 0 (2 ) is given by 2x2 unitary (a unitary m atrix A and its transpose conjugate of the matrix, A \ produces the identity: A^A = I) matrices whth determinant 1 . The 5 0 (2 ) group was first used to describe spin associated with conservation of angular momentum and is now used to describe a variety of phenomenon. Heisenberg discovered that 5 0 (2 ) can be used to describe isospin: the interchange of protons and neutrons in strong interactions as a result flavor independence. Glashow, Weinberg and Salam used 5 0 (2 ) to describe their theory of weak interactions mediated by the W and Z bosons. Every group can be described by group generators, successive applications of which generate all members of the group. For 5 0 (2 ), the group generators are the Pauli matrices, given by, To = T = v 1 o ( 0 -i 1 . T3 = I. 0 56 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (3.6) These generators can also be given as commutation relations, (3.7) [T„,Tt] = 2itawTc where eai,c is the permutation tensor, given by, + 1 if (afcc) is (1,2, 3), (2, 3,1), (3,1,2) £abc={ - 1 0 if (ate) is (3, 2,1), (1,3, 2), (2,1,3) (3.8) > otherwise In this way, complex operations on the Lagrangian can be simplified and catego rized. Although it is not necessary to use group representation to discuss particle physics, it is particularly useful because we can exploit the mathematical prop erties of groups. I will return to the S U (2 ) when discussing specific examples of symmetry breaking related to cosmic strings. Symmetry breaking reduces the number of invariant quantum mechanical op erators of the system. For example, the rotation matrix given above is the matrix representation of the operation of rotating a vector through an angle 6 . If the angle 9 is infinitesimally small such th at the state of the system is effectively unchanged under this small rotation, then the Lagrangian of this system is said to be rotational invariant. In other words, the mathematical group that describes rotations is an invariant group of the system. Any process th at may happen to the system can be described using an operator. Some familiar operators include the aforementioned rotation, spatial translation, time translation, and charge conjugation. Before symmetry breaking, the set of invariant operations, g, on a Lagrangian is defined to be the group G. The vacuum state, labeled |0), gives, (OHO) = rjo 57 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (3.9) where r]0 is a possible value for the vacuum state (e.g. +77 for the double-welled potential). After symmetry breaking, the now smaller set of invariant operations, h, is defined by the group if, a subgroup of G, (3.10) H={ g(EG\D (g) <t >0 = <l>o} where D(g) denote the matrix representation of the operations g, and </>0 is the vacuum state. All of the information about G and H is encoded in the quotient space M, M = G/H. (3.11) M is called the vacuum manifold. In general, we can define a topological space for any group. The mathematical definition of a topology is the following: let A be a set and T a family of subsets of X satisfying the following three conditions, a) the set X and {0} (the singleton containing the element 0) belong to T b) the union of any members of T is a member of T c) the intersection of any finite family members of T is a member of T. T is then a topology for A, and the ordered pair (A, T) is called a topological space. Since both G and H are groups and have topologies, M is a topological space, and I will take the liberty of calling it a manifold without proving that it ’s locally Euclidean (locally flat, like the surface of the earth on human scales). The topology of the vacuum manifold, M , determines the properties of the defect. It is complicated to determine the characteristics of defects from vacuum man ifolds of real phase transitions. Fortunately, our task is simpler if we only want to know whether a given theory will produce cosmic strings, without worrying about the exact structure of the strings. We only need to know the homotopy class of the vacuum manifold. 58 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 3.1.2 Fundam ental H om otop y Group Homotopy is a concept th at can be applied to general m athem atical objects: two objects are homotopic if one can be continuously deformed into another. Thus a circle and a square drawn on a flat sheet are homotopic: the circle can be continuously deformed into the square without any of the points ever leaving the flat sheet, such th at a small change in the input coordinates (of the flat sheet) produces a small change in the path traced by the circle. The homotopy between these two objects is a function of both the objects themselves and the manifold on which the objects are defined. Mathematical objects such as a circle may also be functions which define a path on its manifold. In appropriate cases, this path can be considered a mapping, a function that takes values in the manifold and assigns it different values, representing a transformation between spaces, f i x ) . S ■> l\ (3.12) The mapping, f(x ), takes values given in a space S, the source, and maps them to values in a space T, the destination. The circular and the square paths may both be mappings, and if the topology of their destination is appropriate, these mappings may also be homotopic. As a more rigorous example, consider two functions, / and g. On a manifold M, let / be a mapping of the interval 0 < x < 1 onto M, where the mapping f ( x ) defines a closed loop, / ( 0) = / ( l ) = P , where P is a point on the manifold M. The mapping f ( x ) defines a path on M. If g is also a mapping on M of the same interval, 0 < x < 1, and a loop, g(0) = (/(l) = P , and if the path traced by f ( x ) can be continuously deformed to the path traced by g(x), f and g are homotopic. This is illustrated in Figure 3.2A, where the loops / and g wind around a hole in the manifold. The two loops / and g are homotopic. In contrast, loop c cannot be deformed in any way to form / or g. 59 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. g g Figure 3.2 Figure A: / (the solid lines) and g (the dashed lines) are homotopic to each other, but not to c (the dotted lines). The grey filled circle is a hole in the manifold. Figure B: / and g are freely homotopic, because a path c can be constructed to wind around the defect separating them. We define the product, / ■g (sometimes just written as f g ), as, / ( 2 x), o < x < y2 / ' 9(x) = . g( 2 x - 1 ), y2 < x < 1 (3.13) . In general, two mappings such as / and g do not need to share a base point (P) in order to be homotopic. Mappings are called “freely homotcpic” if they are homotopic but do not share a common base point. The two loops / and g in Figure 3.2B wind around the same hole in the manifold; however, with the addition of a second hole in the manifold near the point P , there is no way to continuously deform / to form g. Yet we know th a t / and g trace the same hole in the manifold. In this case, the two mappings / and g can be said to be freely homotopic, because we can construct a third mapping, c, such that, cfc~l ~ g (3.14) where the loop composition c/cT 1 takes / around the first hole in the manifold, making it conjugate (equivalent) to the loop g. From here on, we make no distinctions between homotopic and freely homotopic. In Figure 3.2B, we have shown / and g to originate from the same point for clarity. In principle, / and g 60 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. are freely homotopic as long as g starts on any point of c. A homotopic group, therefore, is the group of all functions which are the same up to homotopy; / and all loops which can be continuously deformed to / form a homotopy group. Figure 3.2 showed th at homotopy between two functions is dependent on the topology in which they reside. If we disregard the mappings, the homotopy of a topology is a characteristic of the topology itself. Mathematically, the homotopic classification of a topology is based on mappings of the hypersphere, S n: the first homotopic class is based on homotopic mappings to the circle, S'1; the second homotopic class is defined based on homotopies to the sphere, S 2; and so on. They are given designations ivn(T) where n denotes the order of the class, S n, and T denotes the space to which this classification belongs. To clarify exactly how this classification works, we start with the simple case of a continuous, Euclidean space. Pick a point P in this space and consider all loops starting and ending at P. There is essentially only 1 loop that can be drawn, up to homotopy. This loop, in fact, is contractible to P. We therefore say that this space has a trivial first homotopy group. Every simply connected space, Tsc has a trivial vri(Tsc) group; 7ti(M ) is called the fundamental group. This is because on all simply connected spaces, loops are contractible to a point. Note, however, that Tsc may have non trivial 7r2 (Tsc), as in the case of the surface of a hollow sphere. The usual group identities apply to homotopy groups: members of this group include the composite function / • g; the inverse, / - 1 (:z), equivalent to traversing f ( x ) in the reverse direction; and the identity, / , is defined as all loops contractible to the point P. Consider the loop group and the manifold given in Figure 3.2B; there are two holes in the manifold, and since the rest of the manifold is continuous, drawing loops on this manifold is equivalent to drawing circles around the holes. To start 61 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. with, there are three choices, a loop can wind around no holes, one hole, or two holes. A given path may wind around the first hole a number of times before winding around the second hole, and so on. There is an infinite number of possible configurations, and, clearly, this manifold has a non-trivial fundamental group. Similarly, if a manifold contains non-contractible points, they possess a non trivial 7r2(T) homotopy group because all non-contractible points are equivalent to traversing the surface of a sphere, and so on for the higher dimensions. We now look at a more realistic potential for cosmic strings: the Mexican hat potential. It is the 3-dimensional analog of the double-welled potential given in Equation 3.2. The functional form remains the same, except now <p is <p(x, y, z, t), a function in four-space. The vacuum state of this potential is continuously degenerate, defined by the circle of minima at \<f)\ = 77. The topology of the ground state of the Mexican hat potential is isomorphic to the unit circle, S'1. as the rest of the space is energetically forbidden. If we traverse this minima circle, the loops of our path are defined only by the number of times we go around the circle, since partial loops are contractible to the loop origin. Therefore, the only unique characterization of paths in this space is the winding number, given by integers, of how many times around the circle we pass the loop origin. For this vacuum manifold, 7r1 (S’1) is isomorphic to the group of integers, Z. Note that while the vacuum state degeneracy is continuous, the homotopy group of the vacuum manifold isn’t. Each possible value of the vacuum state corresponds to an integer. There may be an infinite number of integers corresponding to an infinite number of states, each state is separate from each other, as the group of integers is discontinuous. This is the reason why the fundamental group is im portant for cosmic strings: this class of defects arises from the non-trivial winding around holes in the vacuum manifold, a result of the topology of G /H . 