Boson stars efficiently nucleate vacuum phase transitions by Thomas John Brueckner A thesis submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Montana State University © Copyright by Thomas John Brueckner (1997) Abstract: In the hot dense early universe, first order phase transitions were possible through the tunnelling of a scalar field. When studying the formation of true vacuum bubbles in the semi-classical approximation, the tunnelling rate depends primarily on the Euclidean action of the bubble configuration. Others have shown that bubble nucleation by compact objects (neutron stars, black holes) proceeds more rapidly than in Coleman's process of bubble formation in empty space. In this paper, I consider nucleation by another kind of astrophysical object, a boson star, the ground state of a self-gravitating scalar field. I model a boson star in a self-interacting potential that also has a term cubic in the scalar field, the so-called 2-3-4 potential. In the limiting case of a "small" star nucleating a "large" bubble, I compare its Euclidean action, SEBubble to the empty space bubble action of Coleman, SE Coleman, and I find that the action ratio SEBubble/SE Coleman decreases significantly from unity as the energy difference between the vacua increases. This decrease from unity enhances the nucleation rate. B O S O N STARS E F F I C IENTLY N U C LEATE V A C U U M PHASE TRANSITIONS byThomas Joh n Brueckner A thesis submitted in partial fulfillment of the requirements for the degree of ./ Doctor of Philosophy in Physics M O N T A N A STATE UNTVERSITY-BOZEMAN Boz e m a n , Mont a n a April 1997 © COPYRIGHT by Thomas John Brueckner 1997 All Rights R e s erved APPROVAL of a thesis submitted by Thomas John Brueckner This thesis has b e e n rea d b y each mem b e r of the thesis committee a n d has b e e n found to be satisfactory regarding content, Engl i s h usage, format, citations, bibliographic style, an d consistency, and is ready for submission to the College of Graduate S t u d i e s . W i l l i a m A. Hiscock (Signature) /(Date) A p p r o v e d for the Department of Physics J ohn C. Hermanson (Signature) (Date) A p p r o v e d for the College of Graduate Studies Robert B rown (Signature) (Datdl I iii STATEMENT OF PERMISSION TO USE In pres e n t i n g this thesis, in partial fulfillment of the requirements for a doctoral degree at Mont a n a State U n i v e r s i ty-Boze m a n , I agree that the Library shall make it available to borrowers under the rules of the L i b r a r y . I further agree that copying of this thesis is allowable only for scholarly purposes, p r e s c r i b e d in the U.S. consistent w i t h "fair use" as Copyright Law. Requests for extensive copying or reproduction of this thesis should be referred to U n i v e r s i t y Microfilms International, Arbor, M i c h i g a n 48106, 300 N o r t h Zeeb Road, A n n to w h o m I have granted "the exclusive right to reproduce an d d i s t r i b u t e ,m y dissertation in and from m i c r o f o r m along w i t h the non-exclusive right to reproduce and distribute m y abstract in any format in whole or in part." Signature Date TTku, 6. M T ? iv TABLE OF CONTENTS Page 1. 2. 3. W H Y SHOULD ONE STUDY B O S O N STARS A S SEEDS F O R V A C U U M PHASE T R A N S I T I O N S ? .................... I W H A T WAS THE E ARLY UNIVERSE L I K E ? .................... 5 The E a r l y Universe Was Hot and D e n s e ............ M a t t e r Fields Are E ffectively Massless at H i g h T e m p e r a t u r e ........... ...... . Cooling of the Universe Leads to the P ossibility of Phase Transitions in the V a c u u m .............................. 5 W H A T A R E THE PROPERTIES A N D CHARACTERISTICS OF B O S O N S T A R S ? ................... Backg r o u n d Concepts for Calculating Boson Star M o d e l s ............................. Details of Calculating Boson Star Models w i t h the Self-Gravitating F i e l d M e t h o d ................................... 4. H O W DO B O S O N STARS AF F E C T THE D ECAY OF THE FALSE V A C U U M ? .................................. Coleman's Th e o r y of the D e c a y of the False V a c u u m ................................... "Add i n g " a Boson Star to the S p a c e t i m e ........... The Case of a Small Star Nucleating a Small B u b b l e .......... ......................... Summary of Simplifications in the Small-X l i m i t .......... .. .................... Methods of Calculation of B ...................... B oson Stars Efficiently Nucleate First Order Phase Transitions, in the SSLB L i m i t ......... ............................. Summary and C o n c l u s i o n s ...................... 5. W H A T A DDITIONAL W O R K REMAINS F O R THE F U T U R E ? ........ 17 20 27 27 34 51 53 67 78 83 86 88 96 99 REFERENCES C I T E D .............................................. 103 APPENDIX, $ C O M PUTER P R O G R A M S ..................... .105 V LIST OF TABLES Table Page 1. Values of % an d 0 for several SSLB combinations ................................... 932 2. Conventional vali d i t y measures for Coleman's thin-wall a p p r o x i m a t i o n ................ 95 vi LIST OF FIGURES Figure 1. Page T u nnelling events for first order vacuum p hase t r a n s i t i o n ......................... ........... 3 2. Sample solution for a (t).............................. 13 3. Shape of an effective potential for the two cases of m 2 > 0 and m 2 < 0 ................. :..... 22 A n effective potential in w h i c h a vacuum p hase transition m a y o c c u r ........................ 25 The effective potential in the ColemanW e i n b e r g p r o c e s s ................................... 33 6. The 2-3-4, p o t e n t i a l .................................... 40 7. Typical solution for the field G(x).................. 44 8. Me t r i c functions for the example b oson s t a r .......... 46 9. The mass function for the example b oson s t a r ........ 46 10. A n example of a trial solution w i t h Q too l a r g e ......................................... 47 11. A n example of a trial solution w i t h Q too s m a l l ......................................... 47 12. A gall e r y of solutions for the scalar field o ( x ) ............... •.......................... 50 13. 54 4. 5. Potential w i t h an unstable e q u i l i b r i u m .............. vii LIS T OF FIGURES (continued) Figure Page 14. Potential in w h i c h a first order phase transition is p o s s i b l e ............................ 57 15. The upside - d o w n p o t e n t i a l ............................. 60 16. Shape of a typical solution (p(p) in the thin-wall a p p r o x i m a t i o n . ................ . 63 17. Traj e c t o r y of the bubble wall. ........................ 66 18. The shape of a smoothly varying bubble s o l u t i o n ............................................ 72 19. A typical b o s o n star solution a* (x).................. 74 20. Shape of a bubble profile that is neither thick- nor t h i n - w a l l e d ............................ 75 21. Graft of a star solution onto an u n ­ p e r t u r b e d bubble s o l u t i o n ........................ 76 22. Tunnelling endpoint c h a n g e s ........................... 77 23. A goo d SSLB c o m b i n a t i o n ............................... 84 24. The relative gai n in efficiency for SSLB vs. the Coleman p r o c e s s ........................... 90 25. The ratio B / B 0 for two different values of A ....... 92 26. Star-bubble combination in w h i c h the bubble is m u c h smaller than the s t a r ........................ 100 27. 6- Star-bubble combination in w h i c h the star and the bu b b l e are of comparable s i z e . ....... .. 101 viii Abstract In the hot dense early universe, first order p h a s e transit­ ions w ere possi b l e through the tunnelling of a scalar field. W h e n studying the formation of true v a c u u m bubbles in the semi-classical approximation, the tunnelling rate depends p r i m a r i l y on the E u c lidean action of the bubble c o n f i g u r ­ ation. Others have shown that bubble n ucleation b y compact objects (neutron s t a r s , black holes) proceeds m o r e rapidly than in Coleman's process of bu b b l e formation in empty space. In this paper, I consider nucl e a t i o n b y another k i n d of astrophysical object, a b o s o n star, the ground state of a self-gravitating scalar field. I m odel a b oson star in a self-interacting potential that also has a term cubic in the scalar field, the so-called 2-3-4 potential. In the limiting case of a "small" star nucleating a "large" bubble, I compare its Euclidean action, SgBubbie, to the empty space bubble action of Coleman, S e Coiemanz and I find that the action ratio g EBubbie/gE Coleman decreases significantly from u nity as the energy difference b e t w e e n the v a c u a i n c r e a s e s . This decrease from u n i t y enhances the n ucleation r a t e . I" CHAPTER I W H Y SHOULD ONE STUDY B O S O N STARS A S SEEDS FO R V A C U U M PHASE TRANSITIONS? In the beginning, the universe was hot a n d dense, v a r i e t y of unusual processes a n d objects existed. and a One of the m o r e unusual processes is that of a first order v a c u u m phase transition in a q u a n t u m field. Of p a rticular interest is the theory of a scalar field that undergoes a v a c u u m phase transition, since the scalar Higgs field is a crucial part of larger theories, like electroweak theory, that u n i f y some of the fundamental forces of n a t u r e . In the early universe's me n a g e r i e of exotic objects are b o s o n stars, field. a self-gravitating configuration of a quantum Eac h b o s o n in the star is in the same q u a n t u m state. These stars, p r e v e n t e d from collapsing b y the Heisenberg u n c e r t a i n t y principle, can range in mass from a few thousand kilograms up to astrophysical size, boson's m a s s . depending u p o n the Some scientists v i e w bosonic m a t t e r as a p o s s i b l e p art of the dark m a t t e r content of the universe. It is logical to ask w h e t h e r b o s o n stars affect first order p h a s e transitions in the scalar field. The mos t basic model of a first order v a c u u m p h a s e transition is one in w h i c h a region of e mpty spacetime spontaneously changes 2 p h a s e ,1 m u c h like drops of rain form spontaneously in a pure w a t e r vapor. bubble. This region of the n e w phase is cal l e d a v a c u u m The scalar field, w h i c h I shall call <p, tunnels from its initial state at a local m i n i m u m of the potential, through the b a r r i e r in the potential V((p), to the true v a c u u m at the global minimum. Figure I shows the tunnelling process for spontaneous bu b b l e formation in e mpty space. A second process, induced nucleation, first order p h a s e t r a n s i t i o n s . can also generate Induced nucle a t i o n is like the m u n d a n e process of u s i n g silver iodide crystals to seed clouds an d form p r e c i p i t a t i o n over a rid regions of land. U s i n g a b o s o n star to seed a p hase transition has a notable advantage over the spontaneous formation p r o c e s s . star is a c o n f iguration of the scalar field, The boson (p(r) , that starts out w i t h a p o s i t i v e central v a l u e , (p(0) > 0. The field in the gr o u n d state decreases g r adually in size as it extends out from the center of the star, asymptotically a pproachi ng zero as radial distance r approaches infinity. In terms of the potential in figure I, the central value of the star is h i g h u p on the "bump" ing to q u a n t u m theory, in the potential. Accord­ the barr i e r is more easily penetrable w h e n the initial v a l u e of (p is h i g h up on the potential barrier. It is therefore reasonable to conjecture that, in compa r i s o n to spontaneous bubble formation in e m p t y space, a 1S. Coleman, "Fate of the false vacuum: Semiclassical theory," Physical Review D, 15., 2929 (1977) . 3 0.001 0.0008- 0.0006- 0.0004- 0 . 0002 - - 0 . 0002 - -0.0004 - 0.02 0.02 0.04 0.06 0.08 0.12 Figure I. Tunnelling events for first order v a c u u m phase transitions. The lower arrow shows the tunnelling event corresponding to spontaneous formation of v a c u u m bubbles in empty space. The upper arrow shows the tunnelling event for a first order p hase transition that a boson star has nucleated. The potential barrier is more easily penetrated in the latter case. 4 b o s o n star m i g h t m o r e read i l y initiate a tunnelling event in the v a c u u m field. The veri f i c a t i o n of that conjecture is the I compared the two processes, and I subject of this thesis. found that a b o s o n star has a significantly greater e f ficiency at nucle a t i n g first order p hase transitions in the case of a small star nucle a t i n g a large bubble of the ne w phase. I shall prov i d e greater detail on wha t m akes a boson star small in r e l ation to a large b u b b l e , h o w to construct a m odel b o s o n star, process. and other aspects of this nucle a t i o n Specifically, in chapter 2, I will discuss the tijne e v olution of the temperature of the universe, in the Robert s o n - W a l k e r cosmological model. Also, I wil l discuss the temperature d e pendence of effective potentials that allow vacuum phase transitions. In chapter 3, I review some of the foundational concepts of b o s o n stars, an d I discuss the calcul a t i o n of b o s o n star models for later use in the bubble n u c l e a t i o n process. In chapter 4, I re v i e w the theory of spontaneous formation of v a c u u m bubbles, "small star-large bubble" limit. and I introduce the I discuss the details of the calculation of a n u cleation rate, an d I conclude chapter 4 w i t h results confirming that b o s o n stars are quite efficient at nucle a t i n g p hase transitions. The thesis ends w i t h a b rief chapter 5 in w h i c h I discuss the p r ospects for future work. 5 CHAPTER 2 W H A T W A S THE E A R L Y UNIVERSE LIKE? The E a r l y Universe Was Hot and Dense. The early universe was hot an d dense in comparison to the p r e s e n t .1 This idea has convincing observational s u p p o r t , p r i n c i p a l l y in the observation of ne a r l y perf e c t isotropy of the remnant 2.7 K cosmic b a c k g r o u n d radiation. To u n derstand the significance of the cosmic b a c k g r o u n d r a diation (CBR), one needs to examine the b i g b a n g theory of the early universe. In 1948, George G a m o w , R alph Al p h e r a n d Robert He r m a n constr u c t e d a b i g b a n g model to study n u c l e o ­ synthesis. T h e y e n visioned a cosmic fireball of neutrons that cooled a d i a batically an d eventually synthesized the lighter nuclei hydrogen, boron. helium, lithium, beryllium, This nucl e a r "soup," consisting of a hot, and dense neutron gas,* 2 that filled the universe was in thermal e q u i l ibrium w i t h all the radiation. Eventually, though, the nucl e a r soup cooled enough that the individual nuclei b egan to capture an d h o l d their ration of e l e c t r o n s . W h e n this i-The discussions in this section are due mainly to P.J.E. Peebles, Principles of Physical Cosmology (Princeton University Press, Princeton, 1993) and to R.M. Wald, General Relativity (University of Chicago Press, Chicago, 1984. 2G. Gamow, "The Evolution of the Universe," Nature, 162. 680 (1948). 6 happened, the d e n s i t y of free electrons, effici e n t l y scattered radiation, w h i c h h a d hitherto d e c reased dramatically. Scattering of r a diation became less frequent, e q u i l ibrium b roke down. an d thermal The r a d iation b egan to stream freely, with o u t further significant interaction w i t h the matter. This d e coupling of mat t e r a n d radiation occurred over a p e r i o d of time, not all at one instant, bu t one refers to this p e r i o d as the "decoupling time." One m a y also say that, at decoupling, the universe became transparent. The light then was m a i n l y in the v i s i b l e and near infrared, w i t h a b l a c k b o d y spectrum at a temperature of a few thousand Kelvins. A l p h e r a n d He r m a n p r e d i c t e d 3 in 1948 that the radiation, like a gas of photons in an enclosed s p a c e , should have expanded w i t h the universe a n d cooled to a temperature today of r o u g h l y 5 K. A f t e r 1948, Britain, other scientists in the US an d the Soviet U n i o n refined the estimate of the pres e n t temperature of the C B R . In 1965 A r n o Penzias and Robert W i l s o n disc o v e r e d an isotropic source of "exce'ss antenna t e m p e r a t u r e " of approximately 3.5 K . 4 P.J.E. Robert D i c k e , Peebles and their collaborators gave the immediate 3R.A. Alpher and R. Herman, "Evolution of the Universe," Nature, 162, 774 (1948). 4a .A. Penzias and R.W. Wilson, "A Measurement of Excess Antenna Temperature at 4080 Mc/s," Astrophysical Journal, 142, 419 (1965). 7 i n t erpretation5 that this "excess antenna t e m p e r a t u r e " was a c t u a l l y the red-sh i f t e d primordial radiation that G a m o w and his collaborators h a d p r e d i c t e d 17 years earlier. d i s c o v e r y implied the hot, This dense nature of the b i g bang: "A temperature in excess of IO10 °K during the h i g h l y c o ntracted p hase of the universe is str o n g ­ ly implied b y a present temperature of 3 . 5°K for b l a c k b o d y r a d i a t i o n . ... If the cosmological solution has a singularity, the temperature w o u l d rise m u c h h i g h e r than IO10 °K in a pproaching the singul a r i t y . " 6 Since 1965, v e r y fine m e a s u r e m e n t s 7 of the cosmic b a ckground r a d i a t i o n hav e y i e l d e d a temperature T = 2.736 ± 0 . 017°K. These observations have fully v i n d i c a t e d the predi c t i o n of G a m o w an d his c o l l a b o r a t o r s . G a m o w b uilt his b i g ban g theory of nucleosynthesis upon the theory of the expanding universe. c alculation of the scale, density, I shall n o w review the and temperature in the early u n i verse b a s e d upo n the Robertson-Walker m odel of an expanding universe. 5R . H . Dicke, et a l ., "Cosmic Black-body Radiation," Astrophysical Journal, 142. 414 (1965). 6Ibid. 7Peebles, Physical Cosmology. 131. 8 M o s t cosmological models assume that the u n i verse is homogeneous an d isotropic — principle. the so-called cosmological This a ssumption is a reasonable one, for a stro­ nomical observations, ■e s pecially of the C B R , r e c o r d a homogeneous a n d isotropic d i s t r ibution of r a diation and m a t t e r in the u n i v e r s e . 8 In the R o b e r t son-Walker model, one envisions the four dimensional spacetime m a n i f o l d as a foliation of spacelike hypersurfaces. A parameter, t, labels each spacelike h y p e r surface of the foliation. The 3 -geometry of each h y p e r surface is homogeneous and isotropic. E ach isotropic observer in the spatial leaf moves upon a w o r l d line that is orthogonal to each leaf. This allows one to call t the p r o p e r time that an isotropic observer w ould measure. It also allows one to synchronize all clocks on each spatial leaf. The imposition of isotropy forces the g e o m e t r y of each leaf to b e that of a space of constant c u r v a t u r e . homogeneous spaces are of three kinds: hyperbolic. spherical, Such flat, and It is customary to refer to the former as a "closed" space-time, to the latter as an "open" spacetime. To get an intuitive grasp of h o w spherical an d h y p e r ­ bolic geometries compare to flat space, circle. In a spherical geometry, one m i g h t examine a the circumference of a 8One must understand this homogeneity and isotropy to be evident on some suitably large scale, certainly larger than galactic. 