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Analytical Characterization ofScalar-Field Oscillons in Quartic
PotentialsA Thesis
Submitted to the Faculty
in partial fulfillment of the requirements for the
degree of
Doctor of Philosophy
in
Physics
by
David Pasquale Sicilia
DARTMOUTH COLLEGE
Hanover, New Hampshire
11 May 2011
Examining Committee
(Chair) Marcelo Gleiser
Robert Caldwell
Miles Blencowe
Noah Graham
Brian W. Pogue, Ph.DDean of Graduate Studies
Abstract
In this thesis I present a series of simple models of scalar field oscillons which
allow estimation of the basic properties of oscillons using nonperturbative an-
alytical methods, with minimal dependence on computer simulation. The
methods are applied to oscillons in φ4 Klein-Gordon models in two and three
spatial dimensions, yielding results with good accuracy in the characteriza-
tion of most aspects of oscillon dynamics. In particular, I show how oscillons
can be interpreted as long-lived perturbations about an attractor in field con-
figuration space. By investigating their radiation rate as they approach the
attractor, I obtain an accurate estimate of their lifetimes in d = 3 and explain
why they seem to be perturbatively stable in d = 2, where d is the number
of spatial dimensions. I also present some preliminary work on a method to
calculate the form of the spatial profile of the oscillon.
ii
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
1 Introduction 1
2 Past & Present Progress in Analytical Characterization 4
3 A Simple Model of Oscillon Amplitude and Energy 7
3.1 Analytical Characterization of Oscillons in Quartic Potentials 7
3.1.1 General Framework . . . . . . . . . . . . . . . . . . . . 8
3.1.2 The Course of Evolution of an Oscillon . . . . . . . . . 11
3.1.3 Oscillon Lifetime . . . . . . . . . . . . . . . . . . . . . 17
4 Frequency Analysis of Oscillons 22
4.1 Linear vs. Nonlinear Dynamics and the Oscillon Mass Gap . . 22
4.1.1 Linear Dynamics . . . . . . . . . . . . . . . . . . . . . 23
4.1.2 Nonlinear Dynamics and Decay Rate . . . . . . . . . . 25
4.1.3 The Attractor Point . . . . . . . . . . . . . . . . . . . 28
4.2 Lifetime of Long-Lived Oscillons: General Theory . . . . . . . 33
4.2.1 The Long-Lived Oscillon Radiation Equation . . . . . . 33
4.2.2 Dynamical Exponents . . . . . . . . . . . . . . . . . . 36
4.2.3 Integration of the Long-Lived Oscillon Radiation Equation 39
4.2.4 Sample Calculation: φ4 Klein-Gordon Field in d = 3 . . 42
4.3 Derivation of Nonlinear Frequency Peak . . . . . . . . . . . . 50
4.4 Analysis of Oscillon Stability as a Function of Time . . . . . . 57
4.5 The Four Oscillon Timescales . . . . . . . . . . . . . . . . . . 62
4.6 Linear Radiation Distribution in d = 2 . . . . . . . . . . . . . 63
4.7 Overlap Function . . . . . . . . . . . . . . . . . . . . . . . . . 65
iii
4.8 Derivation of Stability Function . . . . . . . . . . . . . . . . . 67
5 The Oscillon Profile 69
5.1 Assumption of Partial Separability of Space and Time . . . . . 69
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 An Empirical Relationship between Λ and χ . . . . . . . . . . 72
6 Conclusions and Future Directions 75
iv
List of Figures
3.1 Two curves of constant energy, Eosc = 41.3 (continuous line),
and Eattract = 37.7 (dashed line), together with the parabolic
level curve I(Amax, R) = 0. Vertical dashed line locates the
asymptote of I = 0. . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 The maximum amplitude and radius as a function of time for
a configuration with R0 = 2.86 and A0 = 2 in the symmetric
double well of Eq. 3.10 (continuous lines), and for an asymmet-
ric double well [V (φ) = 12φ2 − 2.16
3φ3 + 1
4φ4] with A0 = 2 and
R0 = 4 (dashed lines). The analytical predictions at the points
(AB, RB) and (AD, RD) are indicated by arrows and dots. The
insets show the minimum radius and frequency for the symmet-
ric double well. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 The dashed line represents the prediction of oscillon energy as
a function of time (Eq. 3.17) for the potential of Eq. 3.10. The
vertical dashed line is drawn to represent the prediction of the
time of decay of the oscillon τmaxlife. The continuous line is
numerical energy from simulation for the longest lived oscillon
in the same potential. . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Stability function I(R,Amax) as a function of time for the longest-
lived oscillon in the potential of Eq. 3.10. . . . . . . . . . . . . 16
3.5 Example of a Lorentzian model of the first peak in the power
spectrum. The width has been greatly exaggerated (by three
orders of magnitude) for easier visualization. Dashed lines show
the frequencies ωmass and ωrad, respectively. . . . . . . . . . . . 18
v
4.1 Schematic description of the linear radiation peak centered on
ωlin being shifted to the left by the presence of nonlinearities.
An oscillon will form if the peak is shifted far enough into the
nonlinear region such that it no longer overlaps significantly
with the linear peak. The curves shown represent neither the-
oretical nor numerical results–instead, they are simply general
pictorial representations of the curves. . . . . . . . . . . . . . 26
4.2 Minimum oscillon energy as a function of core amplitude for a
double-well potential in d = 3. The minimum of this curve is
the attractor point with E∞ ' 37.69. . . . . . . . . . . . . . . 28
4.3 The thick solid line represents the locus of points satisfying the
condition ωgap = Γlin in d = 2 (“line of existence”) calculated
analytically. The thin solid line represents the locus of points
that have the attractor energy E ' 4.44. The dashed line is the
numerical minimum radius based on Gaussian initial configura-
tions. Oscillons can only exist if they have energy E ≥ E∞ and
if they have a core amplitude and average radius lying above the
line of existence. This is why the numerically measured mini-
mum radius follows the line of existence for A & 1 but follows
the line of minimum energy for A . 1. The arrow indicates the
value of the (constant) minimum radius calculated in [8]. The
circle represents the location of the attractor point; since it lies
above the line of existence, oscillons will be absolutely stable in
this system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.4 The thick solid line represents the locus of points satisfying the
condition ωgap = Γlin in d = 3 (“line of existence”) calculated
analytically. The thin solid line represents the locus of points
that have the attractor energy E ' 37.69. The dashed line is
the numerical minimum radius based on Gaussian initial con-
figurations. The arrow indicates the value of the (constant)
minimum radius calculated in [8]. Oscillons can only exist if
their core amplitude and average radius are above the curve
(wherein E ≥ E∞ is automatically satisfied). The circle repre-
sents the location of the attractor point; since it lies below the
line of existence, oscillons will not be absolutely stable in this
system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
vi
4.5 Schematic of an oscillon frequency distribution showing the tail
penetrating the radiation region. The graph is plotted in units
of ωmass. In the inset we show a close-up view of F(ω) and b(ω),
respectively (dashed lines), and their product Ω(ω) (solid line)
for the system given by Eq. 4.1 in d = 3 for typical values of the
various parameters. Note that the curves have been vertically
scaled so that all are visible on the same graph. . . . . . . . . 34
4.6 Analytical calculation of g = ρE − 1 (top curve) and ρA (bot-
tom curve) using Eq. 4.44, plotted against the reference ampli-
tude Ar along the line of existence. This gives the spectrum
of possible values of g and ρA across the various oscillons in
this system. The dots mark the theoretical values of g ' 2.67
and ρA ' .66 assumed by the longest-lived oscillon, which has
Ar ' .92 (thicker line in Fig. 4.8). . . . . . . . . . . . . . . . . 43
4.7 Analytical calculation of γE (top graph) and γA (bottom graph)
using Eq. 4.45, plotted against the reference amplitude Ar along
the line of existence. This gives the spectrum of possible values
of γE and γA across the various oscillons in this system. The
dots serve to mark the theoretical values of γE ' 2.0 × 10−6
and γA ' .21 assumed by the longest-lived oscillon, which has
Ar ' .92 (thicker line in Fig. 4.8). . . . . . . . . . . . . . . . . 44
4.8 Various oscillon trajectories. Note that they all tend to the
attractor point but intersect the line of existence (dashed line)
before doing so. The thicker trajectory marks the longest-lived
oscillon (it intersects the line of existence at Ar ' .92 and Rr '3.25). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.9 Analytical results for critical values of the amplitude (top) and
radius (bottom) along the line of existence (dashed lines) as a
function of lifetime are plotted with several examples of short-
and long-lived oscillons, all with initial amplitudes A0 = 2.