62 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Likewise, different potentials yield different topological spaces for the vacuum state, and we give a summary of the isomorphisms for the fundamental group as found in Vilenkin k, Shellard (1994). In general, for compact connected Lie groups G (Lie group operations are differentiable), the fundamental group 7rx(G) gives, Z tti(G)^< G = U(n)(n > 1), 5 0 (2 ) Z2 G = SO(n)(n > 3) I (3.15) others where Z 2 denotes the discrete group consisting of two elements from Z, and I denotes the identity group, sometimes denoted as {1}. The differences between the double-welled potential and the Mexican hat potential is a good illustration of the differences between two homotopy groups. In the first case, the manifold M consists of two distinct and disconnected components, ± 77. This type of topol ogy belongs to the homotopy group 7r0(M). Manifolds with non-trivial 7t0(M) produced domain walls. The Mexican hat potential, on the other hand, has a non-simply connected manifold with holes around which winding numbers for a defect may develop. This corresponds to a 7Ti(M) homotopy group and there fore produced cosmic strings. Higher dimensions of homotopy groups produce monopoles (non-contractible spheres for 7r2) and textures (non-trivial mapping of 5 3 where 7t3(M) / I). A mathematical theorem allows us to further simplify the condition for a cosmic string topology: Let G be a connected Lie group, with subgroup H , and H may be composed of disconnected parts. We designate H 0 to be the part containing the identity, I, in which case G / H 0 is then simply connected. The quotient group tto(H) = H / H 0 is isomorphic to the fundamental group of 7ti(G/H ) (James 1984, Proposition 5.22 and 5.23). By this theorem, if H is disconnected, then there exists a non-trivial 7ti(G /H ), and one would expect 63 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. cosmic strings to form. We can see the validity of this theorem by considering the reverse: if H is simply connected, and since H is a group, which by definition contains the identity element, all loops within H can be contracted to the identity. However, if H is disconnected, then there exists some loops g(t) in H th at cannot be contracted to the identity. To be more precise, M = 7ii(G/H) = t v0( H ) . (3.16) The existence of cosmic strings boils down to whether the unbroken symmetry group H is connected. If H is unconnected, we expect the formation of cosmic strings. In fact, the homotopic group to which. H belongs delineates the type of topological defects th at are produced in a theory with the given phase transi tion. It is really the topology of the group which describes the vacuum state th at determines the defect characteristic, the most obvious of which is the defect’s dimension. For this reason, objects such as cosmic strings are called topologi cal defects. For the Mexican hat potential, the vacuum manifold is n o n - s i m p l y connected because Z is discrete. We expect cosmic strings to form under phase transitions described by the Mexican hat potential. 3.1.3 C osm ic Strings and G U T s I am particularly interested in using cosmic strings as a possible probe of Grand Unification Theories (GUT). GUTs are complete theories describing the funda mental forces (except gravity) as the result of breaking down one unifying force. The motivation for GUTs has its roots in the successes of the electroweak union. The Glashow-Weinberg-Salam theory (Weinberg, 1967) showed that electromag netism and the weak force are part the same force, known as the electroweak force, at energies slightly larger than the masses of their force mediators. The mass of the W boson is 81.4 GeV, the mass of the Z boson is 91.1 GeV and the 64 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. photon is massless (Hagiwara et ah, 2002). The mass difference of the media tors for the two forces is responsible for the spontaneous symmetry breaking th at occurred when the tem perature of the universe dropped below ~100 GeV. The electroweak force belong to the group S U ( 2 ) x U(l), a union of the two groups formed by the weak force and electromagnetism. Efforts to unite the electroweak with the strong force have had more limited success due to the higher energies involved and therefore the lack of experimental constraints. The strong force, which proceed via Quantum Chromodynamics, is mediated by eight gluons of three colors: red, green and blue, and has a 517(3) symmetry. The current Standard Model is therefore a 517(3) x SU(2\ x U( 1) theory. GUTs (which are not part of the Standard Model) seek to explain this 517(3) x 517(2) x 1/(1) theory as the result of spontaneous symmetry breaking of some fundamental group G. Candidates for G include: the simplest minimal group, 517(5), which has 24 mediators representing the 8 (QCD)x 3 (QED)x 1 (E-M) mediators. For many reasons, including esthetics, 50(10) is more popular. The overall scheme of GUT symmetry breaking is the following, G H ... ->• 517(3) x 517(2) x 17(1) 517(3) x 17(1) (3.17) where the arrows indicate a phase transition, and H indicates a possible inter mediate state between the GUT symmetry group G and the Standard Model. If we believe that a unifying GUT can be constructed, at each point where a phase transition occurs, there is the possibility of forming topological defects. In the absence of other evidence, the existence of topological defects can give us limits on the types of phase transitions that are allowed in the early universe. As an example given in Vilenkin & Shellard (1994), consider the breakdown of an 517(2) symmetry to Z 2 (the subscript 2 denotes two elements of Z), via the 65 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. following mechanism, SU(2) {/(l) Z2, (3.18) where the superscript 3 indicates the phase transition is mediated by a triplet. We break the 517(2) symmetry with a potential similar to the Mexican hat potential, given by V((p,Tp) = ^ ( d ) 2 - r / l ) 2 + ■4>)2 (3.19) where <f> and ijj are in C 3, the three dimensional complex space. The extra 7J)2 + quadratic term is motivated by the fact th at we are using two triplets to break the symmetries of the potential. W ith respect to the first triplet, </>, the ground state occurs when </>2 = rj^. The orientation of this triplet can be in any direction, and for simplicity, we take it to lie along the z-axis, (3.20) 0 V^ Rotations in the z-direction, given by the operator R = exp(—iOT:i) leaves 4>q invariant. Generators of rotations in the z-direction, T3, is given by, T, = i 0 0 0 0 0 (3.21) There remains an unbroken subgroup, 17(1). The vacuum manifold is therefore, G / H = SU(2)/U(l) ~ S 2 (3.22) We see that no cosmic strings form here, because the leftover manifold, 5 2 is a simply-connected sphere. 66 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. This remaining symmetry can be broken again by a second triplet. We set the second scalar field, ip, orthogonal to <fi in the y-direction, i ’o - w (3.23) The remaining 17(1) symmetry is now broken. However, if we look closely, there are two elements, 1 and -1, th at are mapped to the identity I. This means that there is a little group, H = Z 2 (where Z indicates integers) left over. Z 2 contains only two elements, 1,-1, and is disconnected. Cosmic strings will form during this phase transition. Cosmic strings are created as either infinite filaments which span the horizon, or as closed loops. Infinite cosmic strings leave tell-tale gravitational signatures. The metric of a long string in cylindrical coordinates, to linear order in mass per unit length of the string, yu, is given by, ds2 = dt2 — dzA — (1 —h){dr2 + r2d02) (3.24) where h = 87rGpTn(r/r0), r 0 is a constant of integration, and r = (x2 + y2) where x and y are the Cartesian coordinates for a string lying along the 2 -axis. In general, yu is expressed as a dimensionless quantity, G/i, and, (3.25) where m pi is the Planck mass, and is the string symmetry breaking scale Numerical simulations of cosmic string networks have shown that string evolution is self-similar and approaches a scaling solution (see, e.g. Bennett & Bouchet 1990). The correlation length of the scaling solution, £, is set at string symmetry breaking. Strings have length £ in a volume defined by £3. The constant £ grows 67 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. proportional to time, £ = factor of the universe. 