9 circle is less than 27tr. One can u nderstand this b y c o n ­ sidering this circle an d its radii to be confined to the surface of an ordinary sphere. circumference is exactly 2jzr. In a flat geometry, the In a hyperbolic geometry, circumference is m o r e than 27cr. the One can u n d e r s t a n d this b y considering the circle an d its radii to be confined to the surface of a saddle of hyperboloidal shape. Physically, one can say that a closed spacetime contains m a t t e r w h i c h is dense enough to close the universe b a c k on itself. The open and flat spacetimes have sparse matter density, so that they do not close b a c k in on t h e m s e l v e s . The assumptions of h o m o g e n e i t y and isotropy y i e l d significant simplifications of the metric. The metric on a four dimensional m a n i f o l d will in general have ten arbitrary functions of the four-dimensional position. The assumption of ho m o g e n e i t y implies the p r e sence of an isometry, translation, at each point on the manifold. The further assumption of isotropy implies another isometry, rotations. In addition, spatial spatial the isometries reduce the number of independent functions from ten dow n to only one function, a (t ). The mos t convenient form of the Robertson-Walker line element is ds2 = - d t 2 + d£2 , with 10 d\|/2 + sin2\|/ (d0 2+ sin 20 d<()2 ) d£2= a(t)2 jdx|^+x|/2 (d 0 2 + sin20 d<|>2 ) (closed) (flat) (2 .1 ) dx|/2 + Sinh2XjZ(d 0 2+ sin20 d^2) (open) In this m e t r i c , 9 the function a (t) is the cosmic scale p a r a m e t e r ; in the case of a cl o s e d spacetime, it as the size of the universe. one interprets For example, if two galaxies are a pr o p e r distance L 0 apart at present, then at some earlier time t, they w e r e a distance L 0 •a (t ) a p a r t , setting the pres e n t v a l u e of the scale factor to u n i t y for convenience. Therefore, it is imperative to k n o w the b e h a v i o r of a (t) over time: does it get smaller or larger w i t h time? In order to determine the b e h avior of a (t ) , one must solve the Einst e i n field equations for these m e t r i c s . general, In the E i n stein field equations relate the distribution an d m o t i o n of m a t t e r an d radiation to the g e o m e t r y of the spacetime. One uses a stress-energy tensor, T ab, to m a t h e m a t i c a l l y represent the m a t t e r a n d radiation. E i n stein tensor, spacetime. The G ab, represents the curvature of the These two tensors form the Einstein e q u a t i o n s : (2 .2 )* I 9Throughout most of this thesis, I use natural units: c = I, G = I, kB = I and h = I. I employ a metric signature -+++ and follow the sign conventions of C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation (San Francisco: Freeman, 1973). I use abstract index notation; c f . R.M. Wald, General Relativity. 24. 11 The m a t t e r of the universe, on the cosmic scale, at least, is de s c r i b e d b y the stress-energy tensor of a p e r f e c t fluid. (2.3) The. me t r i c tensor of the spacetime is g ^ . In fact, the stress-energy tensor of a perfect fluid is the m o s t general form compatible w i t h the h o m o g e n e i t y and isotropy of the R o b e r t son-Walker m o d e l . The q u a n t i t y p de-notes the energy d e n s i t y of the fluid, P denotes its pressure, fluid's four-velocity. a n d u a is the Focussing on only the matter, approximates it as "dust," w i t h P = O , one w hile for radiation, p = P/3 . To u n d e r s t a n d the evolution of the scale factor, an d the density, ions. p, a(t), one mus t solve the Einstein field e q u a t ­ The two equations relating a (t) w i t h p an d P are (2.4) (2.5) Overdots denote d i f ferentiation w i t h respect to time; k = +1 for cl o s e d geometry, k = O for flat geometry, a n d k = -I for open g e o m e t r y . The first mathematical implication one can d r a w from 12 these equations is that a (t) is not constant. W h e n p and P hav e realistic values p > 0 an d P > 0, the second time d e rivative of a (t), its "acceleration," will b e n e g a t i v e . This in turn shows that the universe must be either expanding or contracting. Indeed, astronomers b e g inning w i t h Edwin Hubble hav e o b s e r v e d 10 that the universe is everywhere expanding. Hubble's famous law, V = HD, recessional velocity, V, of a g a l a x y to its distance, a constant of p r o p o r t i o n a l i t y H. divi d e d b y time, relates the D, w i t h Since speed is distance one m a y state that the H is the inverse of some time interval, H = 1/T. One can interpret T as the total expansion time, w h i c h should be an estimate of the age of the u n i v e r s e . constant, B y convention, H is called the Hubble a n d T is called the Hubble time. It is n o w important to return to the earlier example of the distance b e t w e e n two galaxies, L (t) = L 0*a(t) . Dif f e r e n t ­ iation w i t h respect to time gives 1-f - L q * 3- 311(1 L q Now, a(t) .a(t) •L(t) (2 . 6 ) if one dubs the quant i t y in brackets as H, one has the essential form of the Hubble law, V = H*D, general, H can b e a function of time, except that, i.e., H ( t ) . in Thus 1/H 10E.P. Hubble, "A Relation between Radial Distance and Velocity among Extra-Galactic Nebulae," Proceedings of the National Academy of Sciences, 15., 168 (1929) . 13 is not really the exact age of the u n i v e r s e . M o r e o v e r , if one assumes that the universe has b een expanding such that its " a c c eleration" has always been negative, figure 2, Thus, as shown in then the Hubble "constant" was larger in the p a s t . the Hubble time can overestimate the age of an expand­ ing universe. a (t) today t -0.5 Figure 2. Sample solution for a (t) showing overestimation of the present age of the u n i v e r s e . The diagonal dotted line shows the overestimating effect of extrapolating Hubble's law b a c k in t i m e . Was the scale factor zero at t = 0? To get an answer, one m ust combine the two Einstein field equations. W i t h an 14 algebraic combination of the two equations — p + 3 (p + P )— = 0 — (2 .7 ) one can form a p a i r of c o nserved q u a n t i t i e s . galactic "dust" of matter, P=O, hence 3,(Pn*,a3) = 0 . The q u a n t i t y pa3 is a c o n s t a n t . For the (2.8) Thus the d e n s i t y of matter varies as the inverse cube of the scale factor: J_ P matter For radiation, P =p/3, ~ a3 ‘ ( 2 .9 ) so S t(Pn^ 4) = O . The q u a n t i t y p -a 4 is a constant over time. (2.10) T h erefore the dens i t y of radiation energy varies w i t h the inverse fourth p o w e r of the scale factor. I P rad TT • (2 .11) 15 In the R o b e r t s on-Walker model containing b o t h m a t t e r and radiation, for sufficiently small values of the scale factor, the en e r g y d e n s i t y of radiation will be larger than the energy dens i t y of matter. Later, as a (t) increases, the energy dens i t y of m a t t e r will a p p roach and e ventually exceed the ene r g y dens i t y of radiation. Du r i n g the r a d i ation-dominated pe r i o d one can determine the b e h a v i o r of a (t). Multiplying (2.4) b y a 4 a n d replacing the q u a ntity p*a4 w i t h a constant of integration C, one obtains a n o nlinear differential equation for a ( t ) : a2- + 3k = O . (2.12) a It is easy to determine the b e h avior of a (t) w h e n a (t) becomes rela t i v e l y small. Specifically, w h e n a2 « 87IC/9 |k| , the differential equation becomes (2.13) = O . The approximate solution for small a (t) is a (t) = A 0t1/2, w i t h A 0 b e i n g a constant that incorporates the previous constant C . This thime dependence shows that at some finite time in the past, the scale factor a (t) was tending toward zero as the ) 16 square root of the t ime.11 radiation, The energy density of the p r o p o rtional to the inverse fourth p o w e r of the scale factor, m u s t have b e e n c o rrespondingly large. To s how that the early unive r s e was v e r y hot, sufficient to state that for a r a d iation gas, it is p is proportional to the fourth p o w e r of the temperature, T. p~T4 . (2.14) ? Since the energy dens i t y varies w i t h the inverse fourth power of the scale factor, the temperature varies i n versely wit h the scale factor. T- I a(t) As a (t) tends toward zero, (2.15) the temperature T increases, and w e k n o w indeed that the scale factor was smaller in the past. Therefore, period, w e can say that, during the radiation-dominated the temperature was m u c h hi g h e r than ‘it is today. In summary, isotropy, the implications of homoge n e i t y a n d ' a n d the focus on the r a d i ation-dominated period, lead to the p r e d i c t i o n of a hot, dense, compact early 11Whether one may extend this cosmological model all the way back to a singularity at t = 0 is a matter of current debate. When the energy density of the universe approaches the Planck density, approximately IO93 grams/cm3, one must apply a quantum theory for the gravitational field. This theory is not currently complete. Thus, one may not presume to describe a time when a (t) = 0. For the purposes of this thesis, however, we will not need to resolve this question beyond saying the universe was very compact and dense in its early stage. 17 universe. This is the exact pict u r e for w h i c h Penzias and W i l s o n found observational evidence in 1965 w h e n they disc o v e r e d the cosmic b a c k g r o u n d radiation. M a t t e r Fields Ar e E ffectively Massless at H i g h Temperature. Q u a n t u m theory has b e e n spectacularly successful in d e scribing the b e h a v i o r of ma t t e r in m a n y physical situations, from v e r y low temperatures in .a superconducting q u a n t u m interference device a large p a r ticle collider. (SQUID) to v e r y h i g h energies in W h e n the temperature of the universe is significantly larger than the rest mass of a species of particle, T > > me2 , then the relativistic form of the energy, (2.16) e, for a single particle. (2.17) can b e appr o x i m a t e d w i t h its ultrarelativistic form, (For this section, a n d ft..) e = pc. I b r i e f l y reintroduce explicit constants c 18 C o n sider a degenerate relativistic electron g a s . 12 The number d e n s i t y of electrons in a region of p h a s e space F is 4tcV p 2 dp 2 (2.18) dnP w here V is the volume, "cell-size" a n d Planck's constant h represents the in p hase space. The final factor of 2 accounts for the m u l t i p l i c i t y of states for spin = 1/2 e l e c t r o n s . find the energy of the gas, To one simply integrates the product of E (p) an d Clnp over all m o m e n t a u p to the Fermi limit, p F . The limiting momentum, p F , depends on the number, electrons one can p a c k in the given volume, V, N, of in the following way: N = VTt2Tt3 The total energy, E, (2.19) T P f3 is E = Jedn 1/3 Pf %AcN 0 Stt2N V (2 .20 ) Now, m a k i n g use of standard thermodynamic relationships 12L.D. Landau and E.M. Lifshitz, Statistical Physics (Part I ) . trans. j .B . Sykes and M.J. Kearsley, 3rd ed., (Oxford: Pergamon Press, 1980), 178. 19 relating energy, press u r e and volume. one sees that, P, for h i g h temperature, an d energy density, equation of state, p, form a par t i c u l a r l y simple P = p/3. This is exactly the same equation of state as a p h o t o n gas, addition, the electron pressure, as m e n t i o n e d a b o v e . In one can derive the same equation of state for systems of other relativistic p a r t i c l e s . At some early time, w h e n the temperature was s i g n ificantly greater than the rest mass of the heaviest e l ementary p a r t i c l e (assuming one e x i s t s ) , the equation of state for all species of particles was that of radiation. the temperature decreased, As the radiation-like description for different species of particles b e g a n to fail. The first to lose their r a d i a t i o n - Iike b e h avior wer e the q u a r k s ; their equation of state changed g r a dually to that of "dust," P = O . The n successive species of elementary particle lost their p h o t o n - Iike b e h a v i o r until finally the electrons left the relativistic regime. The time of dominance for radiation was at an end an d the time of ma t t e r dominance b e g a n . * * * * * * * 20 Cool i n g of the U n i verse Leads to the Possibility of Phase Transitions in the Vacuum. In finite temperature q u a n t u m field theory, transitions in the v a c u u m becomes possible. vacuum?" phase W hat is "the ■ This refers to the g r o u n d state of the quantized fields in the absence of s o u r c e s . theory in flat space. Consider a simple cp4 field Its L a grangian density is (2 .22 ) The potential is V((p) = (l/2)m2cp2 + (X/4)cp4 . If m 2 > 0, there is a m i n i m u m at cp = 0. In terms of quantum field theory, m ust instead w o r k w i t h the effective potential, one w h i c h one derives from a Legendre transformation of the classical acti o n . 13 To u n d e r s t a n d the kinds of p h a s e transitions that are possible, one m ust contrast a quan t u m field theory at zero temperature w i t h one at non-zero or "finite" temperature, b o t h w i t h q u a n t u m corrections to leading order in %, the socalled one-loop order. Incorporating the corrections for one-loop q u a n t u m fluctuations results in an effective 13in this section, I follow R.J. Rivers, Path Integral Methods in Quantum Field Theory (Cambridge: Cambridge University Press, 1987), 37, 86 . 21 p o tential that m a y have a different number and location of the extrema, potential. (2.22) . c o m pared to the zero-temperature effective For instance, one m ight consider the (p4 theory of If X > 0 an d m 2 < 0, there will be a second extremum; I discuss this situation below. A l t h o u g h exact details about the location, etc. of the extrema de p e n d upo n w h i c h field theory one is studying, there are a few ideas that are r elatively simple to e x p r e s s : the temperature dependent mass term, internal "hidden" symmetry, spontaneous b r e a k i n g of an a n d a critical t e m p e r a t u r e . These are the areas in w h i c h the differences b e t w e e n zerotemperature and finite-temperature field theories become most important. The effective mass of the quan t u m field can acquire a temperature dependence in a finite temperature field theory. Normally, one interprets the mass of the field v i a the coefficient of the term in the Lagrangian w h i c h is quadratic in the field: efficient. the square of the mass is twice that c o ­ For the Lagrangian of (2.22), the cp field has mass m. W i t h q u a n t u m corrections at finite temperature, however, the mass becomes a function of the temperature, m 2 (T). If m 2 (T) becomes negative, two real-valued extrema occur at cp = 0 an d the global minimum, figure 3 s h o w s . (p = (-m2/X)1/2, as 22 V«p) -0.5 Figure 3. Shape of an effective potential for the two cases m 2 > 0 (upper curve) an d m 2 < 0 (lower curve) . 23 The v a c u u m e xpectation v alue of the scalar field is pre c i s e l y that v alue of the field w h i c h extremizes the p o t e n t i a l . the first case, however, that v a l u e is cp = 0 . In In the second case, there are two values for the v a c u u m , one at cp = 0 an d the other at cp0 = (-m2/X)1/2. In the theory of spontaneous symmetry b r e a k i n g , one mus t first expand the potential about the m i n i m u m at cp0. U s i n g the transformation cp - u = (p — cp,, , (2.23) the potential acquires terms cubic in the field u. V(Cp) - V(u) = - m 2u2 + Xcp0U3 + ^ u 4 + T m 2Cp02 . The u field also acquires a n e w mass: (-2m2)1/2. (2.24) The discrete symmetry, cp —cp , (2.25) w h i c h was m a n ifest in the original Lagrangian does not appear in V ( u ) ; it is a h i d d e n symmetry. state, u=0, H o w e v e r , the n e w ground is asymmetric w i t h respect to the original form of the field theory; the original symmetry is broken. It is p o s s i b l e to express the temperature dependence of 24 m 2 w i t h the concept of a critical t e m p e r a t u r e , T c . Above Tc, m 2 is p o s i t i v e an d the potential has but one extremum, cp = 0. B e l o w T c, m 2 is negative an d the potential has two extrema. The typical form of the temperature dependence is m2 (T) m 2 (2.26) w i t h the exact form of T c depending on the details of the field theory in question. In the simple <p4 theory above, the field cp can "roll" down into the well at cp = cp0 from an initial state at cp = 0, continuous change in the v a c u u m expectation value. a This continuous change from one v alue to the other is a secondorder p hase transition. In electroweak theory, for example, it is p o s sible to have a configuration of coupling constants for the gauge a n d scalar fields such that a second-order ■ p h a s e transition in the scalar f i e l d o c c u r s . A n o t h e r kin d of p hase transition is possible, one in w h i c h there is a discontinuous change in the expectation v a l u e of the scalar field. transition. This is a first-order phase Figure 4 .shows an effective potential in w h i c h a first order p h a s e transition m a y occur. example, In the electroweak a first-order p hase transition is p o s s i b l e under another configuration of the coupling constants, strong gauge coupling constant. a relatively As the temperature decreases 25 Figure 4. A n effective potential in w hich a first order p hase transition m a y occur. 26 toward the critical temperature, effective potential, two wells m a y appear in the as in figure 4. A l t h o u g h the field m ight origi n a l l y be located at the quasi-stable m i n i m u m at <p = 0, that state is unstable in q u a n t u m field t h e o r y . The field m a y tunnel through the potential barrier to cp0, a m uch different process from a second-order phase transition.. F irst-order p hase transitions in quantized scalar fields are the m a i n topic of this t h e s i s . One k i n d of bubble formation process is of special i n t e r e s t : nucle a t i o n b y a n o n - v a c u u m field configuration, a b o s o n star. astrophysical object is the topic in chapter 3. < This exotic 27 CHAPTER 3 W H A T A R E THE PROPERTIES A N D CHARACTERISTICS OF B O S O N STARS? B a c k g r o u n d Concepts for Calculating Boson Star Models The idea of a b o s o n star can trace its a n c estry bac k forty years or m o r e to the geometric-electromagnetic entity that J ohn A. Whee l e r dubbed a "geon." Kugelblitz, the sphere of light. In German, its name is Whee l e r suggested the geon as a "self-consistent solution to the p r o b l e m of coupled electromagnetic a n d gravitational fields."1 His mos t basic geo n m odel was just a stable standing w ave b e a m of light bent into a toroidal shape. Wheeler saw this as a generalization for the concept of material b o d y that was possi b l e wi t h i n the framework of general relativity. The idea of a self-gravitating field configuration is a fruitful one. this way, For example, one can model neut r o n stars in as a self-gravitating spin-1/2 fermionic field configuration. However, one mus t remember the current idea that neut r o n stars m ight not b e simply a huge concentration o nly of neutrons; they might have a shell-like interior structure of exotic m e s o n condensates and a crust of regular baryonic matter. 