From left to right, the initial radii for the oscillons are 2.35,
2.41, 2.53, 2.65, and 2.86. It is quite clear that the oscillons
decay as they cross the critical values computed analytically. . 48
vii
4.10 Oscillon lifetimes vs. decay energy ED = Er. Solid curve is the-
oretical, dashed line is numerical. The theoretical curve (with
an error of ∼ 6% in the horizontal positioning of the peak)
correctly predicts the shape of the distribution and that there
exists a maximum lifetime in this system on the order of ∼ 104. 49
4.11 Comparison of theoretical (continuous line) vs. numerical (dashed
line) radius (top), frequency (middle) and amplitude (bottom)
for an oscillon with initial conditions (A0 = 2, R0 = 2.86),
showing very good agreement [theoretical results are computed
with (Ar, Rr) ' (.92, 3.25)]. . . . . . . . . . . . . . . . . . . . 51
4.12 Comparison of theoretical (continuous line) vs. numerical (dashed
line) values for the energy of the longest-lived oscillon, obtained
with initial conditions (A0 = 2, R0 = 2.86) showing excellent
agreement [(Ar, Rr) ' (.92, 3.25)]. . . . . . . . . . . . . . . . . 52
4.13 Comparison of theoretical (continuous line) vs. numerical (wavy
line) radiation rate for an oscillon with initial conditions (A0 =
2, R0 = 2.86) showing excellent agreement. The inset, which
is plotted on a log scale, makes it clear that the theory cor-
rectly reproduces the rapid initial drop in radiation rate over
many orders of magnitude; the linear scale on the larger graph
shows that the theory correctly reproduces the extremely small
(but finite) radiation rate towards the end of the oscillon’s life
[(Ar, Rr) ' (.92, 3.25)]. . . . . . . . . . . . . . . . . . . . . . . 53
viii
4.14 Comparison of theoretical trajectory (solid curve) vs. numerical
trajectory (dashed curve) for the longest-lived oscillon (A0 = 2,
R0 = 2.86) showing very good agreement during the more stable
phase of the oscillon’s life (A . 1.5). The great increase in den-
sity of data points in the dashed line in the range .9 . A . 1.5
is due to the prolonged period of time spent in this region
by the oscillon (i.e., the “plateau” phase). The point where
the line again becomes dashed (A ' .9, R ' 3.2) signals the
numerical decay point of the oscillon. The thick segment of
the solid line highlights the portion of the theoretical trajec-
tory during the low-radiation plateau phase; the end of the
thick segment (Ar, Rr) ' (.92, 3.24) marks the theoretical decay
point, showing very good agreement. It is interesting to note
that, even after the oscillon decay at A ' .9, the remaining
field configuration continues to tend to the attractor point at
(A∞, R∞) ' (.456, 4.79), as does the theoretical curve. . . . . . 54
4.15 The top graph shows the theoretical calculation of (1− Σ) vs.
time and the bottom shows Σmin, both for the oscillon with
(Ar, Rr) ' (.92, 3.25). The stability measured by Σ is clearly
related to the radiation rate of the oscillon: this kind of stability
becomes greater in time, since the radiation rate decreases. On
the other hand, the stability measured by Σmin is related to
the resistance of the oscillon to decay: as time evolves and the
oscillon moves closer to the line of existence, the buffer against
instability diminishes. . . . . . . . . . . . . . . . . . . . . . . . 60
4.16 The solid curve is the theoretical calculation of Tdecay for (Ar, Rr) '(.92, 3.25). The dashed curve is the numerically measured pe-
riod of the superimposed oscillation, showing very good agree-
ment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.17 The four oscillon time scales plotted over time. From top to
bottom we have plotted Trelax, Tdecay, Tlinear, and Tosc. It can
clearly be seen here that an oscillon is an object governed by
multiple timescales spanning many orders of magnitude. . . . 64
ix
5.1 Plotted are the profiles of the longest lived oscillon in the SWDP
taken at three different times during the oscillon’s life. The
parameters used are (φ(r = 0) = 1.075,Λ = 1.33), (φ(r = 0) =
.96,Λ = 1.27) and (φ(r = 0) = .84,Λ = 1.2). Dashed curves
are numerical and solid curves are theoretical. . . . . . . . . . 71
5.2 The values of q which produce the best fit between theoretical
and numerical profiles as a function of time for longest-lived
oscillon in the SWDP. As can be seen, q(t) is essentially constant
during the stable phase of the oscillon’s life, confirming Eq. 5.6. 73
5.3 Energy as a function of φ(r = 0) after substituting Eq. 5.6
into Eq. 5.4. The curve possesses a minimum at E ' 41.5. The
oscillon “travels” along the curve (from right to left) and decays
when it reaches the minimum. . . . . . . . . . . . . . . . . . . 74
5.4 Radius as a function of φ(r = 0) after substituting Eq. 5.6 into
Eq. 5.4. The curve possesses a minimum at R ' 2.7. . . . . . 74
x
Chapter 1
Introduction
Nonlinear field theories contain a large number of localized solutions that dis-
play a rich array of properties [1]. Of particular interest are those that are
static and stable, that is, that retain their spatial profile as they move across
space or scatter with each other, as is the case of sine-Gordon solitons. The
details of the solitonic configurations are, of course, sensitive to the dimension-
ality of space and to the nature of the field interactions. Long ago, Derrick has
shown that, in the case of models with just a real scalar field, no static solitonic
configurations can exist in more than one spatial dimension [2]. Given that
most models of interest in high energy physics involve more complicated fields
in three spatial dimensions, this restriction was somewhat frustrating. For-
tunately, the subsequent exploration of a variety of models led to a plethora
of static, nonperturbative, localized field configurations. Examples include
topological defects, solutions of models usually involving gauge fields that owe
their stability to the nontrivial topology of the vacuum, such as strings and
monopoles [3], and the so-called nontopological solitons, solutions of models
where a conserved global charge is trapped inside a finite region of space due
to a mass gap condition, such as Q-balls [4] and the models with a real and a
1
complex scalar field of Friedberg, Lee, and Sirlin [5].
In the mid-nineties [6], a new class of localized nonperturbative solution be-
gan to be explored in detail, after being proposed earlier [7]. Named oscillons,
such long-lived solutions have the distinctive and counter-intuitive feature of
being time-dependent. In spite of this, the nonlinear interactions act to pre-
serve the localization of the energy, which remains approximately constant
for a surprisingly long time [8]. During the past few years, oscillons have at-
tracted much interest. Their properties were explored in two [9] and higher
[10] spatial dimensions, in the presence of gauge fields [11], in the standard
model of particle physics [12], and in simple cosmological settings [13]. There
have also been detailed attempts at understanding some properties of oscillon-
related configurations (typically with small-amplitude oscillations), including
their longevity, using perturbative techniques [14, 15]. Also recently, there
have been investigations into the radiation of quantum oscillons [16].
Despite this progress, one still gets the sense that the fundamental ques-
tions about oscillons have not been answered in a way which is satisfying and
sufficiently analytical (i.e., relying minimally on the use of computers). For
example, the question of the existence and energy of an attractor configura-
tion (or a minimum-energy state) has not been derived analytically; the decay
(radiation) rate of an oscillon has not even been estimated analytically (apart
from scaling relations which are not capable, in a practical way, of giving
absolute magnitudes[16]).
On the other hand, a treatment explaining the remarkable longevity of
oscillons in models related to spontaneous symmetry breaking, and thus of
obvious interest in particle physics and cosmology, has been lacking. In this
thesis I will summarize the work done in [17] and [18], which present some ways
in which we might make progress on the fundamental questions mentioned
2
above. First I present a simple model that allows estimation of the relevant
energies and oscillation amplitudes seen in the evolution of the oscillon. In the
second part I make this model a bit more sophisticated and then apply it to
calculate the time scales seen in the evolution of the oscillon, such as lifetime
or radiation rate.
3
Chapter 2
Past & Present Progress in
Analytical Characterization
Although oscillons were originally discovered in the seventies [7], their true
significance was first recognized by Gleiser [6] only in 1994. The abstract of
that paper states, in quite simple terms, what is meant by the term oscillon:
certain non-linear scalar field equations (e.g., non-linear Klein-Gordon equa-
tions) support time-dependent, localized solutions characterized by a critical
energy Ecrit. Below this energy the oscillon is unstable, but above it the os-
cillon stabilizes itself and may exhibit an extremely long life compared with
other time scales in the system. Interestingly, this behavior is observed despite
the non-existence of any obvious stabilizing forces such as conserved charges
or topology.
Further numerical investigations were carried out in [8], which, as stated
in the introduction, form the inspiration for the present work. These authors
establish the robustness of the oscillon solution by demonstrating that oscillons
form in varying potentials and seem to behave as attractor configurations (i.e.,
their properties are somewhat independent of initial conditions). Specifically,
4
provided that the bubble-like field configuration which spawns the oscillon has
a radius larger than a critical value, the bubble will settle into an oscillon
state which is characterized by a near constant energy, radius and maximum
amplitude of oscillation.
Despite its near perfect stability and periodicity, the properties of the os-
cillon exhibit a small, gradual change over the course of its life. Generally
speaking, this is due to the small rate at which the oscillon radiates energy to
its surroundings. The radius and frequency tend to increase while the energy
and amplitude decrease. After having existed for a length of time, the oscillon,
depending on the circumstances at hand, will either experience a sudden decay
or continue to converge to an attractor state, thereby surviving indefinitely.
It has been shown in [9, 10] that oscillons (in quartic potentials) in three spa-
tial dimensions always experience a decay after a time of approximately 104
(in natural units) whereas oscillons in two spatial dimensions appear to live
indefinitely.
Numerous other numerical studies have been conducted on oscillons to
investigate their properties in other contexts and situations. These include
oscillons in varying potentials (with and without gauge fields) [11, 12], inter-
action with heat baths and emergence from thermal backgrounds [12], and
collisions among oscillons [9]. However, the references listed previously are
most relevant to this research in that here we seek to answer (quantitatively)
only the fundamental questions relating to existence and stability of 3-d scalar
oscillons. The difficulty of this task stems from that the fact that we must
study a class of solutions to a nonlinear equation that are nonperturbative and
have no known closed-form analytical representation. Due to these difficulties,
the analysis has initially been limited to the simplest of situations, i.e., one
single, real, spherically symmetric, scalar field oscillon in vacuum.
5
Theoretical work in the oscillon literature (for example, [15, 14, 16]) is typ-
ically based on a series-expansion method initially employed in [14], whereby
time is rescaled according to τ = t√
1− ε2 and field is approximated by an
expansion of the form,
φ =∞∑k=1
εkφk. (2.1)
This method is very accurate and systematic and yields correct expressions
for the scalings of energy, frequency and radiation rate to lowest order in ε for
small-amplitude oscillons. However, as mentioned in [16], it is impractical to
use this method to determine the absolute magnitude of radiation rate which
is needed to calculate lifetime, since this would require carrying the expansion
out to very high-order terms. For this reason, some possible alternate methods
of characterizing oscillons are presented here.