7 where 7 ^ is a constant dependent on the expansion Therefore the energy density of the string is roughly 2The electroweak symmetry breaking scale is around 100 GeV, the energy scale defined by the electroweak mediators, the W and Z bosons whose masses were given earlier. An electroweak string would have G/i would have 77 ~ 1016GeV, and G/j, 10 34; a GUT scale string 10-6 . Therefore, knowing the masses of the strings can give us the energy scale of the phase transition th at created them. Alternatively, the absence of strings would give an upper limit to the energy of possible phase transitions. Strings are created with relativistic velocities. In order for strings to not dominate the energy density of the universe, numerical simulations have shown th at strings could intersect and break off loops in a process called intercommutation. The amount of intercommutation can be tuned so that infinite strings have many intersections. Closed loops can wiggle and emit gravitational radiation, which causes the loop radii to decrease. Eventually, within the lifetime of the universe, string loop radii can reach zero. This is their primary energy dissipation mechanism (see e.g. recent numerical simulations, Moore, Shellard, & Martins 2002) th at prevents cosmic strings from dominating the energy-density of the universe. Cosmic strings grow along with the horizon size, and because they are rela tivistic, string wakes will span the horizon. At last scattering (LS), the horizon has angular size 6h given by, (3.26) where zls is the redshift of LS. Since the CMB freezes gravitational signatures at LS, we expect to find most of the signatures of cosmic strings in the CMB at the horizon scale of LS. However, different numerical simulations give different string 68 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. densities and velocities and there is little agreement on the size and number of strings we should see in the sky. The one fact that all simulations agree on is th at string wakes leave edge-like signatures in the CMB. We emphasize th a t cosmic strings are necessary products of phase transitions whose unbroken symm etry group, H, is disconnected. The detection of a string would certainly be very interesting, but the absence of these defects is equally telling. Were a Z 2 kind of topology to arise from the previous discussion, one defect would be created per causal horizon This leads to a possibility of ~ 13000 cosmic strings in the present day sky. Even if only 0.1 % of these strings survived, there should still be 13 strings in the sky. Given our 2/3 sky coverage (see Section 3), we have the possibility of seeing 8 strings in the WMAP sky. Therefore, a non detection on our part is a genuine limit. The complete absence of the detection of strings would be difficult to reconcile for theories containing disconnected vacuum topologies. 3.2 W M AP Photons produced in the early universe, when the ambient tem perature was high enough to keep hydrogen ionized, remained in equilibrium with the baryons and traced baryon density. At a red-shift of z ~ 1100, electrons and protons re combined and the universe became transparent to photons. These photons have redshifted as the universe expanded and we now observe them as the 3K Cosmic Microwave Background. The photons retained the gravitational signatures of the early universe density distribution. The primary effect of a cosmic string on the CMB is to create temperature anisotropies due to the wake of the moving string. Along a particular line of 69 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. sight, the tem perature anisotropy, 5 T / T induced across a string moving in the plane of the sky is given by, A = SxG vff j. where 7 (3.27) is the Lorentz factor and (3 = v/ c is the velocity of the string, G is the gravitational constant, and n is the string linear mass density.. The Wilkinson Microwave Anisotropy Probe (WMAP) is the latest NASA satellite to measure the tem perature anisotropies of the CMB (see the LAMBDA website for a complete list of publications and public release data products: http://lam bda.gsfc.nasa.gov/product/m ap). WMAP has two back-to-back Gre gorian telescopes which observe two patches of the sky separated by 141°. A set of differencing assemblies obtains the tem perature difference between the two patches. The final data product of WMAP is the CMB temperature anisotropy of the each pixel in the sky, except where there are microwave sources and the pixels are masked. These masked pixels include most of the Galactic plane, the Galactic bulge, and some scattered sources off the Galactic plane. WMAP was launched on June 30, 2001, and it arrived at its L2 orbit on Oct. 1, 2001. The L2 is the second Lagrange point, a quasi-stable point where the gravitational pull of the earth balances the gravitational pull of the sun to. The first year data release in February, 2003, contained data taken by WMAP from Aug. 10, 2001 to Aug. 9th, 2002 . WMAP is still taking data and is expected to last at least until 2005. The following work is based on the first year data. As more data are released in the future, the sensitivity of this project will be improved as the pixel noise is reduced by repeated observations. WMAP observes the sky at five frequencies, from 23 to 94 GHz. Relevant WMAP characteristics are tabulated in Table 3.1. The data products released by the WMAP science team include the thermodynamic temperatures of each 70 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 3.1. Relevant WMAP Characteristics Frequency Resolution Wavelength Sensitivity GHz FWHM, arcmin mm /rK, 0.3° x 0.3° 23 56 13.6 35 33 41 10.0 35 41 32 7.5 35 61 21 5.0 35 94 13 3.3 35 WMAP pixel at all five frequencies. I use the data from the three highest fre quency bands (Q, V, W) in our analysis. The reason for using the highest three frequencies was to eliminate as much Galactic contamination as possible, which the lower two frequencies are more sensitive to. I first present the theoretical limits on the mass of cosmic strings based on WMAP sensitivity, following Vilenkin & Shellard (1994). An ensemble of strings will have strings moving in different directions with different velocities, so it is more useful to consider the root mean squared (rms) tem perature fluctuation from a group of strings. For this, I need to consider the fact th at cosmic strings grew with the horizon size, so that one may have different values for different redshifts. W ith the results of numerical simulations, we can average over string velocities between redshift z and 2z. The rms fluctuation induced in the CMB by string wake, between z and 2z is shown to be, 5oT - j r = 6Gii. 71 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (3.28) and is independent of z. Therefore, I can use this result generally. In real obser vations, one cannot see CMB signatures from before the LS cut-off. I therefore correct for the fraction of strings observable since LS, (3.29) RMS For zls = 103, this gives an rms tem perature fluctuation from an ensemble of strings of 19G/j. WMAP has a maximum beam resolution of full width at half maximum of uj = 13 arc-minutes, so th at the maximum redshift observable for WMAP is zmax ~ (4uj)~2. I replace zLS with zmax in Equation 3.29. WMAP RMS tem perature sensitivity to cosmic strings is 20Gn. As a comparison, the COBE satellite had a field of view of 7 degrees, which resulted in an expected A T /T of 8.6Gji. The observed fluctuations for COBE was, (3.30) RMS which resulted in a limit for cosmic strings with Gji < 1.3 x 10 6. For a similar observed RMS tem perature fluctuation, WMAP improves the limit for the mass per unit length of cosmic strings to Gji < 5.3 x 10“ 7. This is the limit for an ensemble of strings; if we allow the existence of very few cosmic strings, these rare objects can be much heavier than this limit while not having an appreciable effect on the CMB. This paper presents a method of searching for the limits of individual cosmic strings. 3.3 S tr in g Search: T h e E d g efin d er For my string search, I used the WMAP map of tem perature differences, A T /T , of the full sky. At 13 arc-minute resolution, the WMAP sky is divided into 12 x 49 72 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. pixels. WMAP uses the Hierarchical Equal Area isoLatitude Pixelisation of the sphere (HEALPix: http://www.eso.org/science/healpix, Gorski et al 1999) to pixelize the sky. We have designed a digital filter for the WMAP data in search of signatures of filamentous string wakes, and we named this filter the Edgefinder. The Edgefinder took an input pixel and defined a window with radius RA D . Each pixel within this circular window was assigned an [x, y\ pair, with the input pixel at [0,0]. Pixel displacements from the origin, [0,0], are calculated and each pixel within R A D is assigned a normalized (to R A D ) x displacement and a normalized y displacement. The y-axis was defined to be parallel to the line connecting the North and South Galactic poles. Our filter window was small enough that we could consider the sky inside to be flat. For each pixel within the window, the Edgefinder multiplied the filter value, F ( x , y ), by the pixel tem perature at [x, y\ and stored the sum of this product for all pixels in the window at the position of the input pixel. Therefore, the output of this digital filter was a map where the value at each pixel was a sum of the effects of the filter on the surrounding pixels. I called this output the Edgefinder value map, EV for short. For the WMAP data and associated simulations, the Edgefinder had an R A D = 1° window in the sky, which corresponded to the size of the horizon at LS. This window had a radius of 18 pixels and usually contained 260 pixels in total. The filter value, F ( x , y ), was designed to pick out edges in the sky aligned with the y-axis of the filter. The specific values of each of the filter were generated separately, because pixel centers were slightly offset from each other depending on the pixel latitude, and the edges of the filter were ragged due to the diamond grid of the HEALPix scheme. This, coupled with the fact th at I only have 260 pixels in the window, meant that the sum of the filter values didn’t always sum to a perfect zero. We therefore generated the filter values twice at each pixel: the first time to collect the excess, a, for each window, and the second 73 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. time to distribute a evenly among the pixels. This forced filter values in the R A D window to sum to zero exactly. The filter values, F ( x , y ), were given by, F(x, y) = N ( x =f A) x exp ) (3‘31) the minus sign applied to pixels with x < 0 and plus for x > 0, A was the normalized height of the filter, and N was the total normalization factor for the filter, discussed later. The exponential smoothing function ensured th at the filter is smoothed and compact. The normalized radius, r n., is rn = yfx2+ y 2. The filterfunction is designedto be insensitive to a constant value, a gradient, and to be spherically symmetric except for the edge detection. Insensitivity to a constant demanded J F(x, y)dxdy = 0. (3.32) Insensitivity to a gradient demanded J xF (x , y)dxdy = 0, (3.33) and the value of A was adjusted to ensure this. The Edgefinder is depicted in Figure 3.3 as a shaded surface plot with the z axis representing the filter value, F(x, y) . In addition, a plot of the cross section of the filter, is also shown. The filter could be rotated in the sky to detect strings of different position angles with respect to the North-South Galactic alignment. This was achieved by rotating through an angle a by altering the [x,y] values of a pixel to [x',y'] by the following transformation, x, \ y X y \ ( cos a —sin a y s in a (3.34) cos a The filter values are then generated with x' in place of x and y' in place of y. In most simulations, I ran the filter at 20 different a values equally spaced between 74 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. side view - 0 .0 2 0 .0 1 0 .0 0 0 .0 1 0 .0 2 Figure 3.3 A 3-D representation of the Edgefinder. The z-axis represent filter values, and the x and y axis are the pixel numbers. The right bottom inset is a plot of a slice along the y=0 plane of the filter, with the axis in units of pixels. 