1J-A. Wheeler, Nonetheless, a neut r o n star is a w ell-known "Geons, " Physical Review, 97., 511 (1955) . 28 example of a self-gravitating configuration of a quantum field. B o s o n stars also fall into this class of exotic objects, an d they have b e e n the object of extensive study recently.2 The m o s t common m odel is that of a complex scalar field in gravitational equilibrium, w i t h only the u n c e r tainty pri n c i p l e supporting it against gravitational collapse. Before going on to review some of the previous research on different kinds of b o s o n stars, it is important to distin g u i s h the self-gravitating field m e t h o d for construct­ ing a star model, method. from the "traditional" Oppenheimer-Volkoff The distinguishing feature is the use of an implicit equation of state in the former m e t h o d versus the use of an explicit equation of state in the latter.. In the Oppenheimer-Volkoff a p p roach3, one assumes that the star has spherical symmetry an d that there is no time dependence in the solution. Oppenheimer and Volk o f f u sed the S c h w arzschild coordinate system. Their m e t h o d also assumes that the ma t t e r has the stress tensor of a p e r f e c t fluid, s y mbolized in the equations b e l o w as Tap. One m u s t also adopt some e q u at ion of state p = p (P) in order to ob t a i n a solution for the system. Thus, one solves the system (3.1): the 2See recent reviews in P . Jetzer, "Boson stars," Physics Report, 220, 163 (1992), and T.D. Lee and Y. Pang, "Nontopological solitons," Physics Reports, 221, 251 (1992). 3J.R. Oppenheimer and G.M. Volkoff, "On Massive Neutron Cores," Physical Review, 55, 374 (1939). 29 Einstein field equations, fluid, the equation of m o t i o n for the a n d the equation of state, viz. G ap = Stc T ap , V a T ap = 0 , ( 3 .1 ) P = P (P) , for p ( r ) , P (r), an d for the metric functions g tt(r) and ^rr(r )' w h e r e t denotes the time a n d r the Schwarzschild radial coordinate. The physical dimensions of the object are c onstructed from these functions. the object, P(R) R, For e x a m p l e , the radius of is the radius at w h i c h the p r e ssure vanishes, = 0. The self-gravitating field method, w h i c h I shall use, is one w h i c h employs the scalar w a v e equation as the implicit substitute for the concept of an equation of state.4 nin g w i t h the same assumptions about symmetry, same coordinate system, Begin­ a n d using the one considers a scalar field (p with an Euler-Lagrange equation of m o t i o n given b y Dcp + dV/dcp = 0. One still uses the Einstein field e q u a t i o n s , except that now, Tap is the stress-energy tensor for the scalar field cp. The stress-energy tensor is a construction not of p r e ssure and 4r . Ruffini and S. Bonazzola, "Systems of Self-Gravitating Particles in General Relativity and the Concept of an Equation of State," Physical Review, 187, 1767 (1969). 30 density (as w i t h the perf e c t fluid) the scalar field. method, Therefore, but of the Lagrangian of for the self-gravitating field the system of equations one must solve is just a bit simpler, one less equation than for the O p p e n h e i m e r -Volkoff method. For the self-gravitating field method, the system of equations is dV Dtp d<p (3.2) G qP — 8JCT aJ3 . One solves (3.2) those functions, for (p(t,r) , g tt (t, r) and grr(t,r). From one calculates various quantitative properties of the star. For example, one m i g h t find the size of the object b y looking not for v a nishing p r e s s u r e but for the radial distance R at w h i c h the field (p has d e creased to 1% of the central field value. In the self-gravitating field method, the solutions will de p e n d s i g n ificantly u pon the type of scalar p o t ential one studies. U s i n g different potentials in the self-gravitating field m e t h o d is akin to work i n g w i t h different equations of state in the Oppenheimer-Volkoff method. The first b o s o n star models w e r e calculated b y D. J. K aup in 1968 using the free p a r ticle potential for a class­ ical comp l e x scalar field, 31 V(<p) = T m 2|cpf . (3.3) The corresponding Euler-Lagrange equation is the familiar K l e i n -Gordon equation: □ <p = V aVa(p = - m 2(p . (3.4) Kaup calculated size, mass and several thermodynamic qu a n t i ­ ties from his solutions for the Klein-Gordon g e o n . 5 In 1969, Remo Ruffini and Silvano Bonazzola extended the analysis to a qua n t i z e d scalar field of self-gravitating free bosons, as well as to a q u an tized spinor field of free fermions. In their examination of the concept of an equation of state,6 they c onstructed star models and w e r e able to calculate the size, mass an d thermodynamic quantities of b o s o n stars an d neut r o n s t a r s . The case of the complex scalar field w i t h a q u a r t ic self-interaction potential, came u nder the investigative eye of M o n i c a C o l p i , Stuart Shapiro a n d Ira Was s e r m a n in 1986. I In their p a p e r on the gravitational equilibria of self­ interacting scalar fields (hereafter, CSW),7 they u s e d a 5D.J. Kaup, "Klein-Gordon Geon," Physical Review, 172, 1331 (1968). 6Ruffini and Bonazzola, "Self-Gravitating Systems," 1768. 7M. Colpi, S.L. Shapiro, I. Wasserman, "Boson Stars: Gravitational Equilibria of Self-Interacting Scalar Fields," Physical Review Letters, 57., 2485 (1986) . Hereafter, I shall refer to this paper as CSW. 32 potential of the form (3.5) T h e y c alculated the size and masses of b oson star models w i t h this p o t e n t i a l . D e p ending on the mass of the scalar boson in question and on the relative strength of the interaction, b o s o n stars of mass comparable to m a i n sequence stars are possible. T h e y even put together an effective equation of state for the case of r elatively strong self-interaction. Wha t other k i n d of potentials might one use to model ne w varieties of b o s o n stars? In chapter 2, I dis c u s s e d some e arly universe models that call for a scalar potential in w h i c h p hase transitions are possible. A n example from electroweak theory is the Coleman-Weinberg m e c h a n i s m . 8 The effective potential is of the following form: (L i ItPIl 2 ) V c w (tP) = A | ( p |2 + |cpf In — 2 I b2 J (3.6) One sets its parameters A and B using the masses of the W a n d Z particles and w i t h the value of the field at the global m i n i m u m of the potential. In this potential are a pai r of potential wells b e t w e e n w h i c h a first-order p h a s e transition m a y proceed. Figure 5 shows an example of this p o t e n t i a l . 8R 1J. Rivers, Path Integral Methods. 243. 33 0.01 0.008 0.006 0.004 0.002 - 0.002 -0.004 -0.006 -0.008 - 0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 <P Figure 5. The effective potential in the Coleman-Weinberg p r o c e s s . A = 0.1, B = 0.5. 34 A n o t h e r p o s s i b i l i t y is to use a potential that has terms quadratic, cubic an d guartic in the scalar field, cal l e d 2-3-4 p o t e n t i a l .9 the so- This potential. (3.7) is easy to w o r k w i t h and one can force it to have two minima b y setting the three parameters, constant for self-interaction, coefficient T). In fact, b o s o n mass, m, coupling X, and a temperature-related the 2-3-4 potential will b e m y tool in studying the b o s o n stars that nucleate e arly universe v a c u u m p h a s e t r a n s i t i o n s , w h i c h is the topic of the next chapter. Details of Calculating B oson Star Models w i t h the Self-Gravitating Field Method. To construct b o s o n star m o d e l s , I will use the selfgravitating field method, CSW. closely following the procedures in Once I have constructed the models, I will use them to calculate a nucle a t i o n rate for v a c u u m phase t r a n s i t i o n s . To begin, I take the 2-3-4 potential as the model potential for the b o s o n field (p. I ignore m o m e n t a r i l y any 9D. Samuel and W.A. Hiscock, " 1Thin-wall' approximations to vacuum decay rates," Physics Letters B, 261. 251 (1991); A. Linde, Particle Physics and Inflationary Cosmology (New York: Harwood Academic Publishers, 1990), 120. 35 coupling of (p to other fields, even possible gauge fields coupled to (p in the spontaneous symmetry b r e aking p r o c e s s . The Lagr a n g i a n d e n s i t y for the complex scalar field cp is -e = - T g abv acp’ v b c p - v(cp) (3.8) The Euler-Lagrange equation for cp is MJ (3.9) dcp2 F r o m the L agrangian one also constructs the stress energy tensor of the scalar field, w h i c h is t = y g “ [ v e(|>*Vb<p+ V1Cp-V11Cp] (3.10) - y S ; [SedV 1Cp-VdC p tm 2IcpI2 - 2t]|<p|’ + For solution of the scalar w ave equation and the Einstein field equations, it is n e cessary to mak e several decisions about the nature of the solutions for the field a n d about the coordinate system w h i c h has so far remained unspecified. The first decis i o n about the type of b o s o n star solution concerns its functional dependence u pon time. w i t h fixed nu m b e r of bosons, N, Fo r a star the time dependence for the 36 gr o u n d state mus t be of the form exp [-itot] .10 in the discussion, A t this point CO is the u n s p e cified Lagrange m ultiplier w i t h w h i c h ,one minimizes the energy of the system under the constraint of fixed N. A requirement specifically for the ground state is that there m u s t b e no nodes in the gro u n d state solution. Later I will m a k e use of this requirement to determine a specific v a l u e of CO. I also require that the solution b e "localized" in a finite vo l u m e of space. In practice, this wil l happen beca u s e the gr o u n d state solution will have a decaying e x ­ ponential dependence, so that it a s y m ptotically approaches zero far from the center of the star. A n o t h e r specification of solutions is that they be s pherically symmetric. equations, This further simplifies the system of in that the cp solutions depend only on time and the radial coordinate r. Overall, then, b e of the form cp(t,r) = (p(r )e x p [itot] . the solutions <p will This time dependence will simplify the system of equations in that all time d e r i v ­ atives are r e p l a c e d w i t h factors of ±ico. The requirement of spherical symmetry applies to the metric. In addition, since I a m looking for equilibrium solutions for (p, the metric mus t b e static, w i t h no time dependence. In the (t, r, 0, ())) coordinate system, for the line element is 10T.D. Lee and Y. Pang, "Nontopological solitons," 255. the form 37 ds2 = —B(r) dt2 + A (r) dr2 + r 2 dO2 , (3.11) where d ti2 = de2 + sin26 d(|)2 . In this equation, <]) stands for the azimuthal angle; for the c o - latitudinal angle. Far from the star, where gravitational effects have diminished, become flat. B(r) 0 stands the spacetime should In the limit of large r, bot h me t r i c functions and A(r) m u s t approach unity. Later, I will use this asymptotic condition on the metric to scale the metric function B ( r ) . N o w that I hav e chosen a specific coordinate system, it is p o s sible to w r i t e d own the explicit form of the scalar w a v e equation a n d the Einstein field e q u a t i o n s . Before doing that, however, I w ill rescale the field, the potential and the coordinate system so that they are all dimensionless, for convenience in calculation. . In the first rescaling, I replace the field (p(r) w i t h a dimensionless scalar field G(r) such that <P • 'Planck Then, I change to a dimensionless potential such that (3.12) 38 V(Cp) -V (G ) = 2 ----- V(Cp) = G2 - 2r|*G3 + Y A g 4 . (3.13) m MpiaBck The rescaled, dimensionless parameters T|* an d A are 'M Planck Tl , (3.14) 47t m 2 ^ -^Planck A (3.15) 47cm2 H e reafter I will drop the tilde a n d use V(G) to refer to the r e s caled potential. C S W found that, A as far as b o s o n stars are concerned, is a c t ually a b e t t e r meas u r e of the importance of the self­ interaction than X; if m is m u c h smaller than the Planck mass, then A can be large even if X is small. In studying the 2-3-4 potential for first order phase t r a n s i t i o n s , the important par a m e t e r is actually a combination of b o t h A. This par a m e t e r is ^ = Tj* 2/A . consider the extrema of V ( G ) . T|* and To grasp the importance of ^ One extremum is at G = 0; the other two extrema G 1 and G2 are \ / G2,1 4A 3rf 1±. I — 2A I 1 , 9il 2 (3.16) The latter two extrema will hav e real values w h e n £ > 4/9; is the location of the top of the "bump" in the potential. G1 39 and O2 is the location of the b o t t o m of the second w e l l . % = 1/2, the potential is d e g e n e r a t e : V(O) W h e n ^ > 1/2, O2 the extremum at true v a c u u m is at O2, = 0 an d V (O2) W hen = 0. will be a global minimum; the as figure 6 s h o w s . A t the end of chapter 2, I discussed the concept of a critical temperature for v a c u u m p h a s e t r a n s i t i o n s . The par a m e t e r ^ is something like an inverse temperature for the following r e a s o n s . Consider the shapes of the potential curves in figure 6. W h e n % < 1/2, the curve looks like an effective potential above the critical temperature. 1/2, When ^ = the curve looks like an effective potential at the critical temperature. W h e n £ > 1/2, the curve looks like an effective potential b e l o w the critical t e m p e r a t u r e . The anal o g y breaks down w h e n one considers negative values for potential. One is left to interpret a complex-valued Nonetheless, the p a rameter ^ will b e useful as a restr i c t e d indicator of t e m p e r a t u r e . The final two rescalings concern the Lagrange multiplier to an d the radial coordinate r. It is convenient to switch to a dimensionless Lagrange multiplier, D = to/m. Also, a dimensionless radial coordinate x = m r will be h e l p f u l . A f t e r these rescalings, I can write down the system of e q u a t i o n s . The scalar wav e equation in the 2-3-4 potential is a" o" = A Q 21 (A' o - Srfcr2 + A a 3 + (~2A I1-TM B' 2 \ 2B " x y (3.17) 40 0.002 0.0015 V (cp) 0.001 0.0005 -0.0005 0.04 0.08 0.12 Figure 6. The 2-3-4 potential. Curves a, b, c an d d have Z; = 0.42, 0.47, 0.50, 0.53 respectively. A = 300 for all four curves. Note that they are ne a r l y indistinguishable n ear 0 = 0 . 41 The primes denote differentiation b y x. field equations are G tt = SnTt Z A' B ' 2 X2 I 1 " a I - ) <y2 — 2r|*a3 + y A ct4 + -r- ( ct ') zQ 2 — I CT2 + 2ri*CT3 - YACT4 + K - ' ) 2 B I ( Ix } "A xAB an d Grr = 87lTrr , viz. Q 2) PQ 2 "*■ The two Einstein , (3.18) (3.19) A f t e r configuring the system b y setting values for ^ and A, solve equations (3.17), (3.18) and (3.19) b o u n d a r y conditions I will n o w describe. the field, I subject to a set of The central value of CT(O), is left as an unspec i f i e d constant. The v alue of the central field will d istinguish solutions in a given family of s o l u t i o n s . the origin, The field mus t be nonsingular at so the central value of the first derivative, CT' (0), mus t vanish. B (x) is unity. As x approaches infinity the limit of Physically, the dimensionless m ass function jlt(x) of the spacetime M(x) TX (3.20) [‘ " A M m ust v a n i s h nea r the origin faster than x, mass inside a sphere of radius zero. A(O) since there is no This allows m e to set to unity. The asymptotic limit on B(x) wha t v a l u e of B(O) does not, however, to use w h e n starting the solution tell me 42 algorithm. In the fourth-order Runge-Kutta scheme I use, m ust a c t ually specify the value of Q 2Z B ( O ) . I The w a y I resolved this pu z z l e was through the realization that B(O) rela t e d to the central redshift of the star. redshift z(r) In general, is the of a p h o t o n emitted at radius r an d observed at infinity follows from analysis of the constants of the motion along null geodesics in a static spherical s p acetime:11 X(r) 4 m XH ~ Vb h " constant ( 3 - 21) This allows one to calculate z(r), X(°o) -A,(r) z(r) X(r) Therefore, B(°°) B(r) (3.22) no mat t e r h o w the distant observer arranges his scale for B (r) , as long as the quotient B H constant, /B(O) is then he will mak e the same measurement of.the central r e d s h i f t . Thus I m a y set the scale at B(O) p r o c e e d w i t h the Runge Kutta procedure. =1 and The v a l u e I obtain for B (°o) will then a l l o w me to rescale B (x) , so that the r e s caled B (x) approaches u nity as x approaches infinity. It is permis s i b l e to rescale B (x) b y a constant C in order to fit the b o u n d a r y condition at infinity. effect on the system of equations. This has no If one sets B (x) to1 11Misner, Thorne and Wheeler, Gravitation , 659. 43 C - B (x), then the terms containing B ' /B are unchanged. about the terms containing Q 2/B? What This quotient remains the same w h e n I rescale B(x) b y a constant C . Consider the invariant interval As taken along a timelike geodesic. If I rescale B(x) b y the constant C, I m ust rescale the time interval b y a constant C iz2, As = VB(x) At = V C VB(x) (3.23) R e scaling the time interval b y C""1/2 means that frequencies rescale b y C 1/2. Thus the terms containing D 2ZB remain unc h a n g e d b y the rescaling. If I set B(O) = I and rescale B(x), I still m ust come up w i t h the "correct" value for the Lagrange mult i p l i e r D. Before reviewing that task, I mus t present a set of good solutions for a typical b oson star configuration. Comparing b a d solutions against the g ood will show h o w to select D. In figure 7, I show the plo t of the field c(x) typical b o s o n star configuration, G(O)=O.5 G1. respect to x. W i t h it is G' (x) , w i t h A = 300, for a ^ = 0.52 and the first derivative of The radius at w h i c h the G G field has fallen to 1% of its central value is approxi-mately 11.27. One might call this distance the radius of the b oson star. Another m e t h o d is to find the value X max at w h i c h A(x) maximum, wit h is at its a n d calculate an effective radius, X e££: 44 0.02 0.015 0.01 0.005 -0.005 x Figure 7. The upper curve is a typical solution for the field a(x) ; the lower curve is O' (x) . A = 300, £ = 0.52, and a (0) = 0.5 O1. 45 X eff = J V ao o dx . (3.24) 0 This is the spatial distance along a p a t h from x = 0 to X maxIt is n o w apposite look at solutions for A(x) and B (x). In figure 8, I show the plot of the metric functions. has b e e n r e s caled so that B(°°) = I. Figure 3.5 shows the plot of the mass function for the star, although B (x) an d A(x) asymptotic limits, a limit. (x) . N ote that, do not converge rapidly to their the mass function does converge rapidly to In this example, in units of B (x) (Mplanck)2Zm. the mass of the star is 0.003459 If the scalar mass m = 100 G e V , then the mass of the star is 5.0 x IO33 G e V or about 9000 metric tons, the displacement of a' fully loaded guided miss i l e destroyer like D D G - 9 9 3 , the USS Kidd. The evident failure of B (x) a n d A(x) to reach limiting values on the scale of the plot m i g h t lead the reader to complain: w h y not continue the plo t to larger x-values where the asymptotic v alue for B (x) will be evident? that the solution inevitably diverges, positively, either negat i v e l y or due to v e r y slight differences betw e e n the assu m e d v a l u e of the eigenfre q u e n c y , Q*, of £2. The answer is and the true value Figures 10 a n d 11 show these two cases. p r o g r a m uses an estimate, solution for the field, Q*, G*. The computer in order to calculate the trial W hen £2* is slightly larger than 46 1.001 0.999 0.998 0.997 Figure 8. Metric functions for the example b o s o n star. 0.004 0.003 0.002 0.001 Figure 9. The mass function for the example b o s o n star. 47 0.15 0.05 -0.05 0 Figure 10. 