At this point it is worthwhile to point out a few of the many characteristics
of oscillons that a comprehensive theory should predict and clarify. The theory
should begin by explaining the oscillon’s existence and stability. Specifically, it
should explain why the field tends to converge onto a finite-energy configura-
tion by radiating excess energy at an ever decreasing rate. Second, the theory
should then provide a means to compute the various energies associated with
the oscillon as well as ranges of its other parameters such as core-amplitude,
radius. Finally, the theory should allow an estimate of the lifetime of the
oscillon.
6
Chapter 3
A Simple Model of Oscillon
Amplitude and Energy
In this chapter I will present the work published in [17], which provides a
simple model that can be used to estimate the amplitude, radius, and energy
of an oscillon. At the end of this chapter I will give a brief sketch of how the
results can be applied to estimate the oscillon radiation rate. In subsequent
chapters the ideas presented here will be made more sophisticated.
3.1 Analytical Characterization of Oscillons in
Quartic Potentials
In this section I will first outline a theory which is capable of predicting the
values of the oscillon’s energy, radius, amplitude and frequency over the course
of its life. This is achieved by identifying a function which behaves as a stability
indicator. Employing this, it is possible to trace the trajectory of the oscillon
on a “phase diagram” over the course of its life. Then it is a simple matter to
read off values of any parameter of interest.
7
In this chapter we will model the oscillation of the field as a one-dimensional,
nonlinear oscillator whose potential is Veff . The stability indicator will be de-
fined as the second derivative (curvature) of this effective potential evaluated
at the upper turning point of the oscillation. The reason for this is that the
oscillon must probe (for at least part of its oscillation cycle) a part of the
effective potential whose curvature is less in magnitude that the curvature at
the equilibrium point. This is because the stability of the oscillon relies on a
shift of the fundamental oscillation frequency to below the mass frequency. In
order for this to happen, the effective potential modeling the oscillation of the
field must exhibit a decrease in curvature when moving away from the vacuum
in at least one direction.
The phase diagram, however, speaks nothing about time scales associated
with the oscillon’s life. Therefore, the last part of this section presents a
theory capable of addressing the issues of stability, radiation and lifetime.
This is accomplished not by analyzing the dynamics of the oscillon itself but
by analyzing the dynamics of the power spectrum (Fourier transform squared)
of the core amplitude of the oscillon. Resulting from this is a differential
equation governing the energy of the oscillon as a function of time.
3.1.1 General Framework
We begin with the Lagrangian for a d-dimensional, spherically-symmetric real
scalar field in flat spacetime,
L = cd
∫r(d−1)dr
(1
2φ2 − 1
2
(∂φ
∂r
)2
− V (φ)
), (3.1)
where a dot denotes a time derivative, and cd = 2πd/2/Γ(d/2). Inspired by
the numerical solutions, which show that an oscillon is well approximated by
8
a Gaussian undergoing near-periodic motion in its amplitude and with an
effective radius oscillating with small amplitude about a mean value [6, 8], we
approximate the oscillon solution as
φ(t, r) = A(t) exp
(− r
2
R2
)+ φv, (3.2)
where A(t) ≡ φ(r = 0, t)− φv is the displacement from the vacuum, φv. This
approximation works particularly well in d = 2 and d = 3, although it clearly
fails to reproduce the oscillon’s large r behavior, where φ(r) ∼ exp[−mr].
Since oscillons have been observed in a diversity of quartic polynomial
potentials, we write V (φ) as [10]
V (φ) =4∑j=1
gjj!φj − V (φv) , (3.3)
where the gj’s are constants and V (φv) is the vacuum energy. Substituting
Eqs. 3.2 and 3.3 in Eq. 3.1 and integrating over space we obtain,
L =(π
2
) d2Rd
[1
2A2 − Veff(A)
], (3.4)
with
Veff(A) =1
2ω2
0A2 +
4∑n=3
(2
n
)d/21
n!V n(φv)A
n, (3.5)
where V n(φv) ≡ ∂nV (φv)/∂φn and ω2
0 ≡ V ′′(φv)+d/R2 is the linear frequency.
(Primes will also denote partial derivatives with respect to φ.) For potentials
with V ′′(φv) > 0, Veff(A) is the potential energy of a nonlinear oscillator with
at least one equilibrium point (at A = 0) for any value of R. For conservative
9
motion we can write the energy as,
E =(π
2
) d2Rd
[1
2A2 + Veff(A)
]=(π
2
) d2RdVeff(Amax), (3.6)
where Amax is the positive turning point (Amax > 0) of A(t).
We also write the expression for the frequency ω associated with the con-
servative motion between the two turning points,
2π
ω= T =
∫ T
0
dt = 2
∫ Amax
Amin
dA
A, (3.7)
where T is the period of oscillation, A = [2E/cR − 2Veff(A)]12 , cR ≡ (π/2)d/2Rd,
and Amin is related to Amax by Veff(Amin) = Veff(Amax).
It was noted in [8, 10] that the location of Amax relative to the inflection
points of Veff(A) is an accurate indicator of oscillon existence. This can be un-
derstood by noting that since the inflection points separate regions of opposite
curvature in Veff(A), the particular region probed by Amax indicates the degree
to which the field configuration experiences the stabilizing nonlinearities of the
potential V (φ). Following [10], we therefore adopt V ′′eff(Amax) as an indicator
of oscillon stability: increasingly negative(positive) values lead to increased
stability(instability). Defining I(Amax, R) ≡ V ′′eff(Amax),
I(Amax, R) = ω20 +
4∑n=3
(2
n
)d/21
(n− 2)!V n(φv)A
n−2max. (3.8)
In Fig. 3.1 we show the (parabolic) level curve corresponding to I(Amax, R) =
0. Since values inside (outside) the parabola have I < 0 (I > 0), it is easy to
show that, for a given R, there exists a stable region provided that R > Rmin
10
where [10]
R2min = d
[1
2
(23/2
3
)d(V ′′′)2
V ′′′′− V ′′
]−1
. (3.9)
As the figure indicates, when R = Rmin, there is only one value of Amax located
in the stable region, given by A0 = −(4/3)d/2V ′′′/V ′′′′.
3.1.2 The Course of Evolution of an Oscillon
In Fig. 3.1, the continuous and dashed lines show curves of constant energy
(Eq. 3.6) as a function of R and Amax for a symmetric double-well potential,
V (φ) = φ2 − φ3 + φ4/4, (3.10)
that is, g1 = 0, g2 = 2, g3 = −6, g4 = 6, φv = 0 in Eq. 3.5. [Quantities
were made dimensionless with xµ = x′µ(√g2)−1 and the primes have been
suppressed.] We also plot the parabola I(R,Amax) = 0 marking the boundary
between the stable region (I < 0, above the curve) and the unstable region
(I > 0, below the curve). Note that the presence of an oscillon in the unstable
region does not necessarily mean that it will immediately decay; the degree of
stability is given roughly by the vertical distance of a point to the curve I = 0.
Using the curve I = 0 in Fig. 3.1, we can describe the trajectory modeling
the onset of an oscillon from an initial configuration starting at a somewhat
arbitrary point A. On its way to an oscillon, the configuration will radiate
excess energy until reaching point B. To determine the location of point B,
note that the onset of the oscillon phase is marked by a minimum radius.
Taking the minimum radius given by Eq. 3.9, and using that the oscillon stage
should begin in the stable region, fixes the location of point B to A = A0 and
R = Rmin, as shown in Fig. 3.1. As we shall see, this is confirmed by numerical
results. For the potential of Eq. 3.10, Rmin = 2.42 and AB = A0 = (4/3)3/2 =
11
0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2
Maximum Amplitude1.5
2
2.5
3
3.5
4
4.5
Radi
us
I = 0E = 41.3E = 37.7
A
B
C
D
Figure 3.1: Two curves of constant energy, Eosc = 41.3 (continuous line),and Eattract = 37.7 (dashed line), together with the parabolic level curveI(Amax, R) = 0. Vertical dashed line locates the asymptote of I = 0.
1.54, in excellent agreement with the inset in Fig. 3.2.
Since our model approximates an oscillon as having constant energy, we
input these values into Eq. 3.6, to obtain the oscillon energy, Eosc:
Eosc ' E(AB, RB) =(π
2
)d/2RdBV (AB, RB) = (3.11)
1
8
(π2
)d/2(8√
2
9
)d(V ′′′)4
(V ′′′′)3RdB = 41.3
where the numerical value given is for d = 3 and the potential of eq. 3.10.
Compare with numerical results [6, 8] or Fig. 3.3, where, roughly, 45 ≥ Eosc ≥
42 during the oscillon’s lifetime.
In addition to the above estimate for Eosc, we can derive an absolute lower
bound for the oscillon energy. This is found by noting that the dashed line in
Fig. 3.1 shows a critical energy, denoted Eattract, below which the energy curves
12
0 1000 2000 3000 4000 5000 6000 7000Time
0
1
2
3
4
5
Radius
Maximum Amplitude
50 100 150 2002
2.22.42.62.8
3Radius
0 2000 4000 6000 80001.25
1.3
1.35
1.4
1.45Frequency
Figure 3.2: The maximum amplitude and radius as a function of time fora configuration with R0 = 2.86 and A0 = 2 in the symmetric double wellof Eq. 3.10 (continuous lines), and for an asymmetric double well [V (φ) =12φ2 − 2.16
3φ3 + 1
4φ4] with A0 = 2 and R0 = 4 (dashed lines). The analytical
predictions at the points (AB, RB) and (AD, RD) are indicated by arrows anddots. The insets show the minimum radius and frequency for the symmetricdouble well.