75 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0 and 180 degrees. This represented a shift through one pixel at the edge of the filter window, and should pick out edges aligned in all directions. The characteristic Edgefinder behavior around a string horizon is depicted in Figure 3.4. In this figure, the top picture is a simulated input temperature map, where the dark disk is the simulated string horizon. The disk edges are fuzzy due to the noise added to the map (as a real string in the sky would be seen by WMAP). The bottom picture is the Edgefinder response to the input map. The Edgefinder was oriented North to South, and produced a “hot” signal when encountering a rising edge, and a “cool” signal when encountering a dropping edge. If I rotated the filter window by 180°, I would get the opposite signal, meaning that the sign of the EVs was not im portant, so I used absolute values in my statistics. Running through 20 a angles from 0 to 180° really represented running through 40 a angles from 0 to 360°. Visible on the same figure are pale circular disks which indicate masked pix els in the WMAP data, usually where there were strong foreground microwave sources. These regions were not used in my simulations. I used the combined KpO and Kp2 masks (see Bennett et al 2003 for an explanation of the masks) which masked 1109593 out of 3145728 pixels (35.7%). If strings were partially blocked by masked pixels, the Edgefinder can still detect them, but at a reduced sensitiv ity. I ran tests where strings were partially blocked by a mask, with 5 different exposure levels from 100% to 18%, where 100% indicated an unblocked string. These results are tabulated in Table 3.2. For strings which are more than 50% ex posed, there were no differences in the peak EV compared to an unblocked string. For strings th at are less than 50% exposed, the peak EV dropped off rapidly as the string centers moved behind the mask, but were still detectable, their EVs dropping to around 15% of an unblocked string. This test indicated that strings 76 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. with centers located outside of a masked area will be picked up normally by the Edgefinder, and therefore the masks are not blocking more sky than their actual area. Masked Figure 3.4 Edgefinder values around a string horizon. The Edgefinder is oriented North (up) to South (down). The light blue circles are masked pixels in the WMAP data due to foreground sources. 77 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 3.2. Partially Exposed Strings Exposure % Normalized Max EV Non-zero Pixels 100 0.961 575 73.6 0.956 339 48.7 0.569 210 32.6 0.160 135 18.0 0.143 71 The Edgefinder filter values, F (x,y), were calibrated to give a response of Edgefinder value, EV, of 1 for a 1 mK input signal. For example, if the input CMB map had all its southern hemisphere pixels at 1 mK, and northern hemisphere pixels at 2 mK, the Edgefinder would return EV = 1 for equatorial pixels, and EV — 0 at other pixels. The normalization constant for the R A D = 1° WMAP Edgefinder was N = 0.33. Due to fluctuations in the normalization and the excess value collection of the filter, the actual response varied between 0.97 and 1.08, but averaged to 1.0 across the equator. Strictly speaking, the EV is unit-less, but for the sake of clarity, we will sometimes give it units of mK. Unless specifically mentioned, all tem perature units in this paper are in milliKelvin. I created a set of calibrators to verify the gain of the Edgefinder. The cali brating set contained edges, or string horizons, of various tem peratures ranging from A T /T = 1.1 mK to A T /T = 1 nK. The results of the calibration are in Table 3.3. In most cases, the EV was the same magnitude as the input edge to within 2%; we are therefore confident that the Edgefinder had a linear response to the input string horizon over the range of pertinent input temperatures. 78 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 3.3. Edgefinder Gain Calibration n -i a 1s EV G a i n | Q=0b EV G a i n | Q=7r/ 2 1.000000 0.9567 0.9956 0.100000 0.9985 0.9732 0.010000 1.0800 0.9085 0.005000 0.8456 0.9630 0.001000 0.7639 1.0150 0.000500 0.8046 0.9842 0.000100 0.3296 0.8974 0.000050 0.8974 0.9794 0.000010 0.9559 1.0200 0.000001 1.0440 1.0910 aT, is the input cosmic string horizon tem perature bEV Gain is —EV /TS— , here quoted for 2 different a angles. 79 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. o CD o > a —0 CD a — 7v/1 0 O a =ix/b a = 3-n / 1 0 a = 2 7 r/6 Z3 a = 7T/ 2 on -2 0 100 1000 Figure 3.5 The response function of the Edgefinder Filter. Note the peak at I ~ 200, around where the first Doppler peak of the CMB anisotropy angular power spectrum occurs. The different lines correspond to different a angles of the orientation of the filter with respect to the north-south Galactic axis. The size of the Edgefinder was chosen to match the most likely string size that we may be able to find. For WMAP, this was a problem since this scale was also where the CMB Gaussian anisotropy signals were the strongest. The response of the filter to I values from 1 to 1000 is plotted in Figure 3.5 for representative a angles. The largest filter response was around I = 200; the WMAP data indicated th at the first Doppler peak of the CMB anisotropy angular power spectrum is located at I ~ 220. This was undesirable as we wanted to insensitive to as much much of the Gaussian signal as possible so we could focus on the non-Gaussian signals. 80 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. One easy solution would be to make the filter smaller. However, this is not feasible with WMAP data. At R A D = 1°, there were 273 pixels in the filter; at R A D = 0.5°, there would be only 69 pixels in the filter, spanning less than 10 pixels. This resolution was too coarse to run the Edgefinder. For now, I report results of our good enough filter while we work on ways to eliminate the response at i ~ 200. One certain method is to wait until PLANCK data is available, and with the improved resolution, we will get 270 pixels in the filter at R A D = 0.5. This is discussed in Section 6. 3.3.1 Sim ulated M aps I knew from the calibrator set described in the previous section th at the Edgefinder had a linear response to the input edge. 1 next needed to find out how the Edgefinder responded to noisy maps like the ones from WMAP. 1 anticipated th at the Edgefinder would not be able to detect strings when the string hori zon signals became swamped with noise. In order to find this limit, I produced simulated maps to quantify the behavior of the Edgefinder. Once I understood how the Edgefinder responded to maps containing strings of known magnitudes, I could then set detection limits of the Edgefinder for the WMAP data. Because they are so vital, I describe in detail how we generated the simulated maps: 1. Coefficients, Ci, of the CMB angular power spectrum were generated by CMBFAST (see webpage http://www.cm bfast.org) using the cosmological constants derived from the WMAP experiment, a ACDM cosmology, given by: Zmax = 1500, Kmax = 3000, n b = 0.044, Qc = 0.218, Qx = 0.738, H 0 = 71.6, T cmb = 2.725, Yue = 0.24, N^^massiess) = 3.04, tlss = 0.099, and n = 0.955 (Bennett et al, 2003). 2. I used HEALPix associated software SYNFAST to generate a CMB map 81 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. which matched the Ct generated in Step 1. SYNFAST generates random Gaussian fields on a sphere based on the input power spectrum (the CVs). 3. I added string horizons at various temperatures, Ts, into the map by adding the value Ts to circular regions within a string radius R s. 4. I used HEALPix associated software SMOOTHING to convolve three beams with the map made in step 3. A W-Band map was made by convolving a 13 arc-minute Gaussian beam with the cpm’s of the tem perature map created at Step 3, and the new <pm’s were used in SYNFAST to generate a new map. A V-Band map was made from the convolution of a 21 arc-minute beam, and a Q-Band map from a 32 arc-minute beam. For each map generated in Step 3, three maps convolved with beams appropriate for the three WMAP bands were, made. 5. The 3, W-, V-, and Q-Band, maps were averaged to get the final SMOOTHed map. Noise was then added to the map with the following prescription: the an for the noise of each pixel was generated by combining the noise characteristics of the 3 WMAP bands, \JvQ + °V + a W 3 where oq (3.35) is the pixel noise in the Q band, where N q ^ s ”Q = av = -p^L= (3.37) aw = - , \j Nw,obs (3.38) 'obs (3 36) r)Q\ is the number of times the pixel had been observed by the WMAP satellite in the Q band; ctq^ = 2.211, aVt0 = 3.112, ovyo = 6.498 are the noise weights given by the WMAP team. The noise values were generated by a Gaus- 82 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. sian random number generator with a = an. The reason for this noise prescrip tion, and for the convolution with three beams in Step 4, was th at the data I fed to the Edgefinder are a composite of data from the three WMAP bands Q, V and W. Due to the SMOOTHING and the added noise, the final string horizon tem perature was at a slightly