5 10 x 15 20 A n example of a trial solution w i t h too large 0.18 0.16 0.14 0.12 0.08 0.06 0.04 0.02 0 Figure 11. 5 10 x 15 20 A n example of a trial solution w i t h Q too small 48 £2, the trial solution,a*, eventually plunges across the x- axis an d diverges in the negative d i r e c t i o n . ' W h e n £2* is slightly smaller than £2, the trial solution turns a way from the x-axis an d diverges in the p o s i t i v e direction. b i s e c t i o n algorithm, splitting the difference b e t w e e n the oversized £2* value a n d the u n d e r s i z e d £2* value, rapi d l y settle on a goo d value, of £2. £2 v a l u e improves, Using a one can As the a c c uracy of the the solution extends further a n d further along the x-axis be f o r e it diverges. The pre c i s i o n of the computer then limits the search for £2, w h i c h is w h y the sample solution above does not extend into the region w h e r e the metric functions level off. In summary: . I solve the system (3.17), (3.18), .(3.19) in accordance w i t h b o u n d a r y conditions C (O ) = 0 , A(O) = I , (3.25) B(O) = I , a n d I independently select- a central field value in the range 0 < CT(O) CO, . (3.26) I search for a v alue of £2 that gives a nodeless solution e x t ending as far in x as possible. I interpret this v a l u e of £2 as the e i g e n energy of the ground state. I extract an 49 asymptotic v a l u e for B(°°) and rescale the solution for B (x) so that it fits the asymptotic condition B(°°) = I. of the b o s o n star is n o w c o m p l e t e . The model One can then analyze it for various properties such as size a n d m a s s . N o w that I have spelled out the solution method" for the b o s o n star system, in figure 12 I present a collection of field configurations for a range of values of O ( O ) , w ith O(O) expressed as a p ercentage of O 1. It is w o r t h no t i n g that the var i a t i o n in shape of O(x) depends on the central field value. 2-3-4 potential, Speaking in terms of the this means that the spatial v a riation depends on h o w far up the "bump" one starts the solution. One can say that solutions starting near the top of the bump "roll" quic k l y b a c k to O = 0. the well, w i t h O(O) b a c k to O = 0. Solutions that start down in closer to O = 0, roll only v e r y slowly This appeals to one's analogical thinking about a p a r t i c l e in a potential w e l l . In terms of spatial var i a t i o n of the b o s o n star configuration, o(0) one can say that relatively large values of generate solutions that are relatively small in spatial extent. Smaller values of O(O) relat i v e l y large. generate solutions that are This distinction will be of crucial importance in the next chapter w here I consider the nucle a t i o n of v a c u u m p hase transitions in the case of a small star an d a large bubble. 50 0.03 0.025 0.02 0.015 0.01 0.005 0 5 10 x 15 20 Figure 12. A gall e r y of solutions for the scalar field a(x) in the 2-3-4 potential, each solution wit h a different central value, a ( 0 ) . For this set of solutions, A = 300 and £ = 0.60. The central value for each solution is expressed as a percentage of the value of O i , as shown in the legend. N ote h o w rapidly the 90% solution drops off and h o w b r o a d the 10% solution is. 51 CHAPTER 4 H O W DO B O S O N STARS A F F E C T THE D E C A Y OF THE FALSE VACUUM? In chapter 3, I showed h o w one models a b o s o n star in the 2-3-4 p o t e n t i a l . In this chapter, m y objective is to use those models to study first order v a c u u m p hase transitions as a b o s o n star w o u l d nucleate them. To begin, I shall review concepts and techniques for studying v a c u u m p hase transitions. Following that, I shall focus on a special case of "small" b o s o n stars nucleating v a c u u m phase transitions. The chapter concludes w i t h a summary of the effects in this special case. The pict u r e of wha t happens in a first order vac u u m p hase transition is relatively simple. It is similar to the formation of a bubble of steam in hot water. The scalar field <p is initially in the so-called false v a c u u m state, xPfaisez e v e r y w h e r e . Due to quan t u m fluctuations, perturbations to the system, or impurities, other a b u b b l e forms containing the field in the so-called true v a c u u m state, xPtrue- Say that the change in vo l u m e energy d e n s i t y inside the bu b b l e is fE , w h i c h will be negative since the true v a c u u m is at a lower energy density than the false. In addition, say that the bub b l e forms w i t h a positive surface energy d e n s i t y S. One can show, using conservation of energy,, that 52 if the system's total energy change is zero, the bubble forms with radius !R such that 3S R = -I— (4.1) A bubble of at least this size will expand until all the false v a c u u m (or hot water) is converted to true v a c u u m (or steam) . It is important to d istinguish between spontaneous decay of the false v a c u u m an d induced decay. example of w a t e r is u s e f u l . droplets. Here a g a i n , the Consider the formation of cloud A l t h o u g h w a t e r droplets can form spontaneously in a m a s s of w a t e r v a p o r , they form m o r e readily around atmospheric aerosol particles, nuclei or C C N . 1 Similarly, eously, ca l l e d cloud condensation The CC N is an impurity in the vapor. a v a c u u m p hase transition can p r o c e e d spontan­ or a n impurity in the v a c u u m can nucleate a phase transition.2 B o s o n stars can be such an impurity. 1A-S. Arnett, Weather Modification bv Cloud Seeding (New York: Academic Press, 1980), 7, 31. 2d .A. Samuel and W.A. Hiscock, "Gravitationally compact objects as nucleation sites for first-order vacuum phase transitions," Physical Review D, 45., 4411 (1992); V.A. Berezin, V.A. Kuzmin and 1.1. Tkachev, "Black holes initiate false-vacuum decay," Physical Review D 43., R3112 (1990); G. Mendell and W.A. Hiscock, "Gravitational nucleation of vacuum phase transitions by compact objects," Physical Review D .39, 1537 (1989); W.A. Hiscock, "Can black holes nucleate' vacuum phase transit­ ions?" Physical Review D 35., 1161 (1987) . 53 Coleman's The o r y of D e c a y of the False V a c u u m One can liken a first order p hase transition in a q u a n t u m field to the p e n e t ration of a potential barrier b y a particle. Consider a p a r ticle of mass (I in a potential V (x) that has m i n i m a at x = 0 and x = x 2 . classical theory, See figure 13 . In a the point x = 0 is a stable equilibrium, but in a q u a n t u m theory it is not s t a b l e . The particle initially at x = 0 m a y tunnel through the potential barrier and emerge at x = x out w i t h zero kinetic energy. it propagates c lassically toward X 2 . Fro m there The amplitude for this process in the semiclassical approximation is F = A e "B[ l + O(Ji)] . (4.2) The quant i t y B is X o Ut B = J ^ 2(iV(x) dx . 0 a n d A is a n o r m alization constant. constr u c t e d 3 the amplitude (4.2) Banks, Bender and Wu as a p ath i n t e g r a l . The dominant contribution to the total amplitude comes from the region nea r the p a t h that extremizes B, 3T. Banks, C.M. Bender, T.T. W u , "Coupled Anharmonic Oscillators. I. Equal-Mass Case," Physical Review D, R , 3346 (1973). 54 O .OOl 0.0008 0.0006 0.0004 0.0002 - 0.0002 -0.0004 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 x Figure 13. Potential w i t h unstable equilibrium at x = 0. p a r ticle can tunnel quan t u m mec h a n i c a l l y away from x = 0. A 55 5B = O . (4.4) y Equation (4.4) is a special case of the more general v a r i a ­ tional problem. 5 J ^/2|i (E - V ( x ) ) dx = 0 , (4.5) for the m o t i o n of a particle of mass (I w i t h total energy E, m o v i n g in the potential V (x). (4.5) The equation of m o t i o n from is d2x ^ dV = -IbT ' The special variational p r o b l e m (4.4), however, (4.6) corresponds to a p a r t i c l e of mass (X w i t h zero total energy, m o v i n g in an "upside-down potential," - V (x). The classical equation of m o t i o n for the tunneling p r o b l e m has a v e r y significant sign difference w i t h respect to (4.6), viz. d2x dV — V = +-TT (4.7) Cole m a n also interpreted this as the equation of m o t i o n for a p a r t i c l e in the potential V(x) but w i t h a replacement of the 56 time t w i t h an imaginary time, X = it, also k nown as the Euc l i d e a n time. The q u a n t i t y B is the action, in Euclidean s p a c e . Se , calculated The Lagrangian in the Euclidean space. (4.8) This concludes the first half of the analogy b e t w e e n particle tunneling a n d v a c u u m p hase t r a n s i t i o n s . The second half of the analogy focuses on the quantum field, cp. One calculates the amplitude for the p h a s e transition. In g e n e r a l , one calculates the E u clidean action, SE , for the field tunneling from the false v a c u u m at Cpfalse to the true v a c u u m at Cptrue, as figure 14 s h o w s . For a scalar field in flat space, the Euclidean action is (4.9) There are several w a y s 4 to calculate S e . One can solve the E u clidean equations of mo t i o n exactly for the bu b b l e solution, 0(4) cp. symmetry, If one assumes that the bubble solution has then it is useful to transform to an 0(4) radial coordinate, p2 = T2 + r2, w i t h T defined as the 4D.A. Samuel and W.A. Hiscock, " 'Thin-wall' approximations to vacuum decay rates," Physics Letters B261, 251 (1991) . 57 O .OOl 0.0008- 0.0006- 0.0004- 0 . 0002 - true - 0 . 0002 - -0.0004 - 0.02 0.02 0.04 0.06 0.08 0.12 Figure 14. Potential in w hich a first order p hase transition is possible. The field (p can tunnel from the false va c u u m at (Pfaise to the true va c u u m at cptrue- 58 E u clidean time. A f t e r that transformation, the Euclidean equation of m o t i o n for the field becomes + ZJ!! = dV dp2 p dp Once one has obtained a solution, numerically, . (4.10) dcp either analytically or one can calculate the S e integral in a s t r aightforward m a n n e r . Coleman invented another w a y 5 to calculate Se . He d e v i s e d an approximation scheme for getting a solution to (4.10) u nder a special condition. This condition was that the p o tent ial is n e a r l y d e g e n e r a t e . To state this limit more precisely, if one defines e as the energy difference between the vacua, E = V((pfalse) - V(Cptrue), then his approximation is legitimate in the s m a l l -E limit. Coleman's scheme has the a d d e d advantage of giving a c losed-form expression for Se . n a m e d this scheme the thin-wall approximation. a p p r oximation scheme I wil l use, He It is the so it is appropriate to examine it closely. The k e y idea is to interpret p a r ticle motion. (4.10) in an a n a l o g y to Let the reader b e w a r e : this is a different anal o g y from that u s e d at the beg i n n i n g of this section. c l a riify the p a r ticle analogy for equation wrote: 5Coleman, "Fate of the false vacuum," 2932. (4.10), Coleman To 59 If w e interpret (p as a p a r t i c l e p o s ition an d p as a time. Eg. (3.9) is the mechanical equation for a particle, m o v i n g in a potential minus U an d subject ' to a somewhat p e c u l i a r viscous damping force wit h S t o k e 's law coefficient inversely proportional to the t i m e .6 The tunnelling p r o b l e m becomes a trip from p e a k to p e a k in the upside dow n potential, - V (9 ), w h i c h figure 15 shows. In order to get a mor e intuitive grasp of this interpretation of equation (4.10), consider for a moment that it is an ordinary classical mechanics equation of particle motion. That is, substitute x for cp an d t for p, viz. d2x dt2 The first term in + dV dx f3 (4.11) (4.11) is the particle's acceleration. The second term contains the v e l o c i t y d x / d t , and it has a coefficient that is t i m e - d e p e n d e n t , 3/t. of The right-hand side (4.11) has usual the potential gradient, sign is opposite the customary usage, potential's name. important; except that the hence the upside-down The sign of the velocity's coefficient is since it is positive, it corresponds to a force that opposes the m o t i o n of the particle. the particle's s p e e d . That force slows So there are two forces: force from - V (x), the upside-down potential, a gradient an d a damping ■ force opposing the m o t i o n a n d slowing down the particle. One 6Ibid. N.b. In Coleman's notation U((p) is the potential; his equation (3.9) is my equation (4.10). 60 0.0004 0.0002 - 0.0002 -0.0004 -0.0006 - -0.0008 - 0.001 0.02 - 0 0.02 0.04 0.06 0.08 0.1 0.12 9 Figure 15. The upside-down p o t e n t i a l , -V(cp). 61 can conceive of different motions of the p a r ticle for different sets of initial conditions. a critically da m p e d system, For example, there is in w h i c h the p a r ticle starts wit h an initial v e l o c i t y at time t0, from the top of the highest "hill" in the upside d own potential, moves slowly to the left a n d comes to a halt exactly at the b o t t o m of the "valley" b e t w e e n the two h i l l s . w i t h time, Since the damping force diminishes one can conceive another k i n d of motion. some amount of time, to b e negligible. After T 1, the damping force will b e so small as A f t e r that time, the particle w o u l d resp o n d only to the gradient force. If one starts the p a r ticle v e r y near the top of the highest hill, the damping force will keep it n ear the top until about time T 1, whe n it starts to m o v e across the v a l l e y as if there w e r e no damping. In this motion, the p a r ticle will shoot over the top of the smaller hill an d kee p going. third k i n d of motion. of the highest hill, One starts the particle nea r the top and it loiters there amid the damping for an amount of time, elapsed, valley, Betw e e n these two cases is a T < T 1. A f t e r that amount of time has the p a r ticle begins to m o v e rapidly across the b u t it does not shoot over, but coasts p e r f e c t l y to a stop at the top of the smaller hill as t approaches infinity. Taking this intuitive grasp of the p r o b l e m of a particle in an u p s i d e-down potential, theory p r o b l e m at hand. one can apply it to the field W i t h suitably chosen initial "position," cp(0 ) nea r the top of the highest hill, an d wit h 62 initial speed cp' (O) =0, the "particle" can b e r e l e a s e d at "time" p = 0 and it will loiter nea r the top some amount of time p * . elapsed, (at <ptrue) for A f t e r that amount of time has the "damping force" has diminished enough to allow the p a r t i c l e to mak e a "rapid" transition through the va l l e y and r each the secondary p e a k at Cpfalse as p approaches infinity. The tunnelling process from Cpfalse to Cptrue is just the "time-reversal" of this process. Figure 16 shows an example of a tunnelling solution for cp(p) . In terms of the scalar field n e a r l y constant value, cp, cp(p) = Cptrue, the field changes v a l u e rapidly to cp(p) = Cpfalse the solution has a for p < p*. Cpfalse- The n for the false v a c u u m is still present at p*. p > p*, Far b e y o n d p » wall is thin in that the field changes rapidly from Cptrue p = p*, This solution corresponds to a "bubble" of true v a c u u m inside a "wall" of radius approximately this wall, When p*. Cpfalse The to in a r elatively brief interval of "time." The m e t h o d that Coleman employed to get a cl o s e d form for Se took advantage of the b e h a v i o r of cp(p) n ear the bubble wall at p = p *. In the limit of small e, he substituted a degenerate potential V+ (cp) for the actual potential, long as the min i m a of V+ (cp) are v e r y close to an d as long as e is small, Cpfalse (4.10) m a y be neglected. the solution n ear the wall, and As Cptrue, this substitution is legitimate. Since the "time" has already run out long enough, damping term in V(cp) . Cpwall (p) the viscous The result is that obeys an equation that is 63 : <p loiters coasts to a stop -0.5 10 15 20 25 30 35 40 P Figure 16. Shape of a typical solution cp(p) in the thin-wall a p p r o x i m a t i o n . I u s e d a hyperbolic tangent function to plot this curve. (ptrue= 2, cpfaise= 0. Most of the change in <p(p) occurs near p*. 64 slightly different from (4.10), viz. d 2<Pwall = _ d \ _ dp2 (4.12) dcpwa]1 Coleman showed that the action, SEwa11, of this solution (pwall has a simple relationship to the total bubble action, 27 Tt2 (Sg311)4 p Coleman sECol6man: (4.13) w ith t P fa ls c (4.14) S if= J t P tru e Also, Cole m a n showed that the radius w h i c h minimizes the bubble action is R (4.15) Therefore one considers R to b e the radius of the bubble of true v a c u u m at the moment of formation. T=O w i t h radius R. The bu b b l e forms at The wall of the bubble then m oves along a circular trajectory in Euclidean space, T2 + r2 = R 2 . The analytic continuation of this m o t i o n b a c k to Loren t z i a n space indicates that the b u b b l e wall will form at time t = 0 with 65 radius R z followed b y expansion of the bubble wall along a hyperbolic trajectory -t2 + r 2 = R 2 . Figure 17 shows the two trajectories in one combination g r a p h . Note that the bubble wall speed approaches the speed of light in this a p p r o x i ­ mation. .This completes the re v i e w of concepts an d techniques for studying v a c u u m p hase transitions. Before going on, I summarize: I shall b o r r o w from Coleman his p r o cedure for calculating the q u a n t u t y B in the thin-wall approximation. Specifically, I shall use dimensionless versions of thin-wall formulas an d (4.15). (4.13), (4.14) The actions contain dimensionless potential V (a) and V + (G) : Coleman SE (4.16) 0 s Ea1' = J v ^ V + do . (4.17) CT2 For V + (G), I use the degenerate 2-3-4 potential, set ^ to a v alue of 1/2. X w h i c h means I The dimensionless b u b b l e radius is = mR: (4.18) 66 O 10 20 30 r 40 50 60 Figure 17. T r ajectory of the bubble wall. The solid curve is a circle, T2 + r2 = R 2 . The dashed curve is an hyperbola, -T2 + r2 = R 2 . R = 22. The 45° diagonal line is the trajectory of a photon, for reference. For convenience, I have identified t and T in this g raph only. 67 As in chapter 3, a 2 is the location of the true v a c u u m in the dimensionless potential V ( G ) . I shall use the formula for the circular trajectory of the bubble wall, but in a dimensionless version, P2 + x 2 = X 2 , w i t h a dimensionless time, (4.19) p = mT. "Adding" a Boson Star to the Spacetime This n u cleation process w i t h a b oson star serving as the nucle a t i o n site or "seed," is not a simple process compared to spontaneous decay in an empty spacetime. The objective is to calculate a tunnelling rate for a decay process involving bubble nucle a t i o n b y a b o s o n star. tunnelling rate, (4.2) for the F = Ae~B [1+0 (ft) ] , the quantity A contains a difficult functional d eterminant7 . parameters of the potential, However, In formula W h e n I change the there will be a change in A. since small changes in the argument of an exponential function can overwhelm small changes in almost an y other function, it is customary to concentrate on changes in B and neglect smaller changes in A. It is n e cessary to revise B to reflect the presence of 7C.G. Callan and S . Coleman, "Fate of the False Vacuum. II. First Quantum Corrections," Physical Review D, JJl, 1762 (1977). 