13
0 1000 2000 3000 4000 5000 6000 7000Time
20
40
60
80
Energy
0
20
40
60
80
Figure 3.3: The dashed line represents the prediction of oscillon energy as afunction of time (Eq. 3.17) for the potential of Eq. 3.10. The vertical dashedline is drawn to represent the prediction of the time of decay of the oscillonτmaxlife. The continuous line is numerical energy from simulation for the longestlived oscillon in the same potential.
14
do not probe the stable region. Using the I = 0 condition, one can eliminate
R from the expression of the energy (Eq. 3.6), now a function only of Amax.
Remarkably, this function has a minimum, the asymptote energy of Fig. 3.1.
For the potential of Eq. 3.10, Eattract = 37.7 (d = 3) and Eattract = 4.44 (d = 2).
Having arrived at point B (after radiating excess initial energy) the oscillon
proceeds leftward along its line of constant energy, Eosc (Traveling rightward
would cause R to tend to zero, which is unphysical). When the oscillon passes
point C, its distance from I = 0 begins to increase and its instability grows.
To estimate the point of oscillon decay we note that, at point D, the vertical
distance to the I = 0 curve becomes infinite. Mathematically, point D is found
by taking the limit R → ∞ in the expression I = 0 and solving for AD, the
decay amplitude. For a cubic potential this gives AD = −(3/2)3/2(V ′′/V ′′′).
For a quartic potential,
AD = AB
1−
1− 2
(3√
2
4
)dV IV V ′′
V ′′′2
1/2 ' 0.84, (3.12)
where AB was obtained above and the numerical result is for the potential of
Eq. 3.10 in d = 3.
To obtain the final radius RD, we evaluate Eq. 3.11 at point D: Eosc =
(π/2)d/2RdDV (AD, RD). Solving for RD,
g(AD)RdD +
(d
2A2D
)Rd−2D − Eosc
(2
π
)d/2= 0, (3.13)
where g(AD) ≡ V ′′A2D/2 +
∑4n=3(2/n)d/2V nAnD/n!. In d = 3, one gets a
cubic equation for RD. For the potential of Eq. 3.10, with Eosc = 41.3 and
AD = 0.84, we obtain RD = 3.43. In Fig. 3.2, we compare numerical and
analytical values for (AB, RB) (left arrows and dots) and (AD, RD) (right dots)
15
0 1000 2000 3000 4000 5000 6000 7000Time
-0.4
-0.2
0
0.2
0.4
Stab
ility
Func
tion
I(R,A
)
Figure 3.4: Stability function I(R,Amax) as a function of time for the longest-lived oscillon in the potential of Eq. 3.10.
for this potential (continuous lines) and for an asymmetric potential (dashed
lines). The numerical simulation measures R by fitting a Gaussian to the
oscillon and reading off the radius.
Since we know Amax(t) throughout the oscillon’s life, we can use the poten-
tial V (A) to compute Amin and then use Eq. 3.7 to obtain ω(t). For example,
using Amax = AD gives the decay (or critical) frequency, ωD = ωcrit = 1.38 <
ωmass = V ′′(φv) =√
2. In the inset of Fig. 3.2 we compare analytical (dot) and
numerical values. It should be noted that this model correctly predicts that
the oscillon frequency (fundamental frequency of core oscillation) tends to a
maximum (as observed in, for example, [10]) as it approaches decay.
Finally, Fig. 3.4 shows a numerical plot of the stability function I(R,Amax)
as a function of time for the oscillon in the symmetric potential. As can be
seen, its behavior is exactly as predicted by the oscillon trajectory in Fig 3.1.
16
3.1.3 Oscillon Lifetime
If one plots the mod-squared of the Fourier transform in time (henceforth
known as power spectrum) of the core-amplitude of the oscillon one will, not
surprisingly, obtain a series of narrow peaks. The peaks are spaced at regular
intervals and represent harmonics of the fundamental frequency which lies just
below the mass frequency of the field. The peaks are of finite width owing to
the long term time-dependence (from radiation) experienced by the oscillon.
As we will show, the principal reason for the stability of the oscillon is
that the nonlinearity in the field (specifically, the second term in the potential
which creates negative curvature) serves to place large amplitude fluctuations
in the field at frequencies below the mass frequency. Therefore, the large
fluctuation is, in a sense, decoupled from the small-amplitude radiation waves
which would otherwise carry away its energy. We will show below that, in
addition to this effect, a second effect further stabilizes the oscillon: when the
pseudo-stable oscillon emits a small amount of radiation, it only narrows the
width of the harmonics in the power spectrum, thereby stabilizing itself.
To begin the analysis we model the power spectrum of the first peak in
the power spectrum with a Lorentzian (or Breit-Wigner) curve. Fig. 3.5 shows
an example of this (with the width of the peak greatly magnified for easier
viewing–had it been drawn to scale it would have been almost invisible).
Since the power is proportional to the square of the amplitude of a given
frequency component, we write
a2(ω) = K
[(ω − ωosc
12Γ
)2
+ 1
]−1
, (3.14)
where a2(ω) is the square of the amplitude of the frequency component of
φ(0, t) with frequency ω, and K is a constant to be determined. Although
17
1.2 1.3 1.4 1.5 1.6Frequency
0
0.2
0.4
0.6
0.8
1
1.2
Powe
r Spe
ctru
m
Figure 3.5: Example of a Lorentzian model of the first peak in the powerspectrum. The width has been greatly exaggerated (by three orders of mag-nitude) for easier visualization. Dashed lines show the frequencies ωmass andωrad, respectively.
the power spectrum has units of amplitude squared per frequency squared, we
normalize our curve to simply the square amplitude of the oscillon. This is
because we will not be interested in the values of the power spectrum itself, but
instead only physical amplitudes (see below). We therefore have a2(ωosc) =
K ≈ A2max, the oscillon’s core amplitude squared. Since the constant Γ setting
the width of the resonance peak is related to the (inverse) timescale associated
with the instantaneous radiation rate of the oscillon, it is natural to let Γ = dEdE
,
where E is the oscillon energy.
Most of the (small-amplitude) radiation leaking from the oscillon is con-
tained in a mode with wavelength commensurate with the size of the oscillon
– that is, with wavelength ∼ 4R. The frequency of this wave is therefore
ωrad =√m2 + (2π/4R)2, where we’ve employed the dispersion relation for a
linear Klein-Gordon field.
18
Now consider an infinitesimally thin shell of radius R around the oscil-
lon with volume cdRd−1dr, which is filled with the outgoing radiation wave
(ω = ωrad) traveling with speed vrad. Within this thin shell, the wave can
be approximated as a 1d plane wave of constant amplitude; hence its energy
density is A2radω
2rad/2, where Arad is the amplitude of the radiation wave in
the shell. Since A2rad is given by a2(ωrad), the total radiation energy contained
in the thin shell is a2(ωrad)ω2rad
2cdR
d−1dr. The radiation travels outward at a
speed vrad = ωrad/krad, and dr = vraddt. The amount of energy lost by the
oscillon per unit time, −E, is therefore
−E = A2 1
2ω2
radcdRd−1vrad
(ωrad − ωosc
12dEdE
)2
+ 1
−1
. (3.15)
If the oscillon is long lived, ωrad − ωosc dE/dE. In this case, and writing
dEdE
= dEdt
1E
, we have
E2 + 4αE3 = 0, (3.16)
where α ≡ 2(1 − ωosc/ωrad)2[A2cdRd−1vrad]−1. Eq. 3.16 is the main result of
this theory. It is a differential equation for the time dependence of the energy
of the oscillon.
To integrate eq. 3.16, we take α to be constant, a good approximation for
an oscillon. Then,
E(t) =Ei − E∞
α(Ei − E∞)t+ 1+ E∞, (3.17)
where E∞ is the asymptotic energy (to which E would tend as t → ∞ if it
were not to decay) and Ei the initial energy. The oscillon decays at energy
ED > E∞ in a time τlife given by
τlife =1
α
(1
ED − E∞− 1
Ei − E∞
). (3.18)
19
If Ei ∼ ED, the lifetime will be approximately zero, as expected. When
Ei ED, the lifetime becomes independent of Ei and tends to a maximum,
τmaxlife '1
α
1
ED − E∞. (3.19)
We now see that the theory presented here (namely, Eq. 3.16) makes a num-
ber of accurate predictions about oscillons. First, it predicts that the oscillon
should lose energy in time at an ever decreasing rate, tending to some equi-
librium value (attractor state). However, the oscillon’s nature as an attractor
configuration is emphasized even more strongly by the additional prediction
that the time scales associated with the motion (i.e., lifetime) are essentially
independent of initial conditions. In other words, the larger the initial en-
ergy, the faster the energy falls to reach the attractor state in a fixed time
(approximately τmaxlife).
One can show, using Eq. 3.17, that
Γ =E
E∝ α (E − E∞) . (3.20)
Therefore, the width of the peak decreases in time, as can be confirmed nu-
merically. In a sense, the oscillon is a self-organizing structure in that, by
radiating energy, it is capable of concentrating all of its own energy into one
single mode of oscillation (corresponding to a perfectly narrow peak, which it
achieves as t → ∞). Here we are assuming that power contained in higher
harmonics is negligible and that they also contribute negligibly to radiation
due to suppression from the spatial profile of the oscillon.
To evaluate the expression for τmaxlife, note that ED is simply the plateau
energy obtained in Eq. 3.11. For E∞, being the minimum energy the oscillon
may possess, it is most natural to choose E∞ = Eattract, where Eattract is
20
the minimum energy consistent with I ≤ 0 (see Fig. 3.1). For the other
parameters in α, we analytically calculate the average of each quantity over
the stable phase of the oscillon’s life (C → D in Fig. 3.1). For the potential
V (φ) = φ2−φ3+14φ4, we obtain τmaxlife ≈ 6300, which is quite accurate. Fig. 3.3
shows the prediction of energy as a function of time for the the potential
considered here and compares it with the numerical results.