different tem perature than the initial Ts. I sometimes also report, for comparison, the average tem perature of the final input horizon, •which I designated Tf. 3.3.2 Edgefinder Lim its In all, 82 input simulated maps were made according to the prescription given in Section 4.1. Out of these, 15 of the input maps were baseline maps, where no string horizons were inserted. I called the EV output of these maps the No-String sets. The maximum EV of the No-String sets indicated the maximum EV due to background signals (i.e., not from the strings). In other words, the No-String sets produced the noise limit; I considered anything above this limit to be signal. Both the CMBFAST generated CMB signal and noise are basically Gaussian random variables, so I needed to have a range of No-String sets in order to ensure I have proper coverage of the possible noise values. To this end, I created 15 No-String input maps. I then created the 67 “Stringy” maps, which contained different number of string horizons with various sizes and temperatures. The input string horizon temperatures ranged from Ts = 1.0 mK to 0.0001 mK, the inserted string radius, R s, ranged from 1.0 to 4.0 degrees. Most maps were made with one inserted string, but one map contained as many as 60. The Stringy maps are further divided into single and Multi maps, where Multi maps had more than 1 inserted string horizons. There are 16 Multi maps. 83 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0.8 0.6 £ 0 No —String Largest Average No —String No —Strina Smallest .4 X o 0.2 A No —String Values □ Multi String Values .0 0” 6 10 -5 10 ° -4 .-3 LOG Input String Te m pe r a t u r e (K) Figure 3.6 Plot of the input string tem perature Ty vs. the maximum EV of the set. 84 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Comparison between the maximum EV (Max EV) of the Stringy and NoString sets will yield the input string value th at gives a Max EV above the No-String limit. The No-String limit is given by the largest value of the Max EVs of the No-String sets. The average of the maxima of the 15 No-String sets was EV = 0.242 mK; the largest of the 15 sets was EV = 0.269 mK. I plotted the Max EV of all single Stringy sets against the input string temperature, Ts in Figure 3.6. The No-String limits are plotted as the dashed and dotted horizontal lines. A blow up of the region around the No-String limit is in Figure 3.7. The data indicates th at an input string horizon of 0.345 mK and cooler yielded similar Max EVs as the No-String sets. I can consider T =-- 0.354 mK as one of the limits to the sensitivity of the Edgefinder. If I took this limit to be wakes formed by strings traveling at c/y/2 (the mean absolute velocity of strings from numerical simulations), according to Equation 3.27,1 would have limits for the cosmic string at G\x £ 1.37 x 10“ 5. This is the crudest but most robust method of obtaining the sensitivity limit of the Edgefinder. The problem posed by the Edgefinder is th at I am looking for very small non-Gaussian features in a largely Gaussian data set. For an input string of R s = 2°, the number of pixels whose filter window contained the edge constituted 0.0087% of the total pixels. The binning of the EV set data resulted in the non-Gaussian signatures occurring in the edges of the histogram. To improve upon the Max EV limits, I needed a set of descriptive statistics that could pick out small non-Gaussian signals at the outer edges of a Gaussian function. I found the Edgeworth Series to be suitable for my purpose. 85 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0.3 5 N o - S t r i n g Largest Average N o - S t r i n g 0.30 0.20 A No —String Values □ Multi String Value 10 6 10 5 10 -4 10 LOG Input String T e m pe ra tu re (K) 3 Figure 3.7 Detail of the plot of the input string tem perature Tf vs. the maximum EV. Also plotted are the multi-string set data as well as limits from the No-String sets. The stars are the Max EV for each simulated map containing a string of strength shown on the x-axis; the square indicate simulated maps containing multiple strings; the triangles indicate maps containing no inserted strings. 3.3.3 Edgefinder Lim its from E dgew orth C oefficients The Edgeworth Series uses the Method of Moments to obtain coefficients of the terms in the series. It has been proposed as a tool for picking out non-Gaussian signatures in the CMB by various authors (e.g. Cayon et al. 2003). Here, we will only illustrate the derivation of the Edgeworth Series to introduce the components necessary to calculate the coefficient. A full derivation can be found in Kendall (1987), discussions relevant to astrophysics can be found in e.g. Juszkiewicz et al. (1995) and Blinnikov & Moessner (1998). 86 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. The Edgeworth series is an asymptotic expansion of a distribution using the normalized Gaussian function and its derivatives as the basis set. Coefficients of series terms give how well the given distribution approximates a Gaussian. Collection of coefficients of like terms in the Gram-Charlier series results in the Edgeworth series. Specifically, the Edgeworth series is a linear combination of the Chebyshev-Hermite polynomials using cumulants of the distribution as coef ficients of the terms in the series. In this section, we use the abbreviation D r to mean drj d x r. Before beginning the derivation, there are a few things th at need to be intro duced. Let a(x) be the Gaussian distribution, a(x) = j— e~^x2. V2 tt (3.39) The Chebyshev-Hermite polynomials (referred to hereafter as Hermite polynomi als) are generated by successive derivatives of a(x), Da(x) = —xa(x) D 2a(x) = (x 2 — l)a(a:) D 3a(x) = (3x —x^)a(x) (3.40) and we obtain the nth Hermite polynomials, H n(x), as the polynomial resulting from the n th derivation of a(x), (—D)ra(x) = H n(x)a(x). (3-41) Hn{x) = ( - 1 )ne 2 x2D n (e“ ^ 2) . (3.42) ft has the general form, The first 5 Hermite Polynomials are, H, = 1 87 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. #1 x h2 x2 —1 H3 x 3 —3x Ha x 4 — 6x2 + 3. The Hermite polynomials are orthonormal and have the following property which is easily verified, / 0, ?n OO H m(x) Hn(x)a(x)dx n (3.43) nl, m = n -OO The characteristic function of a distribution is the P'ourier transform of the probability density of the distribution. It gives statistical information about the distribution in a compact form and, if it exists, uniquely defines the distribution. The characteristic function, <f>(t), is defined as, m ei,xdF = where dF is the distribution over x. (3.44) For clarity, we may think of dF as the distribution of pixel tem peratures and x as the range of possible temperature values. Taylor expanding <f>(t) in Equation 3.44, we get at t = 0, « (i) ~ i + Y , ^ ( a y (3.45) r —1 where /ir is the rth moment of the distribution $(£). The moments, fi, are more familiarly expressed as, / OO (x — hi )rdF (3.46) -OO where n i is the mean value of the distribution. Taking the natural log of <h(t), we obtain the following Taylor expansion, In <F(f) (3.47) j=i J- R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. where Kj is the j th cumulant of the distribution d>(f). The j th order cumulant, Kj, is defined by, (3.48) The cumulants, k, are related to moments, p, by, (3.49) where the bracket is the binomial bracket. Expanding the terms, the first 4 cumulants expressed in terms of central moments (moments about the mean) are given by (note th at Ki = pi = 0 for central moments), .2 «3 = «5 = A*3 - 1 0 p 3h 2 - Cumulants provide a measure of how the distribution dF differs from Gaussian. For example, k2 is the dispersion (a2) of the distribution; is known as the skewness of dF, which measures the asymmetry of the distribution relative to a Gaussian; k4 is the kurtosis, which measure how quickly the tail of the distribution flattens compared to a Gaussian. The Edgeworth series is essentially a recollection of terms of the GramCharlier series. We introduce here the Gram-Charlier series and motivate the re-collection of terms later. The Gram-Charlier series expands a distribution in terms of the derivatives of the Gaussian function, a(x). Such a series would look like the following, i =o 89 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. where the cjs are coefficients to the Hermite polynomial, H f s. The coefficients, Cj, of this series is obtained by multiplying both sides of the above equation by Hr(x) and integrating from —oo to oo, using the orthonormality condition of the Hermite polynomials (Equation 3.43) we get, (3.51) Substituting in the Hermite polynomials and re-expressing the coefficients in terms of central moments, the series becomes the Gram-Charlier series. The first five coefficients of the Gram-Charlier series expressed in terms of cumulants, are, 1 Cl =0 C2 = (M2 ~ l)./2 C3 = 113 / 6 c4 = (p4 —6 / j ,2 + 3) / 24. (3.52) In essence, this is also the Edgeworth series. The difference is th at the GramCharlier series is not truly asymptotic, because no m atter where the series is truncated, the remainder terms are of similar order due to the mixing of linearorder /i terms in the coefficient cjs. For example, c2 is (fi2 — 1)/2 and c4 is (/i4 — 6 /i 2 + 3)/24, containing both ji2 and /z3 to linear order. We need to re-order the series in such a way th at successive terms are negligible compared to the previous term. Instead of using moments, /r, in the coefficients, the Edgeworth series uses cumulants, k. By transforming the Gram-Charlier series using the relationship between moments and cumulants (Equation 3.49), one obtains coefficients that have the desired properties. 90 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. To show th at such a thing is reasonable involves more work. Consider ob taining the characteristic function of a distribution based, again, on derivatives of the Gaussian function, a(x). This time, however, we will build in the fact that the distribution is expressed as exponents of derivatives of a(x) and consider our starting point the expression, {exp(ftr.Dr )}o;(x). where we have suggestively labeled the coefficients (3.53) k, as we hope to show th at in fact, these are cumulants. Referring to Equation 3.44, the characteristic function of this expression is given by, r°o r°° / e*<x exp(KrD r)a(x)dx = / eltx ^ J —oo J —oo ( kH DrV \ I -J—r.