68 the b o s o n star. The q u a ntity B for this process is not, g e n e r a l , a simple one to calculate. in As Coleman an d De Luccia p o i n t e d o u t ,8 the q u a ntity B is the difference betw e e n the action, SEbubble, for a spacetime containing a bubble which a b oson star has nucleated, and the action, spacetime containing no bubble. Here, SEno bubble, for a the "no bubble" configuration of the spacetime is not empty, as in the earlier c a s e : it contains a b o s o n star. B = s bubble _ s no bubble _ ( 4 . 2 0 ) To compare the induced n u cleation process w i t h spontaneous formation of bubbles in empty space, I shall compute a ratio, B / B 0, w i t h B 0 equal to the E u clidean action in (4.16), bub b l e action in Coleman's thin wal l approximation. the . One might fancy this to b e like taking two equal four volumes, one that is empty of b o s o n stars a n d another that has a b o s o n star in it. W h i c h four-volume will produce a bu b b l e of true v a c u u m m o r e readily, star impurity? the empty one or the one w i t h the boson If B / B 0 < I, then the second four-volume, w i t h the b o s o n star impurity, wins the bubble p r o d u c t i o n race. Since I a m u sing the thin-wall approximation as a co m ­ parison, w h i c h is appropriate onl y in the limit of small 6 , I m ust restrict m y b o s o n star models to the same limit, w i t h 8S. Coleman and F . De Luccia, "Gravitational effects on and of vacuum decay," Physical Review D 21., 3305 (1980) . . 69 potentials "near" the degenerate potential w i t h % = 1/2. At the end of this chapter I shall m a k e this concept of " n e arness" to degen e r a c y mor e precise. G oing toward E1 = 1/2, however, does not guarantee the b o s o n star will nucleate a th^n-wall bubble. Remember that the thin-wall approximation d e scribed spontaneous decay in an empty, bubble, flat spacetime. If a b o s o n star is at the center of a then the spacetime is d e finitely not empty, m i g h t not b e flat. and it One m ust as k if it is legitimate to use the thin-wall approximation for this p r o c e s s . U n d e r what conditions m ight the thin-wall approximation b e legitimate? One can give an answer to these questions after taking a look at the field profiles of the bubble solution an d the n o ­ bubble s o l u t i o n s . G (x) I will refer to the bubble solution as an d to the star solution as G* (x), for clarity. The E u c lidean space vers i o n of the full b o s o n star solution changes onl y in that the imaginary time T = it replaces the real time t. Recall that the time an d spatial dependences separate in the ground state. If one calls the full solution for the star Z*(t,x) , then 2, ( U ) = e-irotG, (x) . (4.21) The analytic continuation of this full solution to Euclidean space simply changes the oscillating behavior of the complex exponential to that of a decaying exponential. replacing T w i t h its dimensionless version, A fter P, the solution 70 has the form E ,( p ,x ) = C-fipO^(X) . (4 .2 2 ) W i t h incre asing time P, the solution Z ile(PzX) decays e x p onent­ ially from its value at P = O . This exponential d e c a y will be important in the next section w h e n I consider the notion of "small" b o s o n stars. The bu b b l e prof i l e will, true v a c u u m at O2. in g e n e r a l , start out near the That is, O(O) -CT 2 - It wil l di p downward toward the false v a c u u m at zero, perhaps in a gradual curve or perhaps in a sharp c u r v e . wall limit, function. limit. For instance, in the extreme thin- the bubble profile w ill have the shape of a stepOther shapes are possible outside the thin-wall Figure 16 shows a function whose shape is nearly that of a step-function. If a b o s o n star nucleates a thin-wall bubble, interior will contain true v a c u u m O = O true. of the bubble, — Outside the wall the spacetime is not filled w i t h false va c u u m it is filled w i t h the b oson star solution, a s y m ptotically approaches the false vacuum. difference. the o* (x), which It is a subtle Instead of approaching the false v a c u u m along a curve like that in figure 16, n o w the bubble solution o(x) approaches the star solution o* (x) , and O* (x) approaches the false v a c u u m at zero. The asymptotic limit of o(x) a n d the asymptotic limit of O* (x) = 0. is O* (x), Figure 18 shows an example of wha t the exact solution should look like. It is 71 not easy to make an exact calculation of a bubble solution which, outside X , will smoothly tend toward the star solution as x approaches infinity. To handle this difficulty, I will make an approximation to <T(x) with the following working assumptions. (i) The interior and wall portions of the bubble solution will vary just as if the boson star were not there. I denote this part of the solution as G 0 (x), which might not have a thin-wall shape. (ii) At x = xt, the bubble solution changes over to the star solution G* (x) in a continuous if not smooth manner. In,mathematical terms, these two assumptions mean that 3 x t: G0(xt) = G,(xt)' G(x) = O0(x), x < x. g (x ) = X > xt . Gt(X), (4.23) Figures 19 and 20 show solutions G* (x) and G 0 (x), respect­ ively. Figure 21 shows how the assumptions (i) and (ii) allow one to graft the tail of G* (x) onto G 0 (x) , forming my approximation to the bubble solution, G(x). 72 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 5 10 15 20 x Figure 18. The shape of a smoothly varying bubble solution. Inside the circle, the standard bubble shape (as in figure 16) smoothly changes toward a star solution, w h i c h then approaches the false v a c u u m at a = 0 as x approaches infinity. 73 This nucl e a t i o n process changes the endpoints of the tunnelling "path." The field a begins tunnelling into the potential barr i e r at Ct instead of zero, chapter I . as I m e n t i o n e d in It still tunnels out to the true v a c u u m at C2. Figure 22 shows this subtle but important c h a n g e . The boson ,star solution <7* (x) reaches zero as x approaches infinity. Therefore, the v alue of O t will b e zero only w h e n the radius of the bub b l e is infinite, w h e n the potential becomes degenerate. For a potential w i t h i; > 1/2, will be positive. It n o w apposite to use Ct to construct a s u p p lementary parameter, m o r e useful than the v alue of Ct the ratio' % = O tZa2 . In one way, % is The two limits £— >1/2 and %— >0 bot h describe the approach to a degenerate potential. however, The latter, has explicit information about the n ucleation b y the b o s o n star a n d about the b u b b l e ; the former knows nothing of an y b o s o n star. * * * * * * * 74 0.035 0.03 0.025 0.02 0.015 0.01 0.005 4- 0 6 8 14 x This is the Figure 19. A typical b oson star solution a* (x) shape of the field at (3 = 0. At later times, P > 0 , the field will have the same shape but it will have b een shrunk b y a factor of e x p (- Q P ) . 75 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 x Figure 20. Shape of a bubble profile that is neither thicknor t h i n - w a l l . The dimensionless bubble r a d i u s , X = 2.882, is w here the curve is at half of its original height. 76 0.09 0.08 0.07 star — g raft 0.06 unperturbed bubble 0.05 0.04 0.03 0.02 0.01 0 2 4 6 8 10 12 14 x Figure 21. Graft of a star solution onto an unperturbed bubble solution. The solid line represents the generic s h a p e , according to assumption (i) and (ii), for an approximate bubble solution O(x) w h e n a boson star has n u cleated the b u b b l e . The b oson star solution a* (x) replaces the empty space bubble solution O q (x), for this e x a m p l e , at about x t = 3.29. Note that this is slightly larger than X = 2.882, because the bubble solution O q (x) is not an extreme thin-wall bubble solution. 77 0.001 0.0008- 0.0006- 0.0004- 0 . 0002 - - 0 . 0002 - -0.0004 - 0.02 0.02 0.04 0.06 0.08 0.12 Figure 22. Tunnelling endpoint c h a n g e s . The upper arrow shows the change that a b oson star impurity produces. The lower a rrow shows the v a c u u m -to-vacuum tunnelling process. 78 The Case of a Small Star Nucleating a Large Rnbblm It is interesting to w o r k out the case of a "small" star that nucleates a "large" bubble. large bubble In the case of small star- (SSLB), "small" means that the star's effective radius is m u c h less than X , the bubble r a d i u s . This is equivalent to a small value for %, as I shall n o w explain. Star solutions that meet this criterion are easy to find. I p o i n t e d out at the conclusion of chapter 3, commenting on figure 1 2 , that relatively large values of CT*(0 ) generate solutions that correspond to relatively small boson stars. If one has the bubble r a d i u s , then one mus t simply compare it w i t h the effective radius of the star. W i t h some potentials, ne a r l y all in their family of star solutions m eet the SSLB criterion. That is, solutions wit h a * (0) = 0.99 (T1 meet the criterions but so do the broadshouldered solutions w i t h a* (0) = 0.10 G 1 and smaller. reasons for this are the following. radius increases without b o u n d as In general, >1/2. The the bubble W h e n the hills of the upside - d o w n potential are n e a r l y equal in h e i g h t , the viscous damping force must be w a i t e d out, for longer "times," before the "particle" can journey across the valley. larger This " time" means the bubble has a larger radius X . the heftiest star, as long as it is localized an d has a finite effective r a d i u s , will be "small" compared to Even 79 dege n e r a c y will always have m a n y SSLB combinations w i t h small values of %. One can also run into the opposite problem: w i t h other potentials, none in their family of star solutions meets the SSLB c r iterion of small-%. in figure 21 has % = 1/9. For instance, the bubble solution I w o u l d not consider that star- bub b l e combination to mee t the SSLB criterion. as ^ increases from 1/2, In general, the bubble radius decreases. When the peaks of the upside - d o w n potential are of significantly different h e i g h t s , the "particle" needs to start across the v a l l e y quickly, w h i l e there is still enough viscous damping to slow it down be f o r e it overshoots the lower peak. Therefore, stars in a potential far from degen e r a c y might fail the small % test. I will b r i e f l y discuss star-bubble combinations like this in chapter 5. The last check for smallness is in the time behavior of the full star solution. In general, the bubble wall traject­ ory in the Euclidean x-|3 plane wil l not n e c e s sarily be a circular one, as it is for the bubble formed in empty space. As s u m e at least that the Euclidean sector t r ajectory of the bub b l e wall, P = ®(x) , is b o u n d e d on the p-axis b y T. 3 T : V x e ( 0, X ), 0(x) < T In other words, . ' one can pencil the trajectory ®(x) (4.24), into a 80 box, c e n t e r e d at the origin, w i t h dimensions 2 X b y 2 T . (If the bub b l e wall oscillates w i t h p e r i o d 2 T in Euclidean space, then the trajectory is not bounded, but one can m a k e a similar argument for a b o x that bounds the first full cycle, 0 < P < 2T.) Recall that the full solution has the time dependence exp (-£2|3) . P=T, This factor is largest w h e n this factor has diminished. of the full solution over time, Q -1 a n d T . practice, If T At To check the "s m a l l n e s s " the quantities to compare are £2-1, then Z* ( T , 0) » P = 0. will b e small. In I found the ratio G = Zjlt( T 7O)Za2 to b e v e r y small indeed for the SSLB case, smaller than %. For example, star-bubble c ombination in figure 23 has X = also has 0 = 0.0001931. the 2 - 3 - 4 potential, one perc e n t of %. 0.002434, the but it Over a large range of parameters in I found that 0 was never m o r e than about Table I shows some specific values of b o t h X an d 0 for a range of b o s o n star s o l u t i o n s . Beca u s e X an d 0 are not n e c e s s a r i l y the same, however, there is some a s ymmetry betw e e n the P- and the x-axes in the SSLB c o n f i g u r a t i o n . This means that 0(4) s y m metry w ould not be a legitimate symmetry of the bubble solution. Samuel and H i s c o c k alluded to the notion of gravity "pulling in" the bubble wall, deforming it from 0(4) spacetime is not exactly f l a t .9 in the symmetry, w h e n the The trajectory of the bubble x - P p lane deforms to an ellipse; the b u b b l e has only 9Samuel and Hiscock, "Compact Objects," 4416. 81 0(3) symmetry. A similar de f o r m i t y might occur in the SSLB case due to a s ymmetry in surface tension. Classically, surface tension in a stretched m e m brane depends u pon a difference in the energy dens i t y betw e e n its equilibrium state an d the stretched s t a t e .10 In bubble nucle a t i o n the surface tension is due to the difference in the scalar field b e t w e e n bu b b l e exterior and bu b b l e interior. the difference in field values at the difference at (T,0). Thus, two points will b e different, 0(4) symmetry. to unity, However, (0, X ) If % > 0, then will b e greater than the surface tensions at those and the bubble will not enjoy if % a n d 0 are b o t h small compared then the surface tensions at (0,X ) a n d at (T, 0 ) will not be substantially different from the tension in the empty space case. Therefore they can each be neglected, though the a s y mmetry is still formally present. case, then, even In the SSLB an 0(4) bubble will b e a good approximation to the actual bubble, a n d the circular trajectory of the bubble wall in E u clidean space will be appropriate. One final m e n t i o n of the ratio %: The integrals for the wall action, tunnelling. (4.14) and (4.17), d e p e n d on the endpoints of A l t h o u g h I shall use the Coleman w all action integral, w h o s e limits are C2 a n d zero, when a b o s o n star nucleates the bubble, the wall action is a c t ually an integral from C2 to G t, even in the SSLB case. 10a .L . Fetter and J.D. Walecka, Theoretical Meohanics of Particles and Continue (New York: McGraw-Hill, 1980) , 271. 82 S?- Jv 2V+ da . (4.25) a0 If one rescales the variable of integration to u = a / a 2, then the wall a ction limits of integration are from u n i t y to %, viz. % (4.26) I In the limit of small %, the u pper limit will b e close to zero. Thus, in the SSLB case, the change in the wall action due to the b o s o n star will be negligible. useful result, This is another for the wall action is identified as the surface tension of the b u b b l e . Introduction of the boson star does not change the surface tension of the 0(4) bubble apprec i a b l y in the SSLB case, so the bubble will b e the same size as the bubble that forms spontaneously in an empty spacetime, equation (4.18). (A greater surface tension might have shrunk the bu b b l e symmetrically, 0(4) s y m m e t r y .) a s sumption while p r e s e r v i n g its This par t i a l l y justifies the use of working (i). * * * * * * * 83 Summary of Simplifications in the Small-Y Limit In summary, the small-% limit describes the SShB type of nucle a t i o n e v e n t . In the SSLB case, it is appropriate to use Coleman's thin-wall approximation for the bubble w h i c h has b e e n n u c l e a t e d b y the b oson s t a r . The small-% limit allows a simpler construction of the bu b b l e solution, solution, <7(x) : graft the tail of the b o s o n star a* (x), onto a step function, the extreme thin-wall limit, at som e v a l u e x t > X the shape of G 0 (x) in doing so pr e c i s e l y at x = X , as in figure 2 1 . not Figure 23 shows the small-% SSLB limit on the shape of the bubble solution. The SSLB simplifies the form of the quant i t y B. integrals are b o t h integrations over all space, the bubble wall, G (x) = G* (x) , b y construction. advantage of this construction, bu t outside To take one divides up S e (G) into an integral over the inside of the bubble wall) (including the bubble a n d an integral over the outside of the bubble, s E (g ) = J d 4x = -Ce (g ) + ' J x > X + x<X viz. £ e (g ) J d 4x x aX The two d 4x £ e (g ) (4.27) 84 0. 11 0.075 - 0.05 a* (0 ) a* (x) 0.025 %=0.002434 Figure 23. A goo d SSLB combination, w i t h % = 0.002434 and 0 = 1x10-6. This is an example of the small-% limit of the m ore general star-bubble combination in figure 2 1 . Here, the parameters of the potential are ^ = 0.54 and A = 300, w i t h the "bump" at Oi = 0.03687 and the true vacuum at Oz = 0.09041. The bubble radius is X = 9.268. The star solution o* (x) has o * (0) = 0.03650, w hich is 99% of O i . The eigenfrequency is Q = 0.827. 85 Similarly, one can divide Se (G*) into a pai r of integrals, a l t hough the integrand is just <7* (x) in b oth pieces, Sn W = J d 4X £ e (g ,) = J d4x £ E(at ) + x<X J d4x £ E(ot ) . viz. (4 .2 8 ) x>X W h e n subtracting Se (G*) from Se (G) , the two integrals over the region outside the bubble cancel exactly in the SSLB case. Therefore, the calculation of the q u a ntity B requires only the integrals inside and including the bubble wall. B = Jd4X l ( G 0) m Jd4X l( G t ) = Sjroleman - Se . in (4.29) (The reader should unde r s t a n d SE* to b e an integral over the bu b b l e interior only.) The calculation of SE* is simple: range of integration on the the circle of radius _ B r> no X. O Coleman oE the x-(3 p l a n e is just the interior of The form of B / B 0 is also s i m p l e : o * C * oE o Coleman aE oE ^ r t Coleman aE ' (4.30) This concludes the summary of simplifications one gains from using the small-% SSLB case. / 86 Methods of Calculation of B N o w I shall explain the details of the calculation of the q u a ntity B for the SSLB case. to calculate, using (4.16). parameters A an d ^ in V the bu b b l e action, Se G iven only the values of (a), (a). The bubble action is easy one can make all calculations of There is some latitude is in the selection of a degenerate potential V + (a). I selected the 2- 3-4 potential w i t h ^=1/2 as the degenerate potential, V + (o) = a2(I - T|*a)2 (4.31) This simplifies the wall action nicely. included, viz. W i t h explicit C t that integral is at wall SE J (c - T|V)da =42 a22[y(%2-l)-yTf o2(x3-l)] (4.32) 0„ In the SSLB case, I will set % to zero in this integral, since it is significantly smaller than unity. This is legitimate because % 2 is the lowest order of % in the integral (4.32). The n substitution of the calculation of Se (C) . and V ( C 2) as in (4.32) into (4.16) One can also calculate completes X from SEwa11 (4.18). The calculation of SE* is a bit trickier, since it 87 requires numerical integration over a two-dimensional region of E u c lidean s p a c e . The full Lagrangian for the b o s o n is 2 m 2M Planck \ B I + A (4.33) v (o.) Because the grav i t y of the b o s o n star is so w e a k in the SSLB combinations I found, w i t h B (r) different from u n i t y b y no m o r e than a few percent in mos t SSLB combinations, I decided to streamline the SE* calculation b y setting A an d B to unity This is equivalent to ma k i n g a flat space calculation. I integrate JHe* over a finite region of the x-|3 plane, the inside of a circle of radius X , u sing x(p) = to describe the circular b o u n d a r y of the region. ( X 2 - |32)1/2 Performing the angular integration leaves a factor of 4jt; converting to dimensionless P an d x integrals supplies a factor of m “4 . The integral for SE* is 2\ ( SE 1 iPlanck 2 V X (P) J dx X2[Q 2Ot (Mplanck)2Zm2 in the expressions for This common factor will cancel from the ratio B / B 0, ma k i n g the results, independent of m, (4.34) o Notice the common factor of g^coieman an(^ g^* _ + v,2 + V (a ,)] as expressed in this ratio, the mass of the scalar field in question. 