Finally, note that Eq. 3.16 admits the solution, E = −α(E − Eattract)2.
Thus, if E → Eattract, E → 0. In d = 2, one can show numerically that the
lowest energy oscillon has Eosc = Eattract = 4.44. We thus see why τmaxlife →∞
in d = 2: the oscillon reaches its attractor value and stops radiating. In d = 3
the oscillon decays at an energy which is greater than E∞, and so it has a
finite lifetime.
21
Chapter 4
Frequency Analysis of Oscillons
In this chapter I will outline the work presented in [18], which improves on
the model presented in the last chapter by incorporating frequency analysis.
The result of this is a better estimation of the decay energy of an oscillon,
the radiation rate, and the time evolution of the amplitude, radius, energy,
and lifetime. It should be mentioned that the theory presented here still relies
upon the Gaussian ansatz. In the next chapter I will present some new work
on deriving the oscillon profile.
4.1 Linear vs. Nonlinear Dynamics and the
Oscillon Mass Gap
In order to introduce some of the basic quantities needed for our theory, it
is instructive to start by reviewing some of the main properties of relativistic
oscillons. We will do so in the context of a simple φ4 model with a symmetric
double-well potential, since this is also the main focus of the present work. To
begin, consider the Lagrangian for a spherically-symmetric, real scalar field in
22
d spatial dimensions,
L = cd
∫rd−1dr
[1
2φ2 − 1
2
(∂φ
∂r
)2
− V (φ)
], (4.1)
where V (φ) = m2φ2/2, and cd = 2πd/2/Γ(d/2) is the unit-sphere volume
in d dimensions. Quantities are scaled to be dimensionless as follows: φ =
m(d−1)/2φ0 and rµ = rµ0/m. We will henceforth only use dimensionless vari-
ables, dropping the subscript “0”.
We start by investigating the linear theory so that we can more easily
contrast it with nonlinear models that give rise to oscillons. Since oscillons have
been shown to maintain their approximate Gaussian-shaped spatial profiles
during their lifetimes, we will write the scalar field as
φ(r, t) = A(t)P (r;R) = A(t)e−r2/R2
. (4.2)
Here, A(t) is the time-dependent amplitude of the configuration and P (r;R)
its spatial profile, which is parameterized by the radial extension R.
4.1.1 Linear Dynamics
As shown in Ref. [8], the d = 3 linear theory with a Gaussian-profile initial
condition has the solution
φ(r, t) =R3
2
A√π
∫ ∞0
ke−R2k2/4 sin(kr)
rcos (ωt) dk, (4.3)
where A is an arbitrary initial amplitude and the dispersion relation is ω =
(k2 + m2)12 , where the mass frequency ωmass =
√2. To calculate Γlin, the
decay width associated with the above solution, we recall that in [8] it was
also shown that Eq. 4.3 can be approximately integrated to obtain (at r = 0,
23
the configuration’s maximum amplitude)
φ(0, t) =A0(
1 + 2t2
R4
) 34
cos
(√
2t+3
2tan−1
[√2t
R2
]), (4.4)
whose envelope of oscillation is given by φ(0, t) = A0/(1 + 2t2/R4)3/4, which
reaches 1/e of its initial value in a time given by Tlinear ' 0.836ωmassR2. This
yields the linear decay width Γlin,
1
2Γlin =
1
Tlinear
' 1.196
ωmassR2' 0.846
R2. (4.5)
In the linear theory, any initial configuration (or excitation above the vacuum)
will quickly decay by emitting radiation. The key difference between the linear
and the oscillon-supporting nonlinear models is that, in the latter case, the
decay modes are strongly suppressed. It is this suppression that gives rise to
the oscillon’s remarkable longevity. Our goal in this chapter is to make this
statement quantitatively precise. To obtain the linear radiation distribution,
which we denote by b(ω), we simply take the k-space representation ke−R2k2/4
of the Gaussian in Eq. 4.3 and express it in terms of ω using the dispersion
relation ω = (k2 + 2)12 :
b(ω) = k[ω]e−R2k[ω]2/4 =
(ω2 − 2
) 12 e−R
2(ω2−2)/4. (4.6)
The function b(ω) is a lopsided distribution with frequencies above ωmass =√
2 and peaked at ωmax = (2 + 2/R2)1/2. Now define ωleft and ωright to be
the frequencies where the distribution b(ω) rises to half of its peak value.
A straightforward calculation gives ωleft ' (2 + 0.203/R2)12 and ωright ' (2 +
7.38/R2)12 . This is found by starting with Eq. 4.6, substituting x2 ≡ R2(ω2−2),
finding the point x where the distribution falls to half its peak value (by
24
computer), then re-expressing in terms of ω. The radiation distribution is
then approximately centered on the frequency ωlin, given by (see Fig. 4.1),
ωlin ≡1
2(ωleft + ωright) (4.7)
' 1
2
(2 +
0.203
R2
) 12
+1
2
(2 +
7.38
R2
) 12
,
which we take to represent the dominant linear radiation frequency (doing
so gives a slightly better fit to the data than using ωmax for the dominant
frequency).
4.1.2 Nonlinear Dynamics and Decay Rate
Imagine now that one or more nonlinear terms are added to the linear po-
tential and that the field is again initialized with the same localized Gaussian
perturbation. Roughly speaking, the nonlinearities will shift the dominant
linear oscillation frequency ωlin, to a new value, denoted by ωnl. If the added
terms serve to decrease the curvature of the potential, then ωnl < ωlin. Such a
situation is depicted qualitatively in Fig. 4.1, where the arrow indicates how
the shift in frequencies occurs.
When nonlinearities are efficient enough that ωnl is lowered substantially
below ωlin, an oscillon may form. As the initial field configuration begins
to oscillate, it will attempt to emit small-amplitude radiation waves in an
effort to dissipate its energy. However, if ωnl is sufficiently less than ωlin, the
bulk of the frequency components composing the oscillation will be unable
to excite small-amplitude radiation waves (since the configuration can only
radiate appreciably in the frequency range ωleft < ω < ωright). This condition
leads to the stabilization mechanism responsible for the formation of oscillons.
It then follows that the oscillon will enter the nonlinear regime if the two
25
Frequencyω ωnl lin
Γlin
Figure 4.1: Schematic description of the linear radiation peak centered onωlin being shifted to the left by the presence of nonlinearities. An oscillonwill form if the peak is shifted far enough into the nonlinear region such thatit no longer overlaps significantly with the linear peak. The curves shownrepresent neither theoretical nor numerical results–instead, they are simplygeneral pictorial representations of the curves.
26
peaks in Fig. 4.1 do not significantly overlap. Mathematically, the right “edge”
of the nonlinear peak, given by ωnl + 12Γnl, must be less than the left “edge” of
the linear peak, given by ωlin− 12Γlin. Therefore, we have ωnl+
12Γnl < ωlin− 1
2Γlin
which, by defining ωgap ≡ ωlin − ωnl as the frequency gap between the linear
and nonlinear peaks, becomes
ωgap >1
2(Γnl + Γlin) . (4.8)
Since nonlinearities tend to increase a configuration’s lifetime and thus de-
crease its decay rate, it follows that 0 ≤ Γnl ≤ Γlin. One can thus state that
if
ωgap > Γlin, (4.9)
then the configuration will be forced into the nonlinear regime. In other words,
Eq. 4.9 is a necessary condition for the formation of an oscillon. As we will
soon see, if an oscillon is formed and, during the course of its time evolution,
reaches a point where Eq. 4.9 is no longer satisfied, it will decay. This implies
that oscillons decay when
ωgap = Γlin. (4.10)
To obtain the nonlinear radiation frequency ωnl and thus ωgap, we substitute
Eq. 4.2 into Eq. 4.1 and integrate, giving
L =(π
2
) d2Rd
[1
2A2 − Veff(Amax)
];
E =(π
2
) d2RdVeff(Amax), (4.11)
where V (A) now includes nonlinear terms and E is the energy, which is found
by taking the appropriate Legendre transform of the Lagrangian and evalu-
ating it at the upper turning point of an oscillation, Amax. The oscillation
27
0.5 1 1.5 2 2.5Amplitude
0
20
40
60
80
100
Ener
gy o
n I=
0 Cu
rve
0.5 1 1.5 2 2.5
0
20
40
60
80
100
Figure 4.2: Minimum oscillon energy as a function of core amplitude for adouble-well potential in d = 3. The minimum of this curve is the attractorpoint with E∞ ' 37.69.
frequency of the oscillon, ωnl, at a given time is given by
2π
ωnl
= Tosc =
∫ Tosc0
dt = 2
∫ Amin
Amax
dA
A, (4.12)
where A = [2E/cR − 2Veff(A)]1/2, cR ≡ (π/2)d/2Rd, and Amin is given by
Veff(Amin) = Veff(Amax).
4.1.3 The Attractor Point
Results from numerical simulations suggest that there exists an attractor point
in configuration space to which the oscillon tends. It was noted in [17] that
one can obtain the energy of this attractor point (in φ4 models) by finding the
minimum energy which has the property that the effective potential Veff(A)
possesses at least one point for which V ′′eff(A) ≤ 0. In order to compute the
28
attractor point, it is easier to work within a specific model. Choosing V (φ) =
φ2 − φ3 + φ4/4, we obtain, using Eq. 4.2 and integrating over all space,
Veff(A) =
(1 +
d
2R2
)A2 −
(2
3
) d2
A3 +A4
2d+42
, (4.13)
and
V ′′eff(A) =
(2 +
d
R2
)− 6
(2
3
) d2
A+ 3A2
2d2
. (4.14)
Equate this to zero and solve for R as a function of A. Then substitute the
result into Eq. 4.11 to eliminate R, yielding energy as a function only of A.