— ) a(x)dx \ J■ (3-54) / Using the definition of Hermite polynomials, and moving terms without x depen dence outside the integral, the characteristic function becomes, 5Z ■] roo I eltx( —i y rH ^ ( x ) a ( x ) d x (3.55) This now resembles a Fourier transform of the expression H r ( x ) a ( x ) . We can derive this starting from the Fourier transform of a(x), V 2 ^ a { t ) = e- ^ 2 = [ elte- L e ~ k2x2dx. J y 27r (3.56) Each derivative of a ( t ) generates an (ix) term in the Fourier transform. So we can write the following, V 2 ^ D ra{t) = ( - l ) r v /2* H r {t)oi{t) = /° ° e ^ ^ L e ^ 2/2) dt. 3—oo \ / 2 tt (3.57) Writing the second equality again, with a little re-arrangement, fOO 1 ir H r (t)a(t) = .— / e~ltx x r a ( x ) d x , V27T 3—oo 91 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. (3.58) which tells us that the Fourier transform of x ra ( x ) is, foo 1 xra(x) = — / . __ e~UxirV27rHr(t)a(t.)dt. (3.59) Z7T 7 - o o Interchange x and t and moving some constants, poo .__________ ( ~ i ) rV2TTtra (t) = / (3.60) e~lxtH r ( x ) a ( x ) d x . J — OO This gives the Fourier transform of H r (x)a{x) to be \ / 2 n i Tt ra{t). Putting this into Equation 3.55, the characteristic function of Equation 3.53 becomes. T 7'! \/2 n { - i ) rn rja(t i = ' = v ¥ r a ( t) ¥ ¥ ( j! v¥ro;(f) exp[«;r (—it)7"]. (3.61) We artificially construct another expression to be considered, based on Equa tion 3.53, and motivated by the form of the Gram-Charlier series, exp 1! 2! 3! 4! (3.62) a{x). The characteristic function of this expression can be obtained by analogy to the characteristic function the Expression 3.53. Again, using the fact that derivatives of a(t) results in another (i x ) term in the Fourier transform, the characteristic function of the previous expression is given by, 4>(t) = y/2na(t) exp (3.63) + ~ { i t f - ^ ( i t f + ^ ( ^ ) 4-2! 3! Looking back to Equation 3.47, the constant that we have labeled k is indeed the cumulant. This is shown by taking the logarithm of Equation 3.63. This means that if k were the cumulants of the distribution $ , then the expression in 3.62 is the proper expansion in powers of derivatives of a(x). We are now justified to take the coefficients of the Gram-Charlier series and turn the moments 92 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. into cumulants. The first 7 Edgeworth series coefficients, E n are given below, coefficients, Ei = 1 (3.64) E2 = ^3 / 6 E3 K 4/24 = Ea = K5 / I 2 0 E5 = (k &4 10K.g)/720 E = (k = (k.s T 56/?5/?3 4 35k^)/40320. q £7 j 4 35/v4«;3)/5040 In its usual form, the Edgeworth series is given by, ff({x)\ = exp 1 1)3 Di a(x) (3.65) or, alternately, using Hermite polynomials, f ( x) = a{x) ( l + — H 3 + — H a + — H 5 -f W V 6 3 24 4 120 — 720 4 J . ((3.66)' It is clear that each successive term contains cumulants of higher order. We tested the Edgeworth Series on the histogram of a perfect Gaussian dis tribution, for which we knew all E n coefficients should be zero. We found th at the biggest effect on the coefficients was how quickly the edges of the Gaussian function went to zero. For example, in two trial runs we used 1 million random numbers to represent a normal Gaussian distribution, with zero mean and unit variance. The one run where we allowed the x variable to go to x = 16, generated zeros at a level of 10-17. The other run where we only allowed x to go to x = 3, had zeros on the level of 10~8. This meant that we had to choose the same set of bins for every EV set instead of binning the data with bin-size set by the maximum and minimum of 93 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. each distribution. We experimented with the number of bins to see what our binsize should be. W ith too few bins, the non-Gaussian aspects of the distribution were hidden. W ith too many bins, the residuals (^(d ata —fi t ) ) increased from errors at the wings of the distribution, and since this was exactly the location of the non-Gaussian signals we wished to detect, we had to be cautious. W ith the number of bins from 500 and 1000, the variation in the coefficients generated were within 10%. The optimal number of bins turned out to be 574, which struck a balance between generating a good zero value for the perfect Gaussian case, as well as having enough room at the wings for the non-Gaussian signals. We binned the absolute value of the EV set for each of the simulated maps into 574 bins, with the uppermost bin at 0.3 mK. The upper bin was chosen because maps containing EV greater than 0.3 mK were well above the Max EV of the NoString set and therefore were already known to contain strings. Since we used the absolute value of the EVs, we mirrored the histogram across the y-axis to create a negative half for the distribution; the whole distribution was then normalized to one. Due to the fact th at wre reflected the distribution across the y-axis, the his tograms were completely symmetrical. This meant th at the odd moments (even cumulants) were zero. Thus, we only needed to look at the odd Edgeworth coef ficients; E n where n is 3, 5, or 7. For our purposes, the 7th Edgeworth coefficient, E -(, was a good discriminator of whether a map contained a cosmic string hori zon. We have included plots of the other coefficients, E 3 and E 5, in Figure 3.8 and Figure 3.9 respectively; note that the y-axis of Figure 3.8 is on a log scale to include all data points. Compared to the E-j coefficient, E 3 did not pick out as many points, so at best, it can used as corroborating evidence. The E 5 coefficient was similar to E 3, but it included more points. 94 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. X No —String Largest Average No —String No —String Sm al les t O O O □ □ CD O 3 o * -3L- o AT JL XC s CD cn ID A No —String Values □ M u l t i - S t r i n g Values X) K) □ q □ At ^ * . - _ XG n ix T f ^ j g * 0 10 —6 -4 ^ —5 1 0 ° 10 LOG Input string t e m p e r a t u r e (K} 10' Figure 3.8 Plot of the input string tem perature Tf vs. the 3rd Edgeworth coeffi cient. 95 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. .0 X * No String Largest ------------- Average ■No —String .............. - N o String s m a ll e s t o o o 0.5 0) o o _c □ □ - - * □ - * □ H—' o 5 o cn TD _ * A A N o - S t r i n g Values □ M u lt i - S t r i n g . V a l u e s x: in - j _g__ 0.0 Lu - - -0.5 10 - 6 10~5 10"4 LOG Input String Te m per at ur e, (K) 10 -3 Figure 3.9 Plot of the input string temperature Tf vs. the 5th Edgeworth coeffi cient. 96 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Out of the 15 sets of No-String maps, the E 7 coefficients ranged from 6.51 x 10- 6 to —3.92 x 1CT6. Inspection of E 7 and other coefficients showed that there was a clear trend: the Stringy sets which contained hot string horizons produced bigger non-Gaussian tails that were picked up by E 7. The fact that the E 7 coefficients of the No-String set were distributed around zero tells us th at a small E 7 indicated a lack of non-Gaussian features like those produced by a string horizon. It also indicated the magnitude at which the E 7 coefficient can be considered noise. This was corroborated by the fact that the higher the Tf, the larger and more positive the E 7 coefficient. Therefore, I could use the No-String E 7 values as a discriminator between maps with strings and maps without strings. I took the second largest No-String (14th out of 15) E 7 value as our limit of the Edgefinder’s single string sensitivity, at EV = 5.06 x 10~ 3 mK. The type of map th at produced the smallest E 7 signals were maps with only one string horizon inserted. I made 66 maps of single string with input tem pera tures ranging from 1 mK to 1 fiK. The E 7 coefficients and input string horizons for all sets are graphed in Figure 3.10. It shows clearly that a large and positive value of E 7 is indicative of the presence of hot string horizons. Figure 3.11 is a blow up of the region around the No-String limit. String horizons with Ts > 0.27 mK were clearly above threshold E 7 value. At Ts = 0.260 mK I encountered the first string horizon with E 7 below the threshold. I therefore set T f = 0.27 mK as our 100% confidence level of string detection. Examining Figure 3.11, I can see th at the E 7 values mostly fall within the boundary defined by the maximum No-String and minimum No-String E 7, with a few peeking above the threshold. The scatter in the data is very constant. Therefore the limit of our detection is firm at 0.27 mK; all input strings cooler this threshold looks like noise to the Edgefinder. For a cosmic string moving at mean simulation velocity, (3 = l / \ / 2 this corresponds to a Gji = 1.07 x 10~ 5 string. 97 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0.6 No —String Largest Average No —String No —String Smallest X O o o 0.4 * 0) o o n 0.2 o 3 0 cn TD 0.0 A No —String Values □ Multi —String Values _c: -o.: .-6 10 J 10 r LOG Input String Tem pe ra tu re , (K) Figure 3.10 Plot of the input string temperature Tf vs. the 7th Edgeworth coefficient. 