88 The gist of m y computer c alculation11 of Se* is as follows. First, I calculate a star solution. M y main p r o g r a m for calculating star solutions is b x r e v i s e .bas. I rescale the star solutions and format them for numerical integration in the p r o g r a m ez_sys.bas. The integration p r o g r a m is a c t n trap.bas uses a simple trapezoidal algorithm. The listings of these three programs are in a p p endix A. In some of these calculations, the star solution began to diverge before reaching the bu b b l e wall, chapter 2. W here this occurred, as I m e ntioned in far out along the x-axis, ought to have b e e n in its asymptotic approach to o = 0. those regions, solution. it In I substituted zero for the v alue of the I judged that the action calculation w o u l d not change significantly u nder this substitution. Boson Stars Efficiently Nucleate First Order Phase Transitions, in the SSLB Limit. Ha v i n g calculated the ratio B / B 0 for a large range of star-bubble combinations, I found that the SSLB nucleation process is mor e efficient at n u cleating bubbles of true v a c u u m than is the spontaneous formation process of Coleman. There are two ways to show this increase in efficiency. 11W-H. Press et a l ., Numerical Recipes in FORTRAN: The Art of Scientific Computing. 2nd ed. (New York: Cambridge University Press, 1992), 704, 708, 130. 89 One w a y to express the increase in e f ficiency is by comparing a set of SSLB combinations at fixed % an d A, p u t i n g B / B 0 for a family of star solutions. Figure 24 displays this comparison w i t h Gir(O) up to 99% of O 1 . figure, co m ­ For that I u s e d star-bubble combinations that me t the SSLB criterion. The curves in figure 24 show that B / B 0 decreases as the ratio a* (O ) /G1 increases. The nucleation of the phase transition is m o r e efficient b y stars w i t h larger values of Gir(O) . These tend to be smaller stars, w i t h steeper field profiles. This is one of the reasons I concentrated on small s t a r s : they are be t t e r at n ucleating v a c u u m p hase transitions. However, than one percent; three percent. wall limit, for ^ = 0.52, Thus, 5=1/2, few p e r c e n t . for % = 0.51, the decrease is less the decrease is less than the gain in efficiency nea r the thin is measurable, an d B differs from B 0 b y a That decrease b y a few percent m i g h t seem u n ­ remarkable until one realizes that B belongs in the exponent­ ial par t of F, a n d a few percent in an exponential can make for a large effect. For example, combination for w h i c h B / B 0 is 97%, consider a star-bubble and consider the coeffic­ ient A to be approximately the same for the SSLB process and the e mpty space process b e 2; then B is 1.94. _T_ = F0 Ae~B A 0e B| (i.e. , A « A 0 .) . Let the value of B 0 The ratio of the n u cleation rates is B —B ~e .0.06 1.06 . (4.35) 90 1.000 0.990- B/B 0.980- 0.970 0.000 0.250 0.500 0.750 1.000 Figure 24. The relative gain in efficiency for SSLB vs. the Coleman process. The upper curve is for star-bubble combinations in a potential w ith E1 = 0.51 and A = 300, the lower curve, for a potential w i t h ^ = 0.52 and A = 300. 91 That means that star-bubble combination induces a mer e 6% gai n in the nucl e a t i o n rate. 200. However, T hen the ratio F Z F 0 is about 403, let the v alue of B 0 be a gain in the n u cleation rate of over forty thousand p e r c e n t ! B o t h curves in figure 24 tend toward B / B 0 = I, as o*(0) tends to zero, as one might e x p e c t . W h e n a* (0) = 0, one no longer has a b o s o n star, pe r se; the tunnelling is from false to true v a c u u m -- the Coleman process. Therefore, in the limit of a* (0) = 0, one ought to expect that B Z B 0 reaches unity. No t i c e that in b o t h curves in figure 24, the biggest decrease in B Z B 0 occurs for higher values of a* (0) Zd1 and for higher values of in ^ a n d A, This p a t t e r n is true over a large range so I deci d e d to look onl y at star-bubble combinations w i t h large fixed values of a* (0)Za1; I selected a* (0) = 0.99 G 1, although some star-bubble combinations with G* (0) values lower than that can mee t the SSLB criterion. w a n t e d to see if B Z B 0 d e c reased as % increased. this is true. I I found that This is the second w a y of showing the increase in the bubble n ucleation rate. Figure 25 shows a pai r of curves of B Z B 0 versus increasing for two values of A. As increases the ratio B Z B 0 decreases significantly from unity. The b o s o n star wins the bubble p roduction race. Recall that one limited w a y to interpret the parameter % is as an inverse t e m p e r a t u r e . In this interpretation, increasing % from 1Z2 corresponds phys i c a l l y to a decrease in 92 0.95 0.85 0.75 0.65 0.55 0.51 0.52 0.53 0.54 0.55 0.56 Figure 25. The ratio B / B q for two different values of A. The efficiency of the SSLB nucleation process is significantly greater than spontaneous bubble formation in empty space. 93 the temperature of the e arly universe. b e l o w the critical temperature, As the universe cools nucleation of a phase transition b y a b o s o n star increases in likelihood. Before going on, I must rate the star-bubble combin­ ations that I u s e d to prod u c e figure 25. Was the value of % I sufficiently small? W hat about the v alue of 0? Table I shows the v a l u e of % an d 0 for several of the solutions from w h i c h I p l o t t e d figure 25. Table I. Values of % a n d 0 for several SSLB combinations. A A = 300 = 10 % X 0 X 0 0.51 < 6 x IO"6 < 10"24 < 8 x 10-6 < IO"22 0.52 < 2 x 10-5 < 10-13 < 9 x ID-5 < H r 12 0.53 9 x 10-5 7 x !O'? 1.6 x IO-4 2 x 10-9 0.54 1.7 x 10-3 6 x !Cr? 2.4 x IQ-3 I x 10-6 0.55 0.01 3 x IQ-5 0.013 5 x IO"5 0.56 0.034 4 x IO-4 0.04 6 x ICT4 In figure 25, I e nded the range of ^ at 0.56, since this corresponds to a v alue of % equal to a few percent. There is a w a y to judge w h e t h e r % is sufficiently small to justify the us e of the approximations in the SSLB limit. integral for the wall action, % as the upper limit, (4.26). Consider the I explicitly included so I denote that integral as SEwall(%). 94 If I h a d set the u pper limit to zero, the integral w o u l d be the wall action for the empty space bubble formation p r o c e s s . I denote this action as SEwall(0) The ratio of these two i n t e g r a l s , Se wal1 (%)/SEwa11 (0) , should approach u n i t y in the small-% limit. I calculated this ratio for each of the star- bubble combinations in table I . For none of those combinations is the ratio less than 0.9912. M ost of the combinations have a ratio larger than 0.9999, w i t h the % = 0. 51 ratios b e i n g the closest to unity. Therefore, the use of the thin wall bub b l e in the SSLB limit is justified. Table 2 shows the dimensionless energy difference b e t w e e n the v a c u a , -V(O2) , for the same solutions as in Table 1. This dimensionless energy difference is the conventional m e a s u r e of a p p roach to d e generacy of the p o t e n t i a l . also sh o w e d 12 that w h e n process is valid. X » Therefore, Coleman I, his thin-wall approximation table 2 also shows the dimensionless bub b l e radius, X . Not i c e in table 2 .that -V(G2) is not larger than 0.03 for an y of the c o m b i n a t i o n s . This verifies that the potentials in use are fairly close to degeneracy, as indeed they mus t be for the use of the thin-wall a p p r o x i m a t i o n . m i n o r caveat: One Samuel a n d Hiscock called the robustness of the thin-wall approximation into q u e s t i o n ,13 comparing Coleman's 12Coleman, 13Samuel, "Fate of the false vacuum," 2933. "Thin-wall approximation," 254. 95 approximate bubble action to the action of the exact bubble solution. T h e y found the thin-wall approximation to be rather un s a t i s f a c t o r y for -V(G2) greater than about 0.02. That caveat is not a m a j o r cause for w o r r y here, though, for -V(G2) is larger than 0.02 in o nly two of the combinations at A = 10. In the A = 300 case, the thin-wall approximation is quite good. Table 2. Conventional v a l i d i t y measures for Coleman's thinwall approximation, over a range of SSLB combinations. A = 10 A = 300 S X -V(O2) X -V(O2) 0.51 46.93 4.2 x 10~3 46.93 1.4 x 10™4 0.52 21.87 8.6 x IO'3 21.87 2.9 x 10™4 0.53 13.48 0.013 13.48 4.5 x IO'4 0.54 9.268 0.018 9.268 6.2 x IO"4 0.55 6.728 0.024 6.728 8.0 x 10'4 0.56 5.026 0.030 5.026 9.8 x IO'4 96 Summary and Conclusions I h ave u s e d the 2-3-4 scalar potential to m o d e l finitetemperature effective potentials in w h i c h first order phase transitions are possible. I examined the process of n ucleation of a p hase transition b y a boson star, a non­ v a c u u m configuration of the scalar field, w h e n the potential is fairly close to d e g e n e r a c y . I focused on a special subset of all possible combinations of b o s o n star and v a c u u m bubble; I call this subset the "small-star-large bubble" this limit, limit. For I crafted an approximation scheme that h a d several convenient simplifications. In that limit I c a lculated the q u a ntity B in the bubble formation rate per unit four-volume, F/V. I have shown that the n ucleation of v a c u u m p h a s e transitions b y b o s o n stars is mor e efficient than the empty space bubble formation process of Coleman. In fact, w h e n the system is supercooled b e l o w the critical temperature, the increase in the bubble formation rate is significant w h e n b o s o n stars are present. It is reasonable to have found a significant increase in the bu b b l e nucle a t i o n rate for b o s o n stars in the SSLB limit. As I m e n t i o n e d earlier in this chapter, found that "impurities" other scientists have in the form of other astrophysical objects h ave an enhancing effect on the n ucleation r a t e . b o s o n star is the latest addition to that list of exotic The 97 astrophysical objects. Another reasonable result is that this increase in nucleation rate is more pronounced for larger values of Z3, which are equivalent to potentials with a deeper w e l l . That is, one would expect the field to tunnel more easily when either a deep well is present or a low barrier is present. One can also say that the presence of the b o s o n star is a w a y of m a k i n g the potential b a r r i e r smaller. Since G t > 0, the field tunnels from part w a y up the potential barrier, instead of from the b o t t o m of the potential well at For this reason, G = 0. one can also say that the results in figure 25 are reasonable. In fact, for a v e r y small investment in %, climbing the potential barrier a v e r y small a m o u n t , one gains a windfall of enhancement in the nucleation rate. In chapter 5, I will have mor e to say about other star-bubble configurations w i t h greater values of %. However, the windfall enhancement of the n u cleation rate only occurs for potentials farther and farther removed from degeneracy. The potential moves awa y from d e generacy as the temperature of the universe drops b e l o w the critical temperature of the field theory in question. W ill the universe cool m o r e rapidly than the field will tunnel? Using the 2-3-4 potential only as a m o d e l , and without a particular field theory (e.g., e l e c t r o w e a k ) , it is not p o s s i b l e to calculate the exact n u cleation r a t e . (One needs to calculate the coefficient A, w h i c h is a thorny matter indeed.) 98 N o n e t h e l e s s , the b o s o n star's enhancement of the nucleation rate presents an interesting effect. For once the temperature of the system dips pas t the critical temperature, an y b o s o n stars present will have acquired a decided advantage in the race to produce bubbles of true v a c u u m . The SSLB limit describes onl y a subset of all the p o s sible bubble - s t a r c o m b i n a t i o n s . There are several interesting classes of star-bubble combinations outside the SSLB limit. Also, there are several refinements one can also m ake to the SSLB calculations I hav e done. will discuss these topics briefly. In chapter 5 I 99 \ CHAPTER 5 W H A T A D D I T I O N A L TASKS R E M A I N FOR THE FUTURE? This thesis covers the SSLB limit, w h i c h is o nly one subset of all p o s sible star-bubble combinations. Several other interesting types of star-bubble combination exist. The first type is n ucleation of a v a c u u m p h a s e transit­ ion b y a b o s o n star that is m u c h larger than the bubble. the bu b b l e is small enough, If then the star solution will be n e a r l y constant over the interior of the b u b b l e . Figure 26 shows a closeup v i e w of such a star-bubble combination. The bu b b l e wal l wil l b e small compared to the star for potentials v e r y far from degeneracy, w h e r e the Coleman thin-wall a p p r oximation loses validity. Therefore, the alternate thin- wall a p p r oximation of Samuel a n d Hisc o c k m ight b e acceptable. If that approximation turns out to be unacceptable, one must calculate the bu b b l e solution exactly, b y integrating (4.9). This type of nucle a t i o n event m a y b e approximated b y a* (x) == a* (O) over the inside of the bubble, the star action quite easy. making the calculation of W i t h-out contributions from the da*/dx t erm in the L a g r a n g i a n , the star action inside the bu b b l e wil l d i m inish m a r k e d l y compared to the SSLB star action, w h i c h was calculated over mos t of the spatial extent of the star. Therefore, this type of star-bubble combination 100 might y i e l d a v e r y significant decrease in B, an d a significant increase in the bubble formation rate. 0.025 0.02 0.015 0.01 0.005 x Figure 26. Star bubble combination in w h i c h the bubble is m u c h smaller than the star. The solid line shows the bubble solution CT(x); the da s h e d lines show the star solution CT* (x) inside the bubble and the remainder of CTo (x) outside the b u b b l e . Notice that the tunnelling will begin from CT = CTt quite close to CT* (0), so % will not be small as in combinations of the SSLB type. Anot h e r interesting type of star-bubble combination is one in w h i c h the star and the bubble are roughly the same size. As Samuel and Hiscock s h o w e d ,1 gravitationally compact objects enhance the bubble nucleation rate w h e n the object and the bubble are of comparable size. 1Samuel and Hiscock, "Compact objects," 4416. This k i n d of 101 nucleation, w i t h star and bubble roughly the same size, does not require a potential w ith a drastic departure from degeneracy. Inside the bubble, value significantly, the star solution will change unlike the previous type. Tunnelling will occur from a = G t, w h i c h will be somewhere between a, (0) and z e r o . this kind. In figure 27, I show a star-bubble combination of The reservations about approximations for the large star-small bubble combination also apply for the case of a star of roughly the same size as the b u b b l e .. 0.09 0 .08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 2 4 6 8 10 12 x Figure 27. A b ubble-star combination in w hich the star and the bubble are of comparable size. 102 In b o t h types of n u cleation events, one will have to mak e a serious m o d i f i c a t i o n to the shape of the bubble wall t rajectory in the x-|3 plane. The reason for this is that the surface tension a s y mmetry will not be negligible in these two types of nucle a t i o n e v e n t s . Thus, one must expect the bubble to d e f o r m significantly from the 0(4) useful in the SSLB l i m i t . onl y elliptical on the expand as rapidly, symmetry that was so If the bubble wall trajectory is x-|3 plane, then the bubble will not a n d the bubble wall will n o t asym p t o t i c ­ all y approach the speed of light b u t some smaller terminal velocity. The bubble wall trajectory might b e even more compli c a t e d — plane. for instance, it m i g h t be periodic on the In a n y case, x-|3 one must develop a quantitative met h o d to ha n d l e the surface tension a s ymmetry and its effect on the bubble w all trajectory. * * * * * * * This concludes m y thesis, in w h i c h I hav e shown how b o s o n stars e fficiently nucleate v a c u u m phase transitions. However, I h ope it is not the end of the study of this small bu t interesting problem. For cosmology and astrophysics are small parts in the general scientific enterprise, the noble quest for u n d e rstanding of the w i d e w o r l d w h i c h G o d created. 103 REFERENCES CITED A l p h e r , R.A . , a n d Herman, R. "Evolution of the U n i v e r s e , " Nature, 162, 774 (1948). Arnett, A . S., W e a t h e r Mod i f i c a t i o n b v Cloud S e e d i n g . Ne w York: A c a demic P r e s s , 1980. Banks, T., Bender,C.M., and W u , T.T., "Coupled Anharmonic Oscillators. I. Equal-Mass C a s e , " Physical R e v i e w D, 3346 (1973). 8., Berezin, V . A . , Kuzmin, V . A . , an d Tkachev, I.I., "Black holes initiate false-vacuum d e c a y , " Physical R e v i e w D 43, R3112 (1990). Callan, C.G., a n d Coleman, S., "Fate of the False V a c u u m . II. First Q u a n t u m C o r r e c t i o n s , " Physical R e v i e w D, 16, 1762 (1977). Coleman, S., "Fate of the false v a c u u m : Semiclassical t h e o r y , " Physical Re v i e w D, 15, 2929 (1977). 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J e t z e r , P., "B oson s t a r s , " Physics Report, 2 2 0 . 163 K a u p , D .J ., "Klein-Gordon G e o n ," Physical Review, (1968) . (1992) 1 7 2 . 1331 104 L a n d a u , L.D., an d L i f s h i t z , E.M., Statistical Physics I ) . trans. J.B. Sykes and M.J. K e a r s l e y , 3rd ed. Oxford: Pergamon Press, 1980. Lee, (Part T . D . , an d Pang, Y., "Nontopological s o l i t o n s , " Physics R e p o r t s , 221, 251 (1992). Linde, A . , Particle Phvsics an d Inflationary C o s m o l o g y . N e w York: H a r w o o d Academic Publishers, 1990. M e n d e l l , G., an d Hiscock, W . A . , "Gravitational n ucleation of v a c u u m p hase transitions b y compact o b j e c t s , " Physical R e v i e w D 39, 1537 (1989). M i s n e r , C.W., T h o r n e , K.S., a n d Wheeler, San F r a n c i s c o : Freeman, 1973. J.A., Gravitation. O p p e n h e i m e r , R., a n d V o l k o f f , G.M., "On Mass i v e Neutron Cores," Physical Review, 55., 374 (1939) . Peebles, P.J.E., Principles of Physical C o s m o l o g y . Princeton: Princeton U n i v e r s i t y Press, 1993. P e n z i a s , A.A., and Wilson, R.W., "A M easurement of Excess A n t e n n a Temperature at 4080 M c / s , " Astrophysical Journal, 142, 419 (1965). { Press, W . H . , et a l ., Numerical Recipes in FORTRAN: The Art of Scientific C o m p u t i n g . 2nd ed. N e w York: Cambridge U n i v e r s i t y Press, 1992. R u f f i n i zR . , an d Bonazzola, S., "Systems of Self-Gravitating Particles in General Relat i v i t y and the Concept of an E q u ation of S t a t e , " Physical Review, 1 8 7 . 