This curve possesses a minimum, shown in Fig. 4.2 for d = 3, the energy
of which yields the correct attractor energy E∞ of the oscillon, which has
numerical values E∞ ' 4.44 in d = 2 and E∞ ' 37.69 in d = 3.
Given the energies calculated above, there is a locus of points in (A,R)
parameter space (the thinner solid lines in Figs. 4.3 and 4.4) which possess
these asymptotic values for the energy; one of them is the attractor. To locate
the attractor point, we need to determine its amplitude coordinate, denoted
A∞. In d = 2, this is most easily determined numerically to be A∞ ' .3. In
d = 3, we cannot determine A∞ numerically since the oscillon decays before
reaching it. Therefore, in d = 3, we must estimate A∞ analytically. To do
this, we choose the point (satisfying E = E∞) which has ωnl ' ωmass, even
though, in reality, its frequency is slightly less than ωmass (never above it). As
we will see, this approximation will suffice for our purposes. From Eq. 4.12, the
amplitude which gives ωnl ' ωmass (in d = 3) has numerical value A∞ ' 0.456.
Given the pair (A∞, E∞) ' (0.456, 37.69) in d = 3 and (A∞, E∞) '
(0.3, 4.44) in d = 2, we can use Eq. 4.11 to obtain R∞. We obtain R∞ ' 4.79
for d = 3, and R∞ ' 5.77 in d = 2. The circles in Figs. 4.3 (d = 3) and 4.4
(d = 2) mark the locations of the attractor points.
29
0 0.5 1 1.5 2 2.5Amplitude
0
2
4
6
8
Radius
0 0.5 1 1.5 2 2.5
0
2
4
6
8
Figure 4.3: The thick solid line represents the locus of points satisfying thecondition ωgap = Γlin in d = 2 (“line of existence”) calculated analytically.The thin solid line represents the locus of points that have the attractor energyE ' 4.44. The dashed line is the numerical minimum radius based on Gaussianinitial configurations. Oscillons can only exist if they have energy E ≥ E∞and if they have a core amplitude and average radius lying above the line ofexistence. This is why the numerically measured minimum radius follows theline of existence for A & 1 but follows the line of minimum energy for A . 1.The arrow indicates the value of the (constant) minimum radius calculated in[8]. The circle represents the location of the attractor point; since it lies abovethe line of existence, oscillons will be absolutely stable in this system.
30
0 0.5 1 1.5 2 2.5Amplitude
0
1
2
3
4
5
6
Radius
0 0.5 1 1.5 2 2.5
0
1
2
3
4
5
6
Figure 4.4: The thick solid line represents the locus of points satisfying thecondition ωgap = Γlin in d = 3 (“line of existence”) calculated analytically.The thin solid line represents the locus of points that have the attractor en-ergy E ' 37.69. The dashed line is the numerical minimum radius based onGaussian initial configurations. The arrow indicates the value of the (con-stant) minimum radius calculated in [8]. Oscillons can only exist if their coreamplitude and average radius are above the curve (wherein E ≥ E∞ is auto-matically satisfied). The circle represents the location of the attractor point;since it lies below the line of existence, oscillons will not be absolutely stablein this system.
31
Given Eqs. 4.5, 4.7, and 4.12 (and their equivalents in d = 2 shown in
section 4.6) we can also calculate the quantities in Eq. 4.9, that is, the oscillon
existence condition, as a function of the parameter pair (A,R). In Figs. 4.3
(d = 2) and 4.4 (d = 3), the thicker continuous lines represent the locus of
points satisfying ωgap = Γlin, defining the boundary line between the region
where oscillons may exist (above the line, and provided that E > E∞) and
where they cannot exist (below the line). We will often refer to this boundary
as the “line of existence.” As a test of the existence condition (Eq. 4.9), we also
plot the numerical result for the “minimum radius” as a function of amplitude
(dashed line), found by pinpointing the minimum initial radius which causes a
configuration to live longer than the linear decay time. The arrows in Figs. 4.3
and 4.4 indicate the values of the minimum radii calculated in [8] which possess
no amplitude dependence and thus provide only limited information.
As can be seen, the attractor point in d = 3 lies below the line of existence
curve, explaining the finite lifetimes of oscillons in that system: the config-
urations decay before reaching the attractor point. On the other hand, the
attractor point in d = 2 lies above the curve, explaining the seemingly infinite
oscillon lifetimes observed in numerical simulations. We can thus interpret the
oscillons as time-dependent perturbations about the attractor point. Those in
d = 3 are unstable, albeit some can be extremely long-lived. Those in d = 2
are at least perturbatively stable.
In situations such as d = 3, where oscillons eventually decay, it is interesting
to compute their lifetimes and see how they depend on their radiation rate. In
a recent work, we presented the basic features of a method designed to do so
[17]. In the next section, we develop the appropriate formalism in detail, based
on the overlap between the nonlinear and linear radiation spectra. We point
out that our formalism is, in principle, applicable to any time-dependent scalar
32
field configuration, offering a much-needed handle on how to compute radiation
rates of nonperturbative configurations in relativistic scalar field theories.
4.2 Lifetime of Long-Lived Oscillons: General
Theory
In the situation that ωgap > Γlin and an oscillon has formed (above the solid
curve in Fig. 4.4) it will begin to radiate small amounts of energy. In this
section, we will derive a general equation governing its radiation rate so that,
in the event that ωgap = Γlin and the oscillon decays, we may calculate its
lifetime. As in the previous section, our general approach will be to compute
the overlap between the nonlinear peak and the linear peak. Since we are
assuming that ωgap > Γlin Γnl for a long-lived oscillon, this overlap will be
small; the amount by which it differs from zero will determine the radiation
rate.
In Fig. 4.5 we plot a possible frequency distribution for an oscillon (the
width of the peak centered on ωnl has been greatly exaggerated for visibility) in
units of the mass frequency (ωmass =√
2). Note how the tail of the distribution
“leaks” beyond the mass frequency. It is this leakage that will determine the
radiation-rate and thus the decay rate of the oscillon.
4.2.1 The Long-Lived Oscillon Radiation Equation
In order to compute the oscillon decay rate, we model oscillons as spherically-
symmetric objects whose radiation obeys a distribution (amplitude per unit
frequency) Ω(ω) which consists of a narrow peak of width δ centered at some
frequency ωrad. The radiation flux Φ (energy per unit time per unit surface
33
0.96 0.98 1 1.02 1.04Frequency
0.98 1 1.02 1.04 1.06 1.08 1.1
Figure 4.5: Schematic of an oscillon frequency distribution showing the tailpenetrating the radiation region. The graph is plotted in units of ωmass. In theinset we show a close-up view of F(ω) and b(ω), respectively (dashed lines),and their product Ω(ω) (solid line) for the system given by Eq. 4.1 in d = 3for typical values of the various parameters. Note that the curves have beenvertically scaled so that all are visible on the same graph.
34
area) emitted by such an object is
Φ ≡ −ES' ρv ' 1
2A2ω2
rad
ωrad
krad
(4.15)
' 1
2δ2Ω(ωrad)2ω
3rad
krad
where S is the surface area of the oscillon, ρ is the radiation-wave energy
density, v = ωrad/krad is the phase velocity of the wave, krad is the wave
number, and A is the amplitude of the radiation wave, which is given by
A =
∫ ∞ωmass
Ω(ω)dω ' Ω(ωrad)δ. (4.16)
The function Ω(ω), which represents the amplitude per unit frequency of the
radiation wave, is simply determined by the “overlap” between the oscillon
and the linear radiation peaks. Taking Ω(ω) (the overlap function) to be
the product of the nonlinear peak and the linear radiation distribution b(ω)
obtained in the previous section, we have
Ω(ω) = αF(ω)b(ω); (4.17)
≡ F(ω)b(ω)
where F(ω) is the nonlinear peak (Fourier transform of the oscillon’s core), α
is a proportionality constant to be determined, and b ≡ αb(ω). The inset in
Fig. 4.5 shows a typical overlap function for an oscillon in d = 3. In section
4.3 we show that, in the tail,
F(ω) ' −√
2
π
A
1 + χ(ω − ωnl)
−2. (4.18)
35
Combining Eqs. 4.15, 4.17, and 4.18 and letting S = cdRd−1, we have
ηdE
dt+
(dA
dt
)2
= 0, (4.19)
where η is a time-dependent parameter given by
η ≡ π(1 + χ)2(ωrad − ωnl)4krad
cdRd−1ω3radδ
2b(ωrad)2. (4.20)
Eq. 4.19 is a differential equation which must be satisfied by a long-lived
oscillon. In section 4.7 we compute both α and δ in general and in the context
of the model of Eq. 4.1.
4.2.2 Dynamical Exponents
The time dependence of the oscillon and all of the various parameters which
describe it is due to a single physical mechanism, namely the emission of
radiation. If the radiation rate is high (low), all parameters will change rapidly
(slowly). In other words, the general nature of the time dependence of a
parameter will follow that of any other parameter.
In analogy with Eq. 4.58, define
ΓX ∝X
X −X∞(4.21)
where X∞ is the value assumed by the parameter X at the attractor point
(obtained numerically). ΓX can be interpreted as the decay width for the
parameter X. Now consider the generic (bulk) parameters X(t) and Y (t)
(which could represent amplitude, radius, frequency, etc.). It is found from
simulation that the long-time, macroscopic time evolution of all basic oscillon
parameters is governed by a single timescale (if one were to fit power laws
36
to any parameter X(t) the coefficient multiplying t, which sets the timescale,
would be the same for any parameter). Mathematically, this implies that ΓX
and ΓY , the rates of change of the parameters X(t) and Y (t), respectively, are
to be proportional with regard to their time dependence,
ΓX ∝ ΓY (4.22)
for any parameters X(t), Y (t). That is, both ΓX and ΓY decay in time like
t−1 on account of the power law behavior of the parameters X and Y . We can
then write, quite generally, ΓX ∝ ΓE for any parameter X, where E denotes
energy. Combining this with Eq. 4.21 yields
X
X(t)−X∞= ρX
E
E(t)− E∞, (4.23)
where ρX denotes the proportionality constant. Integrating both sides of
Eq. 4.23 yields
[X(t)−X∞] = γX [E(t)− E∞]ρX , (4.24)
where γX is the constant of integration.
