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
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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
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/(Date)
A p p r o v e d for the Department of Physics
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(Signature)
(Date)
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Robert B rown
(Signature)
(Datdl
I
iii
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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
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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).
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San F r a n c i s c o : Freeman, 1973.
J.A.,
Gravitation.
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{
Press, W . H . , et a l ., Numerical Recipes in FORTRAN: The Art of
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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
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Wheeler,
J.A.,
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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
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