1767 (1969). S a m u e l , D.A., an d Hiscock, W.A. , "Gravitationally compact objects as n u cleation sites for first-order v a c u u m phase t r a n s i t i o n s , " Physical R e v i e w D, 45, 4411 (1992). Samuel, D . A . , an d Hiscock, W . A . , " 'Thin-wall' approximations to v a c u u m d ecay r a t e s , " Physics Letters B, 2 6 1 , 251 (1991). Wald, R . M . , General R e l a t i v i t y . Chicago: U n i v e r s i t y of Chicago Press, 1984. Wheeler, J.A., "G e o n s ," Physical Review, 92, 511 (1955). 105 APPENDIX COMPUTER PROGRAMS I wrote these programs in the Quick Basic language. The first listing is for bxrevise.bas, the principal program I used for calculating boson star solutions. The second listing is ez_sys.bas, which I used to rescale the boson star solutions and format them for numerical integration. The third listing is for actntrap.bas, which calculated the action integrals. The specifications of a particular boson star solution, i.e. , eigenfrequency £2 and B(°°), are output from bxrevise.bas and input for ez_sys.bas. Data files constructed by ez_sys.bas, containing boson star solutions, were input for actntrap.b a s . =#= PQ 'bxrevise.bas DECLARE SUB headr (eta#, LAMBDA#, OMEGA#, s0#, pass#) DECLARE FUNCTION Sbarl# (s0#, eta#, LAMBDA#) DECLARE FUNCTION BubbleX# (s0#, eta#, LAMBDA#) DECLARE FUNCTION Vbar# (s#, eta#, LAMBDA#) DECLARE FUNCTION F# (x#) DECLARE FUNCTION epsilon# (eta#, LAMBDA#) DECLARE FUNCTION Rbubble# (eta#, LAMBDA#) DECLARE FUNCTION bump# (eta#, LAMBDA#) DECLARE FUNCTION truevac# (eta#, LAMBDA#) DECLARE FUNCTION DVAC# (LAMBDA#, eta#) DECLARE SUB vps2 (xx#(), ss#(), size#) DECLARE SUB SORTN (aa#(), n#) DECLARE SUB vps (xx#(), S S # (), size#) DECLARE SUB SORT4 (d#()) OMEGA#, eta#, DECLARE FUNCTION S41# (x#, S # , V#, A#, sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S42# (x#, s#, v#. A#, B#, OMEGA#, eta#, BO#, sk31#, sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S43# (x#, S#, V#, A#, B#, OMEGA#, eta#, sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S44# (x#,‘s#, V#, A#, B#, OMEGA#, eta#, sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S31# (x#, S#, V#, A#, B#, OMEGA#, eta#, sk22#, sk23#, sk24#, h#) DECLARE FUNCTION S32# (x#, s#, v # , A#, B#, OMEGA#, eta#, BO#, sk21#, sk22#, sk23#, sk24#, h#) DECLARE FUNCTION S33# (x#, S#, V#, A#, B#, OMEGA#, eta#, sk2,2#, sk23#, sk24#, h#) DECLARE FUNCTION S34# (x#, S#, V#, A#, B#, OMEGA#, eta#, sk22#, sk23#, sk24#, h#) LAMBDA#, sk31#, LAMBDA#, sO#, LAMBDA#, sk31#, LAMBDA#, sk31#. LAMBDA#, sk21#. LAMBDA#, sO#, LAMBDA#, sk21#, LAMBDA#, sk21#. 106 DECLARE FUNCTION S21# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, skll# skl2#, skl3#, skl4#. h#) DECLARE FUNCTION S22# (x#, s#. V#, A# j B#, OMEGA#, eta#. LAMBDA#, sO#, B0#, skll#, skl2#, skl3#, skl4#, h#) DECLARE FUNCTION S23# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, skll# skl2#, skl3#, skl4#, : h#) DECLARE FUNCTION S24# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, skll# skl2#, skl3#, skl4#, !h#) DECLARE FUNCTION Sll# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, h#) DECLARE FUNCTION S12# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, sO#, B0#, h#) DECLARE FUNCTION S13# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, h#) DECLARE FUNCTION S14# (x#, s#. V#, A#, B#, OMEGA#, eta#. LAMBDA#, h#) DECLARE FUNCTION DV# (x#, S#, V#, A#, B#, OMEGA#, eta#, LAMBDA#, s0#, BO#) DECLARE FUNCTION DS# (x#, S # , v#. A#, B#, OMEGA#, eta#, LAMBDA#) DECLARE FUNCTION DB# (x#, s#, v # , A#, B#, OMEGA#, eta#, LAMBDA#) DECLARE FUNCTION DA# (x#, s#, v#. A#, B#, OMEGA#, eta#, LAMBDA#) DEFDBL A-Z 'The several arrays for my Runge-Kutta adaptive step size procedure. DIM SHARED x out(0 TO 500): DIM SHARED xdata(0 TO 500) DIM SHARED xlout(0 TO 500): DIM SHARED x2out(0 TO 500) DIM SHARED sout(0 TO 500): DIM SHARED sdata(0 TO 500) DIM SHARED slout(0 TO 500): DIM SHARED s2out(0, TO 500) DIM SHARED v o u t (0 TO 500): DIM SHARED vdata(0 TO 500) DIM SHARED vlout(0 TO 500): DIM SHARED v2out(0 TO 500) DIM SHARED A o u t (0 TO 500): DIM SHARED Adata(0 TO 500) DIM SHARED Alout(0 TO 500): DIM SHARED A2out(0 TO 500) DIM SHARED Bout(0 TO 500): DIM SHARED Bdata(0 TO 500) DIM SHARED Blout(0 TO 500): DIM SHARED B2out(0 TO 500) DIM DIM DIM DIM DIM SHARED SHARED SHARED SHARED SHARED Mdata(0 TO 500) d(l TO 4) dd(l TO 500): DIM SHARED cc(l TO 500) xx(l TO 500): DIM SHARED ss(I TO 500) aa(l TO 500) CLS 'File #1 holds entire solutions for one or mere passes across sO values. ff$ = "a:\dxx26.prn" OPEN ff$ FOR APPEND AS #1 PRINT #1, ff$, DATE$, TIME$ PRINT #1, "parameters = {eps, scales, scalev, scaleA, scaleB, OMEGAtol, passes}" PRINT #1, " {OMEGA, eta, LAMBDA, xi, bump, s (0):bump, X}" 107 PRINT #1, "data = B (infinity)}" {x, s (x), v(x), I - A(x), I - B(x), M(x), I - 'File #2 holds solution parameters and the target OMEGA value. gg$ = "a:\nudata26.prn" OPEN gg$ FOR APPEND AS #2 PRINT #2, gg$, "data summary from bxrevise.has", DATE$, TIME$ PRINT #2, 'energy parameters {OMEGA,eta,LAMBDA} CONST LAMBDA = 100 FOR g = 0 TO 20 'q counts the number of levels of eta of the solutions eta = SQ R (LAMBDA * (.51# + q * .0025)) 'Setting eta = 0 will compare with Colpi, et a l ., Fig.(3) 'Setting LAMBDA = 0 will compare with Ruffini and Bonazzola figures. 'BOUNDARY CONDITIONS: s(0),v(0),A(O),B(O). 'Central redshift is OMEGAA2/B(0); adjust so that B("infinity") -> I. ssl = bump(eta, LAMBDA): ss2 = truevac(eta, LAMBDA) FOR r = 0 TO 0 STEP I 'r counts the number of sO values between 0 and ssl. sO = (.99# + .1# * r) * ssl: BO = I vO = 6: AO = I : xO = 0 1Runge-Kutta adaptive step size parameters. CONST safety = .9: CONST eps = .000002# OMEGAtol = .0000000000000002# h = .2: maxsteps = 300 minshrink = .I : minboost = 1.05: maxboost = 5 scales = sO: scalev = -.7: scaleA = 4: scaleB = .1 PRINT #1, TIME$ 'OMEGA bisection parameters. OMEGAl = .01: OMEGA2 = 1 . 2 OMEGA = OMEGA2 FOR p = l TO 60 'p counts the number of OMEGA bisection passes OMEGACOPY = OMEGA x = x O : s = sO: v = v O : A = A O : B = BO headr eta, LAMBDA, OMEGA, sO, p FOR i = I TO maxsteps 'i counts out the Runge Kutta solution steps, 'including "false" steps. 108 '============================================== '======== RUNGE KUTTA SOLUTION SECTION ======== 'TheI U (i ,j ) are Ull = Sll(x, s , Ul 2 = S12(x, s , U13 = S13(x, s, Ul 4 = S14(x, s , the solution parts after one v. A, B, OMEGA, eta, LAMBDA, v. A, B, OMEGA, eta, LAMBDA, v. A, Bj OMEGA, eta, LAMBDA, v. A, Bj OMEGA, eta, LAMBDA, full step. h) sO, BO, h) h) h) U21 = S21(x. s, V, A, B, OMEGA, eta, LAMBDA, Ull, U12, U13, Ul4, h) U22 = S22(x. s, V, A, B, OMEGA, eta, LAMBDA, sO, BO, Ull, U12, U13, U14, h) U23 = S23(x. s, V, A, B, OMEGA, eta, LAMBDA, Ull, U12, U13, U14, h) U24 = S24(x. s, V, A, B, OMEGA, eta, LAMBDA, Ull, U12, U13, U14, h) U31 = S31(x. s, V, A, B, OMEGA, eta, LAMBDA, U21, U22, U23, U24, h) • U32 = S32(x. s, V, A, B, OMEGA, eta, LAMBDA, sO, BO, U21, U22, U 2 3 , U24, h) U33 = S33(x. s, V, A, B, OMEGA, eta, LAMBDA, U21, U22, U23, U24, h) U34 = S34(x, s, V, A, B, OMEGA, eta, LAMBDA, U21, U22, U23, U24, h) U41 = S41(x. s, v. A, B, OMEGA, eta, LAMBDA, U31, U32, U33, U34 CO O h) U42 = S42(x, s, v. A, B, OMEGA, eta, LAMBDA, BO, U31, U32, U33, U34, h) U43 = S43(x. s, v. A, B, OMEGA, eta, LAMBDA, U31, U32, U33, U34 h) U44 = S44(x. s, v. A, B, OMEGA, eta, LAMBDA, U31, U32, U33, U34 h) 'The arrays siout(), etc. sIo u t (i) S + (Ull + 2 * vlout(i ) = V + (U12 + 2 * Alout(i) A + (U13 + 2 * B + (U14 + 2 * Blout(i) xlout(i) = X + h 'The Y(i,;j) are Yll = Sll(x. s, Y12 = S12(x. s, Yl 3 = S13(x. s, Yl 4 = S14(x. s, are U21 U22 U23 U24 the + 2 + 2 + 2 + 2 solutions after one full step * U31 + U41) / 6 * U32 + U42) / 6 * U33 + U43) / 6 * U34 + U44) / 6 the solution parts after one v. A, B, OMEGA, eta. LAMBDA, v. A, B, OMEGA, eta. LAMBDA, v. A, B, OMEGA, eta. LAMBDA, v. A, B, OMEGA, eta. LAMBDA, half-step. h / 2) sO , BO, h / 2) h / 2) h ,/ 2) 109 Y21 = S21(x. s, v. A, B, OMEGA, eta. LAMBDA, Yll, Y12, Y13, Y14, h / 2) Y22 Y13, Y14, h Y23 h / 2) Y24 h / 2) = S22(x. s, v. A, B, OMEGA, eta. LAMBDA, sO, BO, Yll, Y12, / 2) = S23(x. s, v, A, B, OMEGA, eta. LAMBDA, Yll, Y12, Y13, Y14, = S24(x. s, v, A, B, OMEGA, eta. LAMBDA, Yll, Y12, Y13, Y14, Y31 = S31(x. s, v. A, B, OMEGA, eta. LAMBDA, Y21, Y22, Y23, Y24, h / 2) Y32 = S32(x. s, v. A, B, OMEGA, eta. LAMBDA, sO, BO, Y21, Y22, Y24, h / 2) Y23, Y33 _ S33(x. s, v. A, B, OMEGA, eta. LAMBDA, Y21, Y22, Y23, Y24, h / 2) Y34 = S34(x. s, V, A, B, OMEGA, eta. LAMBDA, Y21, Y22, Y23, Y24, h / 2) Y41 = S41(x. s, V, A, B, OMEGA, eta. LAMBDA, Y31, Y32, Y33, Y34, h / 2) Y42 = S42(x. s, V, A, B, OMEGA, eta. LAMBDA, sO, BO, Y31, Y32, Y33, Y34, h / 2) Y43 = S43(x. s, V, A, B, OMEGA, eta. LAMBDA, Y31, Y32, Y33, Y34, h / 2) Y44 S44(x. s, V, A, B, OMEGA, eta. LAMBDA, Y31, Y32, Y33, Y34, h / 2) 'The values shalf = S + vhalf = V + Ahalf = A + Bhalf = B + xhalf = X + shalf, (Yll + (Y12 + (Y13 + (Y14 + h / 2 etc:. are the solution after one half-step. 2 * Y21 + 2 * Y31 + Y41) / 6 2 * Y22 + 2 * Y32 + Y42) / 6 2 * Y23 + 2 * Y33 + Y43) / 6 2 * Y24 + 2 * Y34 + Y44) / 6 'The W(i,j) are the solution parts after another half-step. Wll = Sll(xhalf. shalf, vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA, h / 2) W12 _ S12(xhalf, shalf. vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA, sO, BO, h / 2) Wl 3 = S13(xhalf. shalf. vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA, h / 2) W14 _ S14(xhalf. shalf^ vhalf, Ahalf, Bhalf, OMEGA, eta. LAMBDA, h / 2) S21(xhalf. shalf. W21 Wll , W1 2 , W 13, W14, h / 2) W22 = S22(xhalf, shalf. sO, BO, W l l , W12, W 1 3 , W14, h / W23 = S23(xhalf. shalf. W l l , W12, W13, W14, h / 2) vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA, vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA 2) vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA HO W24 = S24(xhalf, shalf, vhalf, Ahalf, Bhalf, OMEGA, eta. LAMBDA, W l l , W12, Wl3, W14, h / 2) W31 = S31(xhalf, shalf, W21, W2 2 , W23, W24, h / 2) W32 = S32(xhalf, shalf, sO, BO, W2 1 , W22, W23, W24, h / W33 = S33(xhalf, shalf, W21, W2 2 , -W23, W24, h / 2) W34 = S34(xhalf, shalf, W 2 1 , W 2 2 , W23, W24, h / 2) vhalf, Ahalf, Bhalf, OMEGA, eta. LAMBDA, vhalf, Ahalf, Bhalf, OMEGA, eta, LAMBDA, I 2) vhalf. Ahalf, Bhalf, OMEGA, eta, LAMBDA, vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA, O CO W41 = S41(xhalf, shalf, vhalf. Ahalf, Bhalf, OMEGA, eta. LAMBDA, W31, W 3 2 , W33, W34, h / 2) W42 = S42(xhalf, shalf, vhalf, Ahalf, Bhalf, OMEGA, eta. LAMBDA, BO, W31, W32, W33, W34, h / 2) W43 = S43(xhalf, shalf, vhalf. Ahalf, Bhalf, OMEGA, eta, LAMBDA, W31, W32, W 3 3, W34, h / 2) W44 = S44(xhalf, shalf, vhalf, Ahalf, Bhalf, OMEGA, eta. LAMBDA, W31, W3 2 , W 3 3, W34, h / 2) 'The values s2, etc. s2 = shalf + (Wll + v2 = vhalf+ (W12 + A2 = Ahalf + (W13 + B2 = Bhalf + (W14 + are the 2 * W21 2 * W22 2 * W23 2 * W24 solutionsi after + 2 * W31 + W41) + 2 * W32 + W42 ) + 2 * W33 + W43 ) + 2 * W34 + W44) another■ half:-step. / 6 / 6 / 6 / 6 ADAPTIVE STEP SIZE SUBSECTION 'error estimates scaled by sO, maximum slope in Ruffini and Bonazzola 'Fig.(6), 2 * A(O), and B ("infinity") = I. d(l) d(2) d(3) d(4) = A B S ((s2 = A B S ((v2 = A B S ((A2 = A B S ((B2 - slout(i)) - vlout(i)) - Alout(i )) - Blout(i)) / scales) / scalev) I scaleA) / scaleB) SORT4 d() delta = d(4) IF delta / eps > I THEN hnew = h * safety * (delta / eps) A (-.25) xout(i ) = -I IF delta / eps > (safety / minshrink) A 4 THEN hnew = minshrink * h END IF ELSE Ill IF delta / eps > (safety / minboost) A 5 THEN hnew = minboost * h x = xlout(i) s = slout(i): v = vlout(i) A = Alout(i): B = Blout(i) ELSEIF delta / eps < (safety V maxboost) A 5 THEN hnew = maxboost * h x = xlout(i) s = slout(i): v = vlout(i) A = Alout(i): B = Blout(i) ELSE hnew = h * safety * (delta / eps) A (-.2) x = xlout(i) s = slout(i): v = vlout(i) A = Alout(i): B = Blout(i) END IF xout(i ) = x sout(i) = s : v out(i) = v A out(i) = A: Bout(i ) = B END IF PRINT "h':h = "; PRINT USING "##.#### "; hnew / h; x; s h = hnew END OF ADAPTIVE STEP SIZE SUBSECTION AND THE RUNGE KUTTA SOLUTION SECTION OMEGA BISECTION REDIRECT SECTION 'This section short circuits Runge Kutta when it becomes clear 'that OMEGA fails either low or high. IF v > O THEN 'low failure for OMEGA OMEGAl = OMEGA OMEGA = (OMEGAl + 0MEGA2) / 2 imax = i gogo : -I EXIT FOR END IF IF s < O THEN 'high failure for OMEGA 0MEGA2 = OMEGA OMEGA = (OMEGAl + 0MEGA2) / 2 imax = i 112 gogo = I EXIT FOR END IF '======== END OF OMEGA BISECTION REDIRECT SECTION ======== '========================================================= NEXT i 1Continue Runge Kutta solution. PRINT "gogo = "; gogo 'Test whether successive OMEGA values meet the tolerance setting, OMEGAtol. 'Short circuit the bisection process if tolerance test is m e t . IF A B S (OMEGACOPY - OMEGA) < OMEGAtol THEN makepeace = p PRINT "makepeace @ "; p EXIT FOR END IF NEXT p 'Continue OMEGA bisection search. DATA REARRANGE k = I FOR j = I TO imax IF xout(j) > O THEN xdata(k) = xout(3 ) sdata(k) = sout(j) vdata(k) = vout(j) Adata(k) = Aout(j) Bdata(k) = Bout(j ) Mdata(k) = .5 * xout(j) * (I - I / A o u t (j )) k = k + I END IF NEXT j kmax = k - I Binf = A d ata(kmax) * Bdata(kmax) PRINT "There are kmax; "data points. '======== END DATA REARRANGE ======== ===================—============================================ ======== GRAPH SCALAR FIELD S (x) AND MASS FUNCTION M(x) ======== vps xdata(), sdata(), kmax vps2 xdata(), Mdata(), kmax 113 END SCALAR FIELD AND MASS FUNCTION GRAPH ======== DATA OUTPUT SECTION 'File PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT #2 holds solution parameters and the target OMEGA #2, USING "##.###### ; eta; #2, CHR$(9); #2,' USING "####.###### "; LAMBDA; #2, CHR$(9); #2, USING "##.#### ; eta * eta / LAMBDA; #2, CHR$(9); #2, USING "##.#### ; sO / ssl; #2, CHR$(9); #2, USING "##.##AAAA ; OMEGAtol; #2, CHR$(9); #2, USING "##.#######*########## OMEGA; #2, CHR$(9); #2, USING "##.###### "; Binf #2, CHR$(9); #2, USING "##.####AAAA "; BubbleX(O , eta, LAMBDA) 'File #1 holds entire solutions PRINT #1, "bxrevise.bas output" PRINT #1, USING "##.####AAAA " ; PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA "; PRINT #1, CHR$(9); PRINT■#1, USING "##.####AAAA PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA " ; PRINT #1, CHR$(9); PRINT #1, USING "##.###*AAAA "; PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA " ; PRINT #1, CHR$(9); PRINT #1, USING "##.## PRINT PRINT PRINT PRINT PRINT PRINT PRINT PRINT #1, #1, #1, #1, #1, #1, #1, #1, eps ; CSNG(scales); CSNG(scalev); CSNG(scaleA); CSNG(scaleB); OMEGAtol; makepeace USING "##.################## "; OMEGA; CHR$(9); USING "####.###### "; eta; CHR$(9); USING "####.###### "; LAMBDA; ' CHR$(9); USING "####.###### "; eta * eta / LAMBDA; CHR$(9); 114 PRINT PRINT PRINT PRINT PRINT #1, #1, #1, #1, #1, USING "##.####AAAA "; ssl; CHR$(9); , USING "##.####AAAA "; sO / ssl; CHR$(9); USING "##.####AAAA BubbleX(0 PRINT #1, USING "##.####AAAA "; CSNG(xO); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA "; CSNG(sO); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA " ; CSNG(vO); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA CSNG(AO); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA " ; CSNG(BO); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA 0 LOCATE I, I : PRINT x B(x) " FOR k = I TO kmax PRINT USING "##.### "; : xdata(k); Bdata(k) PRINT #1, USING "##.####AAAA CSNG(xdata(k)); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA " ; CSNG(sdata(k)); PRINT #1, CHR$(9); PRINT #1, USING "##.##*#AAAA "; CSNG(vdata(k)); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA CSNG(Adata(k) - I); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA CSNG(Bdata(k) - I); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA " ; CSNG(Mdata(k)); PRINT #1, CHR$(9); PRINT #1, USING "##.####AAAA CSNG((Adata(k) * Bdata(k)) NEXT k PRINT #1, PRINT #1, PRINT #1, NEXT r NEXT q solutions. CLOSE #1 CLOSE #2 END 'Select another value of sO for a new solution. 'Select another value of eta for a new range of 115 'SUB PROCEDURES AND FUNCTIONS OF BXREVISE.BAS FUNCTION BubbleX (sO, eta, LAMBDA) 1Dimensionless radius of the true vacuum bubble. epsilonO = Vbar(s 0, eta, LAMBDA) + epsilon(eta, LAMBDA) BubbleX = 6 * Sbarl(sO, eta, LAMBDA) / epsilonO END FUNCTION FUNCTION bump (eta, LAMBDA) 1Field value at the top of the bump in the potential V bar. xi = eta * eta / LAMBDA IF xi >= .5 THEN bump = 1.5 * xi * ( I - SQR( 1 - 4 / (9 * xi))) / eta ELSE bump = 12.3 END IF END FUNCTION FUNCTION DA (x, s, v. A, B, OMEGA, eta, LAMBDA) 1First derivative of A(x) IF x = 0 THEN DA = 0 ELSE DAl = A * V(1 - A) / X DA2 = (OMEGA * OMEGA / B + I) * x * A * A * S * S DA3 = -2 * eta * x * A * A * s * s * s DA4 = LAMBDA * x * A * A * s * s * s * s / 2 DA = DAl + DA2 + DA3 + DA4 + x * v * v * A END IF END FUNCTION FUNCTION DB (x, s, v. A, B 7 OMEGA, eta, LAMBDA) 'First derivative of B (x) IF x = 0 THEN DB = 0 ELSE DBl = B * ( A - I ) / x DB2 = (-1 + OMEGA * OMEGA / B) * x * A * B * s * s DB3 = 2 * eta * x * A * B * s * s * s DB4 = -LAMBDA * x * A * B * S * s * S * s / 2 DB = DBl + DB2 + DB 3 + DB 4 + x * v * v * B END IF END FUNCTION 116 FUNCTION DS (x, s, v. A, B, OMEGA, eta, LAMBDA) 'First derivative of the scalar field DS = v END FUNCTION FUNCTION DV (x, S, V, A, B, OMEGA, eta, LAMBDA, sO, BO) 1Second derivative of the scalar field, .-.the scalar wave equation! IF x = 0 THEN DVl = (I - OMEGA * OMEGA / BO) * sO / 3 - eta * sO * sO DV = DVl + LAMBDA * sO * sO * sO / 3 ELSE DV2 = - (2 / x + DB (x, s , v. A, B, OMEGA, eta, LAMBDA) / (2 * A) ) * V DV3 = D A (x, s , v. A, B, OMEGA, eta, LAMBDA) * v / (2 * A) DV4 = (I - OMEGA * OMEGA /.B) * A * s - 3 * eta * A * s * s DV = DV2 + DV3 + DV4 + LAMBDA * A * s * s * s END IF END FUNCTION FUNCTION epsilon (eta, LAMBDA) 'This block defines the difference between the false and the true vacua, 'epsilon. (In my notes it is epsilon with a tilde!) xi = eta A 2 / LAMBDA IF xi > .5 THEN epsilon = -Vbar(truevac(eta, LAMBDA), eta. LAMBDA) ELSE epsilon = 3030 END IF END FUNCTION SUB headr (eta, LAMBDA, OMEGA, sO, pass) PRINT "eta, LAMBDA and xi "; PRINT USING "###.#### " ; eta; LAMBDA; eta * eta / LAMBDA PRINT "bump, truevac and s (0)/bump: "; PRINT USING "##.####### "; bump(eta, LAMBDA); truevac(eta, LAMBDA); sO / bump(eta, LAMBDA) PRINT "Energy difference and bubble radius: "; PRINT USING "##.####### "; epsilon(eta, LAMBDA); BubbleX(0, eta, LAMBDA) PRINT "OMEGA = "; OMEGA, "pass "; pass END SUB 117 FUNCTION Sll (x, S , v. A, B, OMEGA, eta, LAMBDA, h) 'The S (i ,j ) are various shots in the Runge Kutta calculation. 