We will now derive an expression for the general exponent ρX . First define
the new variable
νX ≡ ln(X −X∞), (4.25)
which we will henceforth employ, instead of X, to describe the oscillon. When
combined with Eq. 4.23 this gives
ρX =dνXdνE
. (4.26)
During the evolution of an oscillon, the change in the coordinate νX is given
37
by
dνX = ∇νX · d~ν, (4.27)
where d~ν is a differential vector which lies tangent to the trajectory of the
oscillon in ~ν space and ∇νX is the vector gradient of νX whose direction lies
perpendicular to the “level” curves associated with νX . To calculate d~ν we
note that, by virtue of the attractor-like nature of oscillons, the oscillon will
evolve according to a trajectory that runs perpendicular to the level curves
associated with the radiation rate. Mathematically, we can write
∇νE × d~ν = 0. (4.28)
To proceed, we will make use of our initial assumption in Eq. 4.50 that the
oscillon (and hence any oscillon parameter X or νX) can be taken to be a
function of two degrees of freedom. These two degrees of freedom can be
arbitrarily chosen to be any two independent oscillon parameters (so long as
the state of the oscillon is uniquely defined by the coordinate pair). Choosing
the coordinate pair ~ν = (νE, νA) Eq. 4.28 becomes
∂νE∂νE
dνA −∂νE∂νA
dνE = 0 (4.29)
Combining this with Eq. 4.27 and dividing by dνE we have
dνXdνE
=
∂νE∂νE
∂νX∂νE
+∂νE∂νA
∂νX∂νA
∂νE∂νE
. (4.30)
38
Combining Eqs. 4.36 and 4.26 we have
ρE = 2− 2ρA + ρη (4.31)
=d
dνE(2νE − 2νA + νη) =
d
dνEνE,
which implies that, up to a constant, νE = 2νE − 2νA + νη. Substitution of
this expression for νE into Eq. 4.30 and combining with Eq. 4.26 yields the
desired result:
ρX =
(2 + ∂νη
∂νE
)∂νX∂νE
+(∂νη∂νA− 2)∂νX∂νA(
2 + ∂νη∂νE
) . (4.32)
Eq. 4.32 gives the power to which E−E∞ shall be raised in order to obtain the
time dependence of the function X(t) (see Eq. 4.24). This is essential because
it will allow the time dependence of any oscillon parameter to be expressed in
terms of the energy E(t). For example, given an equation for the radiation rate
of the form E = f (A,R,E, ω), where f is some function of the arguments,
we can express all quantities on the right hand side in terms of the energy E
using Eq. 4.32, thereby permitting a simple integration of the equation.
4.2.3 Integration of the Long-Lived Oscillon Radiation
Equation
In this section, we will attempt to integrate Eq. 4.19 to obtain the oscillon
energy as a function of time. This process is not straightforward since Eq. 4.19
contains derivatives of two different quantities (amplitude and energy) and the
parameter η possesses a complicated time dependence which is not known.
However, the results of the previous section can be used to solve this problem.
The main result of that section which we will employ below is Eq. 4.24.
First, write Eq. 4.24 for the cases X = A and X = η and substitute into
39
Eq. 4.19, obtaining,
E =−1
ρ2Aγ
2A
[η∞ + γη(E − E∞)ρη ] (E − E∞)2−2ρA . (4.33)
Now, consider the situation where η∞ ' 0 (which is the case in the system
we are studying here since, in d = 3, the attractor point satisfies ωnl ' ωmass,
leading to ωrad ' ωmass, which causes η to be zero there). In this situation,
Eq. 4.33 is somewhat simplified:
E = −γE (E − E∞)ρE , (4.34)
where the constant γE is given by
γE ≡γηρ2Aγ
2A
, (4.35)
and the exponent ρE is (using that [E]∞ = 0)
ρE = 2(1− ρA) + ρη. (4.36)
Eq. 4.34 is a constant-coefficient, ordinary differential equation governing the
oscillon energy as a function of time, and can be easily integrated:
E(t) = E∞ +Ei − E∞
[1 + γEg(Ei − E∞)gt]1g
, (4.37)
where g ≡ ρE − 1 and Ei is the energy at t = 0. Eq. 4.37 is the energy of an
oscillon as a function of time.
As shown in Fig. 4.4, in d = 3 an oscillon will always decay before reaching
E = E∞. The decay is quite sudden, a burst of scalar radiation. As stated in
section II, this occurs when ωgap = Γlin; hence the decay energy, denoted ED,
40
is given by
ED = E|[ωgap=Γlin]. (4.38)
To calculate the lifetime, denoted Tlife, which is defined as the amount of time
taken for the oscillon to decay from a sufficiently high initial energy down to
ED, we invert Eq. 4.37 to yield time as a function of energy and evaluate at
ED:
t(ED) =1
γEg
[1
(ED − E∞)g− 1
(Ei − E∞)g
]. (4.39)
When Ei−E∞ E−E∞, the function t(E) tends to a finite, maximum value
leading to
Tlife =1
γEg
1
[ED − E∞]g. (4.40)
This means that, when the initial energy Ei is much larger than the energy
ED in question, the time it takes for E(t) to fall from Ei to ED becomes
independent of the initial condition (i.e., from Ei). One can then say that, in
a restricted sense, the long-lived oscillon is decoupled from initial conditions:
if the necessary conditions for its existence are satisfied, a variety of initial
configurations will approach an oscillon. Recent studies that have observed
the emergence of oscillons from stochastic initial conditions after a fast quench
offer strong support for this claim [19].
Eqs. 4.38 and 4.40 together give the lifetime of an oscillon and can be
considered the main results of this chapter. Before moving on, we note that,
for a long-lived oscillon, Γnl is given by
Γnl ≡ ΓA = − A
A− A∞= −ρA
E
E − E∞(4.41)
= ρAγE(E − E∞)g,
where we’ve used Eqs. 4.58, 4.23, and 4.34. This is related to the lifetime by
41
(combine Eqs. 4.40 and 4.41)
Tlife =
(ρAg
)Γ−1
nl
∣∣∣∣E=ED
. (4.42)
4.2.4 Sample Calculation: φ4 Klein-Gordon Field in d =
3
We now apply the above results to the system given by the Lagrangian in
Eq. 4.1 for d = 3, supplemented by the nonlinear potential of Eq. 4.13. We
will begin with the existence condition shown in Fig. 4.4. In section II.c we
calculated the coordinates of the attractor point and obtained (A∞, R∞) '
(.456, 4.79). Comparison with Fig. 4.4 reveals that the attractor point lies
below the curve, and thus that ωgap/Γlin < 1 there. As we noted before,
stable oscillons will not exist in this model. However, there are still long-lived
oscillons obtained by initializing the field sufficiently far from the attractor
point.
As these structures radiate energy, their amplitude A and radius R will
change in time. Hence, they will trace out trajectories in the (A,R) plane
of Fig. 4.4, all of which will eventually intersect the line of existence. To
calculate these trajectories, begin by writing Eq. 4.24 for A(t) and R(t), ob-
taining [A(t)− A∞] = γA[E(t)− E∞]ρA and [R(t)− R∞] = γR[E(t)− E∞]ρR ,
respectively. Substituting the first into the second to eliminate the energy, we
obtain
R(t) = R∞ + γRA[A(t)− A∞]ρRA , (4.43)
where γRA ≡ γR/γρRAA and ρRA ≡ ρR/ρA. Note that when A = A∞, R = R∞.
Since each possible trajectory will intersect the line of existence, every point
along the line of existence is a point along some trajectory. Thus, we can choose
42
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3Amplitude of Reference Point
0
1
2
3
4
Expo
nent
s
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0
1
2
3
4
Figure 4.6: Analytical calculation of g = ρE − 1 (top curve) and ρA (bottomcurve) using Eq. 4.44, plotted against the reference amplitude Ar along theline of existence. This gives the spectrum of possible values of g and ρA acrossthe various oscillons in this system. The dots mark the theoretical valuesof g ' 2.67 and ρA ' .66 assumed by the longest-lived oscillon, which hasAr ' .92 (thicker line in Fig. 4.8).
43
0.6 0.7 0.8 0.9 1 1.1 1.2 1.30
2x10-6
4x10-6
6x10-6
8x10-6
0.000010.6 0.7 0.8 0.9 1 1.1 1.2 1.3
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3Time
0
0.1
0.2
0.3
0.4
0.50.6 0.7 0.8 0.9 1 1.1 1.2 1.3
Figure 4.7: Analytical calculation of γE (top graph) and γA (bottom graph)using Eq. 4.45, plotted against the reference amplitude Ar along the line ofexistence. This gives the spectrum of possible values of γE and γA across thevarious oscillons in this system. The dots serve to mark the theoretical valuesof γE ' 2.0× 10−6 and γA ' .21 assumed by the longest-lived oscillon, whichhas Ar ' .92 (thicker line in Fig. 4.8).
44
an arbitrary point on this line (which we will call the “reference point”) and use
its coordinates, labeled (Ar, Rr), to calculate all of the dynamical exponents
(ρA, ρR, ρE, ρω, ρη, etc.) associated with the trajectory that intersects that
point. Eq. 4.32, reproduced below, was evaluated numerically for X = ρE and
X = ρA, respectively:
ρX =
(2 + ∂νη
∂νE
)∂νX∂νE
+(∂νη∂νA− 2)∂νX∂νA(
2 + ∂νη∂νE
) . (4.44)
Fig. 4.6 shows the results for the exponents g ≡ ρE − 1 (top curve) and ρA
(bottom curve) as a function of Ar.