98 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0.02 c 0) o Q) o 0.0 o 0.00 sz 0.0 A N o - S t r i n g Values 0.02 L _ - 0 10 ° 10 -4 10 3 Input String Tem per at ur e, (K) Figure 3.11 Detail of the plot of the input string tem perature Tf vs. the 7th Edgeworth coefficient. Also plotted are the multi-string set data as well as limits from the No-String sets. 99 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 3.3.4 M ultiple S tring M aps 17 simulated maps were made with multiple string horizons inserted in the map, ranging from 1 to 60 horizons a,nd R s ranging from 1 to 4°. The number of string horizons and their sizes are summarized in Table 3.4. The purpose of the multiple string sets were twofold, first to determine if multiple strings had a similar effect on the Edgeworth coefficients as the single strings, and if so, determine the number of weak strings needed to generate an E 7 signal above the No-String discriminant. The Max EV and the E 7 coefficients for the multiple sets are plotted as letters A to P, according to their set name in Table 3.4, in Figures 3.7 and Figure 3.11, respectively. The bold items in the table represent sets with E 7 coefficients larger than the No-String limit. From the results of Sets A to C, I can see that the Ma.x EV of multiple string sets are generated by individual strings, and therefore from the Max EV alone, I cannot tell if there is more than one hot string in the data. However, the E 7 coefficient of multiple strings are cumulative. This means that m a situation where I have a small Max EV, and a large E 7, I know th at the sky contains multiple strings. However, looking at the results of Sets D, E, and F, I can see that if the strings are cooler than the detection threshold, their E 7 signatures are small enough th at a few strings (less than 10) will not show detection. It takes quite a few cool strings for the cumulative effect to show up. For very cool strings (Ts ~ 0.15mK), it takes 20 input strings for the E 7 coefficient to be above the threshold. The size of the string has some effect on the coefficients, but it is secondary to the strength of the edge. Compare Sets D to J and G to K; these sets have similar number of input strings, and similar Ts, but Sets J and K have twice 100 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 3.4. Number and Strength of Input Strings for Multi-String Simulated Maps. D ata Set # of Strings RAD T 1S Max. EV £V(10“6) M ulti A 2 2 .0 0.30 0.290 84.4 M ulti B 4 2 .0 0.30 0.310 185 Multi C 10 2 .0 0.030 0.231 2 .8 6 Multi D 4 2 .0 0.030 0.236 Multi E 30 2 .0 0.030 0.257 3.59 Multi F 60 2 .0 0.030 0.243 -2.42 Multi G 5 2 .0 0.015 0.231 2.09 Multi H 10 2 .0 0.015 0.245 4.76 M u lti I 20 2 .0 0.15 0.244 8.77 Multi J 2 4.0 0.015 0.249 4.49 Multi K 2 4.0 0.030 0.263 0.609 M u lti L 2 1 .0 0.30 0.265 46.2 Multi M 5 2 .0 0.15 0.234 0.394 Multi N 10 2 .0 0.15 0.230 2.54 M u lti O 5 1 .0 0.30 0.294 120 M u lti P 10 1 .0 0.30 0.294 114 - 2 .2 1 101 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. the R A D , which means 4 times the number of pixels. The E 7 coefficients of the sets are similar and show little trend reflecting the change in inserted string size. Furthermore, consider Set L, where the inserted string had R s = 1 degree; this set had an E 7 coefficient significantly above the detection threshold. Both these phenomenon point to the fact th at the Edgefinder is most sensitive to the temperature jum p at the edge of a string horizon, rather than the size of the string. The results of the multiple string sets shows th at the Edgefinder limit is firmly set at Ts ~ 0.27 mK. I can detect cooler strings, but they need to be numerous: looking at sets M and N shows that I need more than ~10 strings of 0.15 mK for a detection. For very cool strings at Ts < 0.03 mK, the data are very noisy. The multiple strings sets indicated that the most im portant criterion for string detection is the temperature of the input string. If the string is above the detection threshold, within a reasonable range of sizes, it will be picked up by the Edgefinder 3.4 W M A P R e su lts The WMAP data I ran through the Edgefinder is an average of the data in three of the WMAP bands: Q, V and W. I called this the QVW composite map. Pertinent statistics of the WMAP QVW composite map are in Table 3.5, including the Max EV and its E 7 coefficient. The resulting EV set was binned in the same manner as the simulated maps. Comparing the QVW composite map and the No-String sets, statistics such as mean, median, standard deviations are similar. Looking at the results of the WMAP composite QVW data, the maximum EV 102 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 3.5. Select Properties of the WMAP composite QVW Data. WMAP Band Max Map T Max EV 0 .6 6 6 0.258 QVW e 7 -3.82 x 10~ 6 is below the threshold set from the max EV of the No-String set, and is in fact, below the maximum EV of half of the No-String simulated data sets. This means that there are no hot string horizons in the WMAP data at the level of noise set by the No-String sets. The £7 coefficient leaves no doubt that there are no non- Gaussian signatures in the tail of the distribution of the WMAP data produced by string horizons similar to those I inserted into the simulated maps. I can confidently say that the Edgefinder did not find any evidence of string wakes in the CMB data measured by WMAP, to the single string limit of Gfi < 1.07 x L0“5. The 20 string limit of less 0.15 mK strings gives G/i < 5.97 x 10-6. I mention again the fact that in the actual WMAP data, due to Galactic and foreground contamination, about 1/3 of the pixels were masked. I have reproduced this masking in our simulated data so that I have the same number of pixels per map as the actual WMAP data. I caution that any strings hidden behind these masked pixels would not be picked up by the Edgefinder. 3.4.1 W M A P 2nd Year Sim ulation I have performed the analysis necessary in anticipation of the WMAP second year data release. All of the simulated maps were done in the same manner as the first year data, with the exception th at the noise is now 1 / y/2 time the noise 103 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. of the first year data. The second year simulated maps produce limits at max EV = 0.283 mK and E 7 = 0.257 mK. The E 7 threshold yields a single string limit of Gfx < 1.02 x 10-5 . Most of the noise in our string search is due to the first acoustic peak in the CMB angular power spectrum; the Edgefinder is cosmic variance limited, so reduced radiometer noise has little effect. When the second year WMAP data is release, I will run the filter and if the result is negative, this is the detection limit. 3.5 A C o sm ic S tr in g C a n d id a te? Sazhin etal. (2003) reported a discovery of an object which contains two sources of identical isophotes, color, and fitted 2 -D light profiles in the Osservatorio Astronomico ai Campodimonte Deep Field (OACDF). In addition, spectra of the sources are identical with a confidence level higher than 99%. Morphological ar guments led them to propose th at this object is a background galaxy lensed by acosmic string. They have named this object the Campodimonte-Sternberg-Lens candidate 1, or CSL-1. The red-shift of both sources in the object is 0.46T0.008; the separation of the two sources is 2 Precise finder charts for the object were not available. I found this object by visually inspecting the OACDF deep field and comparing it to the Palomar All Sky Survey plates. I found CSL- 1 to be located at (J2000) RA 12:23:30.72, Dec -12:38:57.8. There may be some small uncertainty about the location of CSL-1. I examine the WMAP data at this location to see if the Edgefinder can detect a string. As WMAP data has a resolution of 13 arcminutes, by including the four nearest pixels to th at coordinate, I believe I have covered this object in our search. 104 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 3.5.1 String Search From the image separation, I can derive a string mass per unit length, /i. In a flat universe where matter = 1, from G ott (1985), I find th at the image separation Ad of lensing by a cosmic string is related to D, the deficit angle of the conical space around the string, by, Ad = D cos a 1 — (1 4 xs) -1/2 l - ( l + ^)-V 2 (3.67) where zs is the red-shift of the string, zg = 0.48 is the red-shift of the background object (galaxy), and a is the angle of the straight string with respect to the plane of the sky. If I assume th at the string is in the plane of the sky, which means a = 0, there are two limiting results: for zs = 0, I find Ad = D; for zs = 0.4, Ad = 0 .1 0 2 D. For a flat universe, D , is related to the string mass per unit length, G/j, by, D = 87 tG u . (3.68) Therefore, with a maximum D of 9.7 x 10~ 6 radians, I get G/j = 3.86 x 10~7. A string will cause a jum p in the value of the tem perature of the CMB due to Doppler shifting. The change in tem perature is given by Equation 3.27. I have processed WMAP data at the position of the CSL- 1 object and com pared the Edgefinder values to Edgefinder values of regions with similar sky coverage and galactic latitude. The CSL- 1 object was small enough that it was within one WMAP pixel (on pixel 968549). However, I include results from the surrounding 4 pixels in the event th at I have misjudged the position of CSL-1. In addition, the alignment of the two images contains some uncertainty, thus I also ran the Edgefinder along two separate position angles, a = 0 and a = 7t / 2 0 . The results are in Table 3.6. 105 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. Table 3.6. Pixel # Edgefinder Values for CSL-1 WMAP QVW T EV | Q =0 /% -tile EV |Q=7r/2 o/%-tile 968549 -0.144015 0.07336 / 95.30% 0.08686 / 97.61% 968550 -0.0735366 0.08322 / 97.11% 0.08280 / 97.05% 968548 0.0418802 0.1248 / 99.76% 0.08621 / 97.53% 968527 0.0717491 0.1234 / 99.74%. 0.09720 / 98.65% Note. — EV of CSL-1 and 3 immediately adjacent pixels. Vahies listed are for a = 0 and a = tv / 20 The Edgefinder values of the 4 pixels are all above the 95th percentile, es pecially pixels 968548 and 968527, which have very high E.V.’s above 99th per centile. However, these high percentiles can be misleading. First, they are all significantly under the No-String detection limit. Second, as an upper limit, if I allow the WMAP tem perature at those pixels to be entirely caused by the pres ence of a string, I can say th at the EV is exactly the A T due to the string motion, due to the Edgefinder gain being 1 . W ith the Gfi given above, this means for an E.V. = 0.08686 mK, the string needs /Ty ^ 3.3, or v = 0.957c, to account for the tem perature jum p. The ranges of string velocity for the E.V. in Table 3.6 is from v = 0.941c to v = 0.979c. These high string velocities makes the case for the existence of cosmic strings at this location more unlikely, as the rms string velocity is v ~ 0.7c. I therefore cannot say that I have a significant detection of a cosmic string at the location of CSL-1 in the WMAP data. 106 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. A recent paper (Shlaer & Tye, 2005) showed th at a correction factor is nec essary for measuring the angle deficit of a relativistic string, given by, A9rei = + 0 cos(a)) A 6t 7(1 (3.69) where a isthe angle between the emitted photons and the perpendicular plane of the cosmicstring. If I assume an optimal alignment, where cos (a) = —1, the measured angular separation is the true separation modulated by the correction factors 7 (1 —0). This significantly modifies our result for the CSL-1 object. The corrected tem perature difference is given by, A 9rel = - P )A 9 (3.70) A9rei — 7 ( 1 —(3)&7rG[i (3-71) 7(1 A 9rei 7(1 - P) = 8 irGfi (3.72) and by putting in Equation 5, I obtain, S T 7 o n r , = 8i W S T 7 t A ( ) rel a = ^ ST 1 T A9 ,0 ^ 7 0 1-/3 ,3,3) (3.74) For our EV = 0.08686 mK, I get v = 0.765. The range of velocities is 0.756 < v < 0.785. Given a rms string velocity of v ~ 0.7c, these numbers are quite reasonable. I caution th at given the fact there is only one double image, the optical evidence for an edge is only suggestive, not conclusive; after all, many close pairs of galaxies do exist. A better method was suggested by (Shlaer & Tye, 2005) , who showed that partial images with abrupt edges would be caused by cosmic string lensing. That is not the case here. 107 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 3.6 S im u la te d P L A N C K R e s u lts Looking forward, I also made simulated maps of CMB data th at will be gathered by the European Space Agency’s PLANCK satellite. PLANCK is scheduled to launch in 2007. Most of PLANCK is still being constructed, so I do not have data on the noise characteristics of PLANCK. W hat I do know is th at PLANCK will have a 5 arc-minute resolution, with a tem perature sensitivity of 4 x 10~ 6 K (more information is available from the PLANCK website: h ttp :/www.rssd.esa.int/ in dex.php?project=PLANCK). I expect PLANCK will have about 1/10 the noise of WMAP. For our simulated PLANCK maps, I followed the same procedure outlined in Section 4, except in step 5, where for the added noise, I used 1/10 the average WMAP noise value for the an of each PLANCK pixel. In the HEALPix scheme, the improved PLANCK resolution means having 12 million pixels in the sky, a 4 fold increase on WMAP. Running each simulated map at 20 a angles became computationally infeasible. Therefore, I only ran the PLANCK maps through 6 different a angles, spaced equally between 0 and 180'L The increased resolution of the PLANCK satellite will allow us to bypass the problems of being sensitive to the first Doppler peak at I ~ 200. W ith the PLANCK Edgefinder at R A D = 0.5°, I still had ~ 270 pixels in the filter window, which was comparable to the number of pixels in the WMAP Edgefinder window of R A D = 1.0°. The transfer function for the R A D = 0.5° PLANCK Edgefinder filter is plotted in Figure 3.12. For the R A D = 0.5° Edgefinder, the peak filter sensitivity is at i ~ 500, well away from the first Doppler peak. I performed the bulk of our simulation using the R A D = 0.5° PLANCK Edgefinder. At this resolution, the PLANCK filter had a similar number of pixels as the WMAP R A D = 1° filter, so the gain from the filter was comparable. As with the WMAP set, I created calibrators for the PLANCK Edgefinder and 108 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0.5 o 0.0 D CD D D 0.5 > CD O 5 zs (/; 2.0 10 100 1000 10000 Figure 3.12 Transfer function for the PLANCK simulation; the filter had R A D 0.5. 109 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. tuned the filter to return EV = 1 for a 1 mK edge. Noise-only, A-CDM-only (a flat, cold dark m atter universe with the a cosmological constant dominating the total density of the universe) maps were put through the PLANCK Edgefinder for calibration purposes. 1 made a total of 30 Stringy PLANCK maps with only one string horizon per map. In addition, 6 baseline No-String maps were generated. Since the range of EV for PLANCK is expected to be wider, the optimum number of bins for the PLANCK data is 704. The total number of data points remain similar to the WMAP simulation, so the comparison of the two statistics is valid. The plots of the string tem perature, Ts, of a map vs. the max EV, and vs. E j are in Figure 3.13 and Figure 3.14, respectively. The limit obtained from the max EV was very similar to the the limit obtained from the E 7 coefficient, most probably as a result of the fact th at I have eliminated a lot of the Gaussian signal by going to a smaller Edgefinder window. This strengthened our confidence in these limits. Both the Max EV and E7 methods indicated th at the detection limit for the PLANCK Edgefinder occurred at T = 0.14 mK. This represented a factor of 2 increase in the single string Edgefinder sensitivity compared to the WMAP data. This is, however, very similar to the multiple strings limit for the WMAP data. For a cosmic string moving at mean simulation velocity, (3 = 1 /\/2 this input string corresponds to a string mass of Gji = 5.77 x 10-6 . 110 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0.8 No —String Largest Average No —String No —String Sm al les t 0.6 A > LU X ¥ o 0.4 >K z a z < _i Q_ — OK 0.2 Z r\_ _ ?K - M O—— M O— /IS X j/_ 7 ix • M T \- _ Z \._ ................ ' A No —Strinq Values 0.0 0 “ 10 —6 10 ^ - 4 ^ 10 -3 ' Input String T e m pe ra tu re , (K) Figure 3.13 Input string tem perature vs. the maximum EV of the EV set for simulated PLANCK data. Ill R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. 0 .0 2 0 c CD o 0 .0 15 (5 0 .0 1 0 ° 0 .0 05 --— No —String Largest Average No —String No —String Sm al le s t * W * 4r CD _________ tL____________ - 4 . ... CD S o.ooo ttt -0.005 - %------ A No —String Values 0.010 0 -6 , —5 10 10 " -4 10 -3 Input String T e m p e r a t u r e , (K' Figure 3.14 Blow up of input string temperature vs. the 7th Edgeworth coefficient of the EV set for simulated PLANCK data. 112 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. CHAPTER 4 C on clu sion s and F u tu re W ork I have constructed and calibrated a digital filter for the WMAP CMB data to search for cosmic strings. By comparing the WMAP data and a control set cheated from CMBFAST and SYNFAST, we have concluded th at the CMB data returned by the WMAP satellite do not contain single strings to the limit of Gjj, 1.37 x 10~ 5 using the max EV as the threshold and Gn < 1.07 x 10- 5 using the 7th Edgewort h coefficient as the threshold. This limit may be more stringent if we allow the sky to have multiple strings, to a limit of G\i < 5.97 x 10 if there are more than 20 string horizons in the sky. We caution that WMAP effectively examined 2/3 of the visible sky so it is possible th at we are missing strings in our analysis. W ith the second year WMAP data, we can improve this limit by about 5%. This improvement is a result of an expected 30% improvement in the WMAP radiometer noise. However, because the Edgefinder filter window was 2 degrees, the first CMB Doppler peak generated a large background that cannot be filtered out, and therefore a large reduction in radiometer noise resulted in a relatively small improvement in the discrimination threshold. I have also investigated claims of a possible cosmic string detection of the object CSL-1, and I found little evidence of a string at this position. For the proposed string mass of Gfi = 3.86 x 10-7, WMAP CMB temperatures at the location of CSL- 1 would require the string to have been moving very relativistically, with v ~ 0.96c. We conclude that this possibility is unlikely and that much 113 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. more sensitive and higher angular resolution data would be needed for a critical test of CSL-1. I ran the Edgefinder method through simulated PLANCK data. As more information about the PLANCK satellite becomes available, we can do a more realistic modeling of the noise characteristics and therefore get better limits on the strength of cosmic strings PLANCK can detect. Currently, I have a projected limit of Gfj, <; 5.77 x 10~6, a factor of two better than the single string limit from WMAP. The predicted detector noise of PLANCK is too small to affect the filter appreciably, and the true limit of the PLANCK data for the Edgefinder method will be cosmic variance. The factor of two improvement over the first year WMAP data is mostly a result of the smaller filter window which reduced signals from the first CMB Doppler peak. W ith more and more sensitive all sky CMB surveys, we can begin to set firm experimental limits on the existence of GUT scale cosmic strings, and thereby iimiting the types of allowed phase transitions. Relic products such as cosmic strings currently provide the only viable tool for probing conditions of the very early universe. Limits such as those presented here are useful in the absence of testable conclusions. Cosmic strings only exert an influence on CMB photons through their gravi tational field. Therefore, if we know the string spatial distribution, their length scale, and their energy density (the values tabulated in Chapter 3), we should be able to completely characterize the behavior of CMB photons within a network of strings. All of the above parameters were explored with numerical simulations by various groups, e.g. Bennett k, Bouchet (1990), Albrecht & Turok (1985), Allen et al. (1997). I have laid down the calculations necessary for calculating a Monte Carlo initial string configuration. A possible future direction would be in 114 R eproduced with perm ission o f the copyright owner. F urther reproduction prohibited w itho ut perm ission. writing a multi-variable, ray-tracing code to produce the final CMB sky modified by the gravitational potential of the strings. I have also demonstrated, in the Chapter 2 , th at buckyballs are viable candi dates for an anomalous component of the CMB foreground. Buckyballs have high frequency (20 to 40 GHz) emissions generally with a power spectral index (j3) of around 2, and low frequency (1 to 20 GHz) emission of spectral index 1. The peak emission frequency across the simulated regions spans 5 to 37 GHz, where the anomalous free-free emission is observed. Buckyball emission from our simulation was dominated by collision density effects. The composite spectra indicated that the dominant emission from buckyballs comes from the C$o molecule, while the highest frequency emissions comes from C2oIf we assume buckyballs are responsible for the anomalous free-free excess emission in the 35 to 45 GHz range, they would need to comprise 0.5% of the total: amount of Galactic carbon in order to explain the observed e m iss iv ity . We believe this is a reasonable number. While the spectral index of the buckyball emission is not “free-free” , i.e. p ~ 0 , the observations can still be explained by an emission mechanism th at peaks in the range of the anomalous emission window between bright synchrotron at low frequencies and vibrational dust at high frequencies. A composite buckyball spectrum is close enough given the modest signal-to-noise ratio of the observations to date. 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