'j = I denotes shot for calculating s (x) v(x) 'j = 3 A (x) 'j=4 B (x) 511 = h * D S (x, s , v. A, B, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S12 (x, S, v. A, B, OMEGA, eta, LAMBDA, sO, BO, h) 512 = h * D V (x, s , v. A, B, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S13 (x, S, V, A, B, OMEGA, eta, LAMBDA, h) 513 = h * D A (x, S , v. A, B, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S14 (x, S, V, A, B, OMEGA, eta, LAMBDA, h) 514 = h * DB(x, s , v. A, B, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S21 (x, s, v. A, B, OMEGA, eta, LAMBDA, skiI, skl2, skl3, skl4, h) 521 = h * D S (x + h / 2, s + skll / 2, v + skl2 / 2, A + ski3 / 2, B + ski4 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S22 (x, s, v, A, B, OMEGA, eta, LAMBDA, sO, BO, skll, ski2, skl3, skl4, h) 522 = h * DV(x + h / 2, s + skll / 2, v + skl2 / 2, A + sk!3 / 2, B + ski4 / 2, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S23 (x, s, v. A, B, OMEGA, eta, LAMBDA, skll, skl2, skl3, skl4, h) 523 = h * D A (x + h / 2, s + skll / 2, v + sk!2 / 2, A + skl3 / 2, B + ski4 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S24 (x, s, v. A, B, OMEGA, eta, LAMBDA, skll, ski2, skl3, skl4, h) S24 = h * D B (x + h / 2, s + skll / 2, v + sk!2 / 2, A + sk!3 / 2, B + ski4 / 2, OMEGA, eta, LAMBDA) END FUNCTION 118 FUNCTION S31 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk21, sk22, sk23, sk24, h) 531 = h * D S ( x + h / 2 , s + sk21 / 2, v + sk22 / 2, A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA) -END FUNCTION FUNCTION S32 (x, S, v. A, B, OMEGA, eta, LAMBDA, sO, BO, sk21, sk22, sk23, sk24,.h) 532 = h * D V (x + h / 2, s + sk21 / 2, v + sk22 / 2, A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S33 (x, s, v, A, B, OMEGA, eta, LAMBDA, sk21, sk22, sk23, sk24, h) 533 = h * D A (x + h / 2, s + sk21 / 2, v + sk22 / : , A + sk23 / 2, B +, sk24 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S34 (x, s, v, A, B, OMEGA, eta, LAMBDA, sk21, sk22, sk23, sk24, h) 534 = h * D B ( x + h / 2 , s + sk21 / 2, v + sk22 / : , A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA) END FUNCTION ' FUNCTION S41 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk31, sk32, sk33, sk34, h) 541 = h * D S (x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S42 (x, s, v. A, B, OMEGA, eta, LAMBDA, sO, BO, sk31, sk32, sk33, sk34, h) 542 = h * DV(x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S43 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk31, sk32, sk33, sk34, h) 543 = h * D A (x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S44 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk31, sk32, sk33, sk34, h) 544 = h * DB(x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA) END FUNCTION 119 FUNCTION Sbarl (sO, eta, LAMBDA) 'Bounce action bb2 = truevac(eta, LAMBDA) Sbarl = -.5 * (sO A 2 - bb2 A 2) + (eta / 3 ) * (sO A 3 - bb2 A 3) END FUNCTION SUB SORT4 (d () ) FOR j = 2 TO 4 dd = d (j ) FOR i = j - I TO I STEP -I IF d(i) <= dd THEN kk = I EXIT FOR ELSE d(i + I) = d(i) END IF NEXT i IF kk < I THEN 1 = 0 d(i + 1 ) = dd NEXT j END SUB SUB SORTN (aa(), n) FOR j = 2 TO n dd = a a (j ) FOR i = j - I TO I STEP -I IF aa(i) <= dd THEN kk = I EXIT FOR ELSE a a (i + I) = ■a a (i ) END IF NEXT i IF kk < I THEN i = 0 a a (i + I) = dd NEXT j END SUB FUNCTION truevac (eta, LAMBDA) 'Field value at the true vacuum xi = eta * eta / LAMBDA IF xi >= .5 THEN truevac = I .5 * xi * ( 1 + SQR(I ELSE truevac = 99 END IF END FUNCTION 4 / (9 * xi))) / eta 120 FUNCTION Vbar (s, eta, LAMBDA) 'Dimensionless 2-3-4 potential Vbar = s A 2 - 2 * eta * s A 3 + .5 * LAMBDA * s A 4 END FUNCTION SUB vps (xx(), s s (), size) 'Graphs the arrays (xx(), ss ()) in the upper right corner. FOR i = I TO size dd(i) = xx(i): cc(i) = ss(i) NEXT i SORTN dd(), size SORTN C f c (), size Lx = dd(l): Ux = d d (size) Ly = cc(I): Uy = c c (size) ww = Ux - Lx: hh = Uy - Ux CLS SCREEN 2 VIEW (300, 5)- (600, 80), , 0 IF Ly < 0 THEN WINDOW (0, Ly)-(1.05 * Ux, 1.05 * U y ) ELSE WINDOW (0, 0)-(I.05 * Ux, 1.05 * U y ) END IF LINE (0, 0)-(Ux, 0) FOR i = I TO size PSET (xx(i), ss(i)) NEXT i END SUB SUB vps2 (xx(), s s (), size) 'Graphs the arrays (xx(), s s ()) in lower right. FOR i = I TO size dd(i) = xx(i): cc(i) = ss(i) NEXT i SORTN dd(), size SORTN c c (), size Lx = dd(l): Ux = dd(size) Ly = cc(I): Uy = c c (size) ww = Ux - Lx: hh = Uy - Ux VIEW (300, 105)- (600, 180), , 0 121 IF Ly < 0 THEN WINDOW (0, Ly)-(I.05 * U x , 1.05 * U y ) ELSE WINDOW (0, 0)-(I.05 * U x , 1.05 * U y ) END IF LINE (0 0)-(Ux, 0) FOR i = I TO size PSET (xx(i ), ss(i)) NEXT i END SUB 'end of bxrevise.bas 122 'ez_sys.has DECLARE FUNCTION Vbar# (s#, eta#, LAMBDA#) DECLARE FUNCTION Sbarl# (sO#, eta#, LAMBDA#) DECLARE FUNCTION BubbleX# (sO#, eta#, LAMBDA#) DECLARE FUNCTION F# (x#) DECLARE FUNCTION epsilon# (eta#, LAMBDA#) DECLARE FUNCTION Rbubble# (eta#, LAMBDA#) DECLARE FUNCTION bump# (eta#, LAMBDA#) DECLARE FUNCTION truevac# (eta#, LAMBDA#) DECLARE FUNCTION DVAC# (LAMBDA#, eta#) DECLARE SUB vps2 (xx#(), ss# (), size#) DECLARE SUB SORTN (aa#(), n#) DECLARE SUB vps (xx#(), SS#(), size#) DECLARE SUB S0RT4 (d#()) DECLARE FUNCTION S41# (x#, s#. v # , A#, B#, OMEGA#, eta#, LAMBDA#, sk31#, sk32#, sk33*, sk34#, h#) DECLARE FUNCTION S42# (x#. S#, v # , A # , B#, OMEGA*, eta#. LAMBDA#, sO#, BO#, sk31#, sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S43# (x#, s#, v # , A#, B#, OMEGA#, eta#. LAMBDA#, sk31#, sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S44# (x#. S#, v # , A#, B#, OMEGA#, eta#. LAMBDA#, sk31#. sk32#, sk33#, sk34#, h#) DECLARE FUNCTION S31# (x#. s#, v # , A#, B#, OMEGA#, eta#. LAMBDA#, sk21#, sk22#, sk23#, sk24#, h#) DECLARE FUNCTION S32# (x#. s#. v # , A#, B#, OMEGA#, eta#. LAMBDA#, sO#, BO#, sk21#, sk22#, sk23#, sk24#, h#) DECLARE FUNCTION S33# (x#. s#, v#. A#, B#, OMEGA#, eta#, LAMBDA#, sk21#, sk22#, sk23#, sk24#, h#) DECLARE FUNCTION S34# (x#, s#, v # , A#, B#, OMEGA#, eta#, LAMBDA#,' sk21#, sk22#, sk23#, sk24#, h#) DECLARE FUNCTION S21# (x#. s#. v # , A#, B#, OMEGA#, eta#, LAMBDA#, skll#, skl2#, skl3#, skl4#, h#) DECLARE FUNCTION S22# (x#, s#. v # , A#, B#, OMEGA#, eta#, LAMBDA#, sO#, BO#, skll#, skl2#, skl3#, skl4#, h # ), DECLARE FUNCTION S23# (x#. s#, v#. A#, B#, OMEGA#, eta#, LAMBDA#, skll#, skl2#, skl3#, skl4#, h#) DECLARE FUNCTION S24# (x#. s#, v # , A#, B#, OMEGA*, eta#. LAMBDA#, skll#, ski2#, skl3#, skl4#, h#) DECLARE FUNCTION Sll# (x#. s#, v # , A # , B#, OMEGA#, eta#, LAMBDA#, h#) DECLARE FUNCTION S12# (x#, s#. v # , A#, B#, OMEGA#, eta#, LAMBDA#, sO#, BO#, h#) DECLARE FUNCTION S13# (x#, s#. v # , A#, B#, OMEGA#, eta#. LAMBDA#, h#) DECLARE FUNCTION S14# (x#, s#. v # , A#, B#, OMEGA#, eta#. LAMBDA#, h#) DECLARE FUNCTION DV# (x#. s#. v # , A#, B#, OMEGA#, eta#. LAMBDA#, sO#, BO#) DECLARE FUNCTION DS# (x#, S#, V#, A#, B#, OMEGA#, eta#, LAMBDA#) DECLARE FUNCTION DB* (x#, S#, v#. A#, B#, OMEGA#, eta#, LAMBDA#) DECLARE FUNCTION DA# (x#, s#, V#, A#, B#, OMEGA#, eta#, LAMBDA#) DEFDBL A-Z S 123 'The several arrays for my Runge-Kutta procedure. DIM SHARED xlout(0 TO 500) DIM SHARED slout(0 TO 500) DIM SHARED vlout(0 TO 500) DIM SHARED Alout(0 TO 500) DIM SHARED Blout(0 TO 500) DIM SHARED Mdata(0 TO 500) DIM SHARED dd(l TO 500): DIM SHARED cc(l TO 500) DIM SHARED xx(l TO 500): DIM SHARED ss(I TO 500) DIM SHARED aa(l TO 500) ff$ = 'This gg$ = 'This "a:\ezss284.pr" file will be used for graphing on Power Mac/Clarisworks "a:\sys284.pp" file will be input for actntrap.bas OPEN ff$ FOR OUTPUT AS #1 OPEN gg$ FOR OUTPUT AS #2 PRINT #1, ff$ PRINT #1, "Tab delimited output from ez-sys.bas", DATE$, TIME$ PRINT #1, PRINT #2, gg$ PRINT #2, "Straight output from ez-sys.bas", DATE$, TIME$ PRINT #2, 'These data will come from newdata().prn summary. OMEGAbar = .8672246631207673# Binf = 1.167474# 'energy parameters {OMEGA,eta,LAMBDA} CONST LAMBDA = 1 0 eta = S Q R (LAMBDA * (.5600000000000001#)) 'Setting eta = 0 will compare with Colpi, et a l ., Fig. (3) 'Setting LAMBDA = 0 will compare with Ruffini and Bonazzola figures. 'BOUNDARY CONDITIONS: s (0),v(0),A(O),B(0). 'Central redshift is OMEGAA2/B(0); adjust so that B ("infinity") -> I. ssl = bump(eta, LAMBDA): ss2 = truevac(eta, LAMBDA) sO = .99# * ssl: BO = I / Binf vO = 0: AO = I : xO = 0 'Runge-Kutta fixed step size parameters, h = BubbleX(0, eta, LAMBDA) / 100 'h = .070644# scales = s O : scalev = -.7: scaleA = 4: scaleB = .1 124 x = x O : s = sO: v = v O : A = AO: B = BO xlout(O) = x O : slout(O) = sO: vlout(O) = vO Alout(O) = AO: Blout(O) = BO OMEGA = OMEGAbar / SQR(Binf) CLS O A sratio = sO / ssl PRINT "eta, LAMBDA and xi "; PRINT USING "###.#### "; eta; LAMBDA; eta * eta / LAMBDA . PRINT "bump, truevac and s (O)/bump: "; PRINT USING "##.######* "; ssl; ss2; sratio PRINT "Energy difference and bubble radius: "; PRINT USING "##.####### "; epsilon(eta, LAMBDA); BubbleX(0, eta, LAMBDA) PRINT "OMEGA = "; OMEGA i = O 'FOR i = I TO maxsteps UNTIL S < O OR v OR i = 100 i = i + I 'The 'U(i,j) are Ull = Sll(x. s, U12 = S12(x. s, Ul 3 = S13(x, s, Ul 4 = S14(x. s, the parts of the: Runge Kutta v. A, B, OMEGA, eta. LAMBDA, v. A, B, OMEGA, eta. LAMBDA, v. A, B, OMEGA, eta. LAMBDA, v. A, B, OMEGA, eta. LAMBDA, procedure h) sO, BO, h) h) h) U21 = S21(x. s, v. A, B, OMEGA, eta, LAMBDA, Ull, U12, Ul3 , U14, U22 = S22(x. s, v, A, B, OMEGA, eta. LAMBDA, sO, BO, Ull, U12, , U14, h) U23 = S23(x. s, v. A, B, OMEGA, eta. LAMBDA, Ull, U12, Ul3 , U14, U24 _ S24(x. s, v. A, B, OMEGA, eta. LAMBDA, Ull, U12, Ul3 , U14, h) U31 = S31(x. s, v. A, B, OMEGA, eta, LAMBDA, U21, U22 , U23, U24 CO h) O BO, U21, U22, U32 _ S32(x. s, v. A, B, OMEGA, eta, LAMBDA, U23, U24, h) U33 = S33(x, s, v, A, B, OMEGA, eta, LAMBDA, U21, U22 , U23, U24 h) U34 = S34(x. s, v, A, B, OMEGA, eta, LAMBDA, U21, U22 , U23, U24 h) U41 S41(x. s, v. A, B, OMEGA, eta, LAMBDA, U31, U32 , U33, U34 h) O CO BO, U31, U32, U42 S42(x, s, v. A, B, OMEGA, eta, LAMBDA, U 3 4 , h) U33, U43 = S43(x. s, v. A, B, OMEGA, eta, LAMBDA, U31, U32 , U33, U34 h) 125 U44 = S44(x, s , v. A, B, OMEGA, eta, LAMBDA, U31, U32, U33, U34, h) 'The arrays slout(), etc. slout(i) = S + (Ull + 2 * vlout(i) = V + (U12 + 2 * Alout(i) = A + (U13 + 2 * Blout(i) = B + (U14 + 2 * xlout(i) = X + h are U21 U22 U23 U24 the + 2 + 2 +• 2 + 2 solutions. * U31 + U41) * U32 + U42) * U33 + U43) * U34 + U44) / / / / 6 6 6 6 x = xlout(i) s = slout(i) v = vlout(i) A = Alout(i) B = Blout(i) PRINT i , x 'NEXT i LOOP maxsteps = i - I PRINT "Maxsteps = maxsteps, gg$ Mdata(O) = O FOR j = I TO maxsteps Mdata(j ) = .5 * xlout(j) * (I - I / Alout(j)) NEXT j FOR i maxsteps + 'This loop xlout(i) = slout(i) = vlout(i ) = Alout(i) = Blout(i) = Mdata(i) = I TO 100 pads out the data files with ones and zeros, h * i 0 0 I I Mdata(maxsteps) NEXT i PRINT #1, eta; CHR$(9); LAMBDA; CHR$(9); sratio; CHR$(9); OMEGA; CHR$(9); PRINT #1, maxsteps O TO 100 PRINT #1, PRINT #1, PRINT #1, PRINT #1, PRINT #1, PRINT #1, PRINT #1, PRINT #1, PRINT #1, PRINT #1, xlout(k); USING "##..### CHR$(9); USING "##.#####AAAA " ; CSNG(slout(k)); CHR$(9); USING "##.#####AAAA " ; CSNG(vlout(k)); CHR$(9); Alout(k); USING "##.##*###### CHR$(9); Blout(k) ; USING "##.######### CHR$(9); 126 PRINT #1, USING "##.#####AAAA CSNG(Mdatafk)) NEXT k PRINT #2, eta, LAMBDA, sratio, OMEGA, maxsteps 0 TO 100 PRINT #2, PRINT #2, PRINT #2, PRINT #2, PRINT #2, PRINT #2, USING USING USING USING USING USING "##.### "##.#####AAAA "##.#####AAAA "##.######### "##.######### "##.#####AAAA "; -xlout(k); " ; CSNG(slout(k)); " ; CSNGfvlout(k)); "; Alout(k); " ; Blout(k); "; CSNG(Mdatafk)) NEXT k 'vps xlout(), slout(), maxsteps END 'SUB PROCEDURES AND FUNCTIONS OF EZ_SYS.BAS FUNCTION BubbleX (sO, eta, LAMBDA) epsilonO = V bar(sO, eta, LAMBDA) + epsilon(eta, LAMBDA) BubbleX = 6 * Sbarl(sO, eta, LAMBDA) / epsilonO END FUNCTION FUNCTION bump (eta, LAMBDA) xi = eta * eta / LAMBDA IF xi >= .5 THEN bump = 1.5 * xi * ( I - S Q R ( 1 - 4 / ELSE bump = 12.3 END IF END FUNCTION (9 * xi))) / eta FUNCTION DA (x, s, v, A, B, OMEGA, eta, LAMBDA) IF x = 0 THEN DA = 0 ELSE DAl = A * ( I - A ) / x DA2 = (OMEGA * OMEGA / B + I) * x * A * A * s * s DA3 = -2 * eta * x * A * A * s * s * s DA4 = LAMBDA * x * A * A * s * s * s * s / 2 DA = DAl + DA2 + DA3 + DA4 + x * v * v * A END IF END FUNCTION 127 FUNCTION DB (x, s, v. A, B, OMEGA, eta, LAMBDA) IF X = 0 THEN DB = 0 ELSE DBl = B * ( A - I ) / x DB2 = (-1 + OMEGA * OMEGA / B) * x * A * B * s * s DB 3 = 2 * eta * x * A * B * s * s * s DB4 = -LAMBDA * x * A * B * s * s * s * s / 2 DB = DBl + DB2 + DB3 + DB4 + x * v * v * B END IF END FUNCTION FUNCTION DS (x, DS=V END FUNCTION S, A, B, OMEGA, eta, LAMBDA) V, FUNCTION DV (x, S, V, A, B, OMEGA, eta, LAMBDA, sO, BO) IF X = 0 THEN DVl = (I - OMEGA * OMEGA / BO) * sO / 3 - eta * sO * sO DV = DVl + LAMBDA * sO * sO * sO / 3 ELSE DV2 = -(2 / x + D B (x, s , v. A, B, OMEGA, eta, LAMBDA) / (2 * A)) * v DV3 = D A (x, s , v. A, B, OMEGA, eta, LAMBDA) * v / (2 * A) DV4 = (I - OMEGA * OMEGA / B) * A * s - 3 * eta * A * s * s DV = DV2 + DV3 + DV4 +•LAMBDA * A * s * s * s END IF END FUNCTION ) FUNCTION epsilon (eta, LAMBDA) 'This block defines the difference between the false and.the true vacua, 'epsilon. (In my notes it is epsilon with a tilde!) epsilon = -Vbar(truevac(eta, LAMBDA), eta, LAMBDA) . END FUNCTION FUNCTION Sll (x, Sll = h * D S (x, S END FUNCTION S, , A, B, OMEGA, eta, LAMBDA, h) A, B, OMEGA, eta, LAMBDA) V, V, FUNCTION S12 (x, s, v. A, B, OMEGA, eta, LAMBDA, sO, BO, h) 512 = h * D V (x, s , v. A, B, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S13 (x, s, v. A, B, OMEGA, eta, LAMBDA, h) 513 = h * D A (x, s , v. A, B, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S14 (x, S , v. A, B, OMEGA, eta, LAMBDA, h) 514 = h * DB(x, S , V , A, B, OMEGA, eta, LAMBDA) END FUNCTION 128 FUNCTION S21 (x, s, v. A, B z OMEGA, eta, LAMBDA, skll, ski2, skl3, skl4, h) 521 = h * D S (x + h / 2, s + skll / 2, v + skl2 / 2, A + skl3 / 2, B + ski4 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S22 (x, s, v. A, B, OMEGA, eta, LAMBDA, sO, BO, skll, skl2, skl3, sk!4, h) 522 = h * D V (x + h / 2, s + skll / 2, v + skl2 / 2, A + ski3 / 2, B + ski4 / 2, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S23 (x, s, v. A, B, OMEGA, eta,, LAMBDA, skll, skl2, skl3, skl4, h) 523 = h * D A ( x + h / 2, s + skll / 2, v + sk!2 / 2, A + skl3 / 2, B + skl4 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S24 (x, s, v. A, B, OMEGA, eta, LAMBDA, skll, skl2, skl3, skl4, h) S24 = h * DB(x'+ h / 2, s + skll / 2, v + skl2 / 2, A + skl3 / 2, B + skl4 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S31 (x, s, v, A, B, OMEGA, eta, LAMBDA, sk21, sk22, sk23, sk24, h) 531 = h * D S ( x + h / 2 , s + sk21 / 2, v + sk22 / 2, A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S32 (x, s, v, A, B, OMEGA, eta, LAMBDA, sO, BO, sk21, sk22, sk23, sk24, h) 532 = h * D V (x + h / 2, s + sk21 / 2, v + sk22 / 2, A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S33 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk21, sk22, sk23, sk24, h) 533 = h * D A (x + h / 2, s + sk21 / 2, v + sk22 / 2, A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S34 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk21, sk22, sk23, sk24, h) 534 = h * D B (x + h / 2, s + sk21 / 2, v + sk22 / 2, A + sk23 / 2, B + sk24 / 2, OMEGA, eta, LAMBDA) END FUNCTION 129 FUNCTION S41 (x, s, V, A, B, OMEGA, eta, LAMBDA, sk31, sk32, sk33, sk34, h) 541 = h * DS(x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S42 (x, s, v. A, B, OMEGA, eta, LAMBDA, sO, BO, sk31, sk32, sk33, sk34, h) 542 = h * D V (x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA, sO, BO) END FUNCTION FUNCTION S43 (x, s, v. A, B, OMEGA, eta, LAMBDA, sk31, sk32, sk3 3, sk34, h) 543 = h * D A (x + h, s + sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION S44 (x," s, V, A, B, OMEGA, eta, LAMBDA, sk31, sk32, sk33, sk34, h) 544 = h * DB(x + h, s +_ sk31, v + sk32, A + sk33, B + sk34, OMEGA, eta, LAMBDA) END FUNCTION FUNCTION Sbarl (sO, eta, LAMBDA) dd2 = truevac(eta, LAMBDA) Sbarl = -.5 * (sO A 2 - dd2 A 2) + (eta / 3 ) END FUNCTION SUB SORT4 (d () ) FOR j = 2 TO 4 dd = d(j) FOR i = j - I TO I STEP -I IF d(i) <= dd THEN kk = I EXIT FOR ELSE d(i + I) = d(i) END IF NEXT i IF kk < I THEN 1 = 0 d(i + 1 ) = dd NEXT j END SUB SUB SORTN (aa(), n) FOR j = 2 TO n dd = a a (j ) FOR I = j - I TO I STEP -I IF aa(I) <= dd THEN * (sO A 3 - dd2 A 3) 130 kk : I EXIT FOR ELSE aa (i + D aa(i) END IF NEXT i IF kk < I THEN i = 0 aa (i + I) = dd NEXT j END SUB FUNCTION truevac (eta, LAMBDA) xi = eta * eta / LAMBDA IF xi >= .5 THEN truevac = 1.5 * xi * (I + S Q R (I ELSE truevac = 99 END IF 4 / (9 * xi))) END FUNCTION FUNCTION Vbar (s, eta, LAMBDA) 'Scalar potential Vbar = s A 2 - 2 * eta * s A 3 + .5 * LAMBDA * s A 4 END FUNCTION SUB vps (xx(), s s (), size) FOR i = I TO size dd(i) = xx(i): cc(i) = ss(i) NEXT i SORTN dd(), size SORTN c c (), size Lx = dd(l): Ux = dd(size) Ly = cc(l): Uy = c c (size) v j w = Ux - Lx: hh = Uy - Ux CLS SCREEN 2VIEW (300, 5)-(600, 80), , 0 IF Ly < 0 THEN WINDOW (0, Ly)-(I.05 * Ux, 1.05 * U y ) ELSE WINDOW (0, 0)-(I.05 * Ux, 1.05 * Uy) END IF LINE (0, 0) -(Ux, 0) / eta 131 FOR i = I TO size PSET (xx(i), ss(i)) NEXT i END SUB SUB vps2 (xx(), s s (), size) FOR i = I TO size dd(i) = xx(i): cc(i) = ss(i) NEXT i SORTN dd(), size SORTN c c (), size Lx = dd(l): Ux = dd(size) > Ly = cc(I): Uy = c c (size) ww = Ux - Lx: hh = Uy - Ux VIEW (300, 105)-(600, 180), , 0 IF Ly < 0 THEN WINDOW (0, Ly)-(I.05 * U x , 1.05 * Uy) ELSE WINDOW (0, 0)-(I.05 * U x , 1.05 * U y ) END IF LINE (0, 0)-(Ux, 0) FOR i = I TO size PSET (xx(i), ss(i)) NEXT i END SUB 132 'actntrap.bas DECLARE FUNCTION BubbleX# (ssO#, eta#, LAMBDA#) DECLARE FUNCTION truevac# (eta#, LAMBDA#) DECLARE FUNCTION Sbarl# (ssO#, eta#, LAMBDA#) DECLARE FUNCTION epsilonbar# (ssO#, eta#, LAMBDA#) DECLARE FUNCTION Vbar# (s#, eta#, LAMBDA#) DEFDBL A-Z CLS maxcuts = 50 ff$ = "a:\sysl02.pp" OPEN ff$ FOR INPUT AS #1 INPUT #1, a$, b$, C$ INPUT #1, eta, LAMBDA, sratio, OMEGA, maxsteps DIM x (0 TO 100): DIM s(0 TO 100): DIM v(0 TO 100) DIM a(0 TO 100) DIM b (0 TO maxcuts): DIM xq(0 TO maxcuts): DIM j j (0 TO maxcuts) INPUT #1, x(0), s(0), v(0), aO, bO, mO PRINT USING "###.###### x (0); s(0); v(0);*a0; b 0 ; mO FOR i = I TO 100 INPUT #1, x(i), s (i), v(i), aa, bb, mm PRINT USING "###.###### x(i); s(i); v(i) NEXT i 'These two step size lines can be adjusted, the;first being the 'standard default value of X/100. hh = BubbleX(0, eta, LAMBDA) / 100 'hh = .070644# FOR i = 0 TO 100 a (i) = (x (i) ) A 2 * (OMEGA A 2 * (s(i) ) A 2 + (v(i) ) A 2 + V b a r (s (i ), eta, LAMBDA)) NEXT i xq(0) = hh * 100: jj (0) = 100 betastep = hh * 100 / maxcuts FOR i = I TO maxcuts - I beta = S Q R ((hh * 100) A 2 - (betastep * i) A 2) j = FIX(beta / hh) PRINT j ; PRINT USING " ##.## x(j); PRINT ".."; j j (i) = 3 xq (i ) = x (j ) NEXT i PRINT FOR j = 0 TO maxcuts - I q = ]](]) 133 accumulate = hh * .5 * (a(O) + a(q)) FOR i = I TO q - I accumulate = accumulate + hh * a (i) NEXT i PRINT j ; "Number of steps:"; q; PRINT USING "-->##.########## accumulate b(j) = accumulate NEXT j actionaccumulate = .5 * betastep * (b(0) + b(maxcuts - I)) FOR j = I TO maxcuts - 2 actionaccumulate = actionaccumulate + betastep * b(j) NEXT j PRINT a$ PRINT b$ PRINT c$ PRINT "specs: eta, LAMBDA, sratio PRINT " "; OMEGA, maxsteps PRINT ff$ PRINT "Boson star action ="; actionaccumulate / 2 ratiol = (epsilonbar(0, eta, LAMBDA)) A 3 / (Sbarl(0, eta, LAMBDA)) A 4 ratio = actionaccumulate * ratiol / (216 * ATN(I)) PRINT ".Bubble action:"; 108 * ATN(I) / ratiol PRINT "Bounce action:"; Sbarl(0, eta, LAMBDA) PRINT "Energy difference:"; epsilonbar(0, eta, LAMBDA) PRINT "Action ratio:"; ratio END 'SUB PROCEDURES AND FUNCTIONS FOR ACTNTRAP.BAS i FUNCTION BubbleX (ssO, eta, LAMBDA) BubbleX = 6 * Sbarl(ssO, eta, LAMBDA) END FUNCTION / epsilonbar(ssO, eta, LAMBDA) FUNCTION epsilonbar (ssO, eta, LAMBDA) epsilonbar = V b a r (ssO, eta, LAMBDA) - V b a r (truevac(eta, LAMBDA), eta, LAMBDA) END FUNCTION FUNCTION Sbarl (ssO, eta, LAMBDA) ss2 = truevac(eta, LAMBDA) Sbarl = A B S (.5 * (ssO A 2 - ss2 A 2) - (eta / 3) * (ssO A 3 - ss2 A 3)) END FUNCTION I 134 FUNCTION truevac (eta, LAMBDA) xi = eta * eta / LAMBDA' IF xi >= .5 THEN truevac = 1.5 * xi * ( 1 + SQR(I - 4 / ELSE truevac = 99 (9 * xi))) / eta END IF END FUNCTION FUNCTION Vbar (s, eta, LAMBDA) Vbar = s, A 2 - 2 * eta * s A 3 + .5 * LAMBDA * s A 4 END FUNCTION MONTANA STATE UNIVERSITY LIBRARIES 3 1762 10318466 7