Given values for ρX , we can calculate γX by evaluating each side of Eq. 4.24
at the reference point (wherein X assumes the value Xr = X[Ar, Rr]) and
solving for γX , obtaining
γX =Xr −X∞
(Er − E∞)ρX. (4.45)
Fig. 4.7 shows the results of such a calculation for γE (top graph) and γA
(bottom graph).
Having computed ρX and γX for the parameters A and R allows us to
calculate the coefficients in Eq. 4.43 for each trajectory (i.e., compute Eqs.
4.44 and 4.45 for various reference points along the line of existence). The
result of this is shown in Fig. 4.8 for selected trajectories. Note that they
all asymptotically tend toward the attractor point but intersect the line of
existence before doing so.
As an example, consider the trajectory which intersects the line of existence
at Ar ' .92 (this will be shown to correspond to the longest-lived oscillon
in the system of Eq. 4.1). From Figs. 4.6 and 4.7 we have g ' 2.67 and
45
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Amplitude
2
2.5
3
3.5
4
4.5
5
Radius
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
2
2.5
3
3.5
4
4.5
5
Figure 4.8: Various oscillon trajectories. Note that they all tend to the at-tractor point but intersect the line of existence (dashed line) before doing so.The thicker trajectory marks the longest-lived oscillon (it intersects the line ofexistence at Ar ' .92 and Rr ' 3.25).
46
γE ' 2.0 × 10−6. If this oscillon were initiated at an energy of Ei ' 82.5
(which will be the case for the numerical simulation we will be comparing to)
then Eq. 4.37 for that oscillon becomes
E(t) ' 37.69 +44.8
[1 + (0.136)t]0.375 , (4.46)
which, for times t & 100 is approximately
E(t)− E∞ '94.77
t0.375. (4.47)
Since the decay energy for this oscillon is ED = Er = Er(Ar, Rr) ' 41.0963,
Eq. 4.40 yields
Tlife '1
[2.0× 10−6][2.67]
1
[41.10− 37.69]2.67(4.48)
' 7100.
Given Eq. 4.46 we can write, for example, an expression for the oscillon am-
plitude as a function of time. From Figs. 4.6 and 4.7 we have ρA ' 0.67 and
γA ' 0.21. Combining this information with [A − A∞] = γA[E − E∞]ρA and
Eq. 4.46 we have
A(t) ' 0.456 +2.58
[1 + (0.136)t].249 . (4.49)
Carrying out the above calculations for several trajectories along the line of
existence and plotting the lifetime vs. Ar, Rr, and Er, results in Figs. 4.9 and
4.10, respectively. Fig. 4.9 shows the analytical decay amplitude and radius
as a function of lifetime (dashed lines) plotted against several long-lived and
short-lived oscillons, with excellent agreement: oscillons decay as they cross
the coordinates specified by the line of existence. Fig. 4.10 shows the computed
47
0 1000 2000 3000 4000 5000 6000 70000
0.5
1
1.5
2
Amplitude
0
0.5
1
1.5
2
0 1000 2000 3000 4000 5000 6000 7000Time
0
1
2
3
4
5
6
Radius
0 1000 2000 3000 4000 5000 6000 7000
0
1
2
3
4
5
6
Figure 4.9: Analytical results for critical values of the amplitude (top) andradius (bottom) along the line of existence (dashed lines) as a function oflifetime are plotted with several examples of short- and long-lived oscillons,all with initial amplitudes A0 = 2. From left to right, the initial radii for theoscillons are 2.35, 2.41, 2.53, 2.65, and 2.86. It is quite clear that the oscillonsdecay as they cross the critical values computed analytically.
lifetime as a function of decay energy (ED = Er), reproducing the maximum
lifetime on the order of 104 which is characteristic of oscillons in this system.
The figure compares the analytical computation of lifetime (continuous line)
with the numerical results (dashed line). The small disparity in the center of
the peak (of order ∼ 6%) is probably due to the Gaussian ansatz we use to
describe oscillon configurations.
As was done in Eqs. 4.46 and 4.49, our method allows us to investigate
an oscillon evolving along a particular trajectory in detail. As an example,
we consider the longest-lived oscillon in this model, obtained with the initial
parameters (A0 = 2; R0 = 2.86). Using the information from the curve
in Fig. 4.10, we can find the trajectory whose lifetime corresponds to this
48
0 20 40 60 80Decay Energy
0
2000
4000
6000
8000
10000
Life
time
0 20 40 60 80
0
2000
4000
6000
8000
10000
Figure 4.10: Oscillon lifetimes vs. decay energy ED = Er. Solid curve istheoretical, dashed line is numerical. The theoretical curve (with an error of∼ 6% in the horizontal positioning of the peak) correctly predicts the shapeof the distribution and that there exists a maximum lifetime in this system onthe order of ∼ 104.
49
oscillon (Tlife ' 7100), marked as the thicker line in Fig. 4.8 (the coordinates
of the reference (or decay) point are Ar ' 0.92 and Rr ' 3.25). We then
use Eqs. 4.44 and 4.45 to calculate ρX and γX for any parameter of interest.
Then, combining these values with Eqs. 4.37 and 4.24, we can compute the
amplitude (Eq. 4.49), radius, frequency, energy (Eq. 4.46), and radiation rate
as functions of time and compare the results with the numerical values. The
results are plotted in Figs. 4.11, 4.12, and 4.13, showing excellent agreement.
We can also plot the theoretical prediction for the trajectory of this oscillon
in the (A,R) plane (shown by the thicker line in Fig. 4.8), and compare it to
the numerical value, as shown in Fig. 4.14.
In the next section, we will complete the characterization of this system by
deriving an expression for the frequency of the superimposed oscillation ob-
served in, for example, Fig. 4.9, which seems to be connected with the oscillon
decay process: the larger the amplitude of the superimposed oscillation, the
shorter the lifetime.
4.3 Derivation of Nonlinear Frequency Peak
In this thesis we consider spherically symmetric oscillons that can be accu-
rately modeled by an oscillating field configuration whose spatial profile and
amplitude of oscillation vary little over the course of an oscillation, i.e.,
φ(r, t) ' Ac(t)P (r;R) ≡ Aosc(t)A(t)P (r;R) (4.50)
where φ is the field, r is radial position, t is time, P (r) is the spatial pro-
file of the oscillon normalized so that P (r) = 1 at the origin, R is a time-
dependent measure of the spatial extent of the oscillon (i.e., the “radius”)
which is assumed to vary little over the period of a single oscillation, Ac(t) is
50
0 1000 2000 3000 4000 5000 6000 7000Time
0
1
2
3
4
Amplitude/Frequency/Radius
0 1000 2000 3000 4000 5000 6000 7000
0
1
2
3
4
Figure 4.11: Comparison of theoretical (continuous line) vs. numerical (dashedline) radius (top), frequency (middle) and amplitude (bottom) for an oscillonwith initial conditions (A0 = 2, R0 = 2.86), showing very good agreement[theoretical results are computed with (Ar, Rr) ' (.92, 3.25)].
51
0 1000 2000 3000 4000 5000 6000 7000Time
0
20
40
60
80
Energy
0 1000 2000 3000 4000 5000 6000 7000
0
20
40
60
80
Figure 4.12: Comparison of theoretical (continuous line) vs. numerical (dashedline) values for the energy of the longest-lived oscillon, obtained with initialconditions (A0 = 2, R0 = 2.86) showing excellent agreement [(Ar, Rr) '(.92, 3.25)].
52
0 1000 2000 3000 4000 5000 6000 7000Time
0
0.002
0.004
0.006
0.008
0.01
Radi
atio
n Ra
te (L
inea
r)
0 1000 2000 3000 4000 5000 6000 7000
0
0.002
0.004
0.006
0.008
0.01
0 500 1000 1500 20000.0001
0.001
0.01
0.1
1
10
Figure 4.13: Comparison of theoretical (continuous line) vs. numerical (wavyline) radiation rate for an oscillon with initial conditions (A0 = 2, R0 = 2.86)showing excellent agreement. The inset, which is plotted on a log scale, makesit clear that the theory correctly reproduces the rapid initial drop in radiationrate over many orders of magnitude; the linear scale on the larger graph showsthat the theory correctly reproduces the extremely small (but finite) radiationrate towards the end of the oscillon’s life [(Ar, Rr) ' (.92, 3.25)].
53
0 0.5 1 1.5 2Amplitude
0
1
2
3
4
5
6
Radius
0 0.5 1 1.5 2
0
1
2
3
4
5
6
Figure 4.14: Comparison of theoretical trajectory (solid curve) vs. numericaltrajectory (dashed curve) for the longest-lived oscillon (A0 = 2, R0 = 2.86)showing very good agreement during the more stable phase of the oscillon’slife (A . 1.5). The great increase in density of data points in the dashed linein the range .9 . A . 1.5 is due to the prolonged period of time spent in thisregion by the oscillon (i.e., the “plateau” phase). The point where the lineagain becomes dashed (A ' .9, R ' 3.2) signals the numerical decay pointof the oscillon. The thick segment of the solid line highlights the portion ofthe theoretical trajectory during the low-radiation plateau phase; the end ofthe thick segment (Ar, Rr) ' (.92, 3.24) marks the theoretical decay point,showing very good agreement. It is interesting to note that, even after theoscillon decay at A ' .9, the remaining field configuration continues to tendto the attractor point at (A∞, R∞) ' (.456, 4.79), as does the theoretical curve.
54
the time-dependent oscillon core [φ(0, t)], A(t) is its time-dependent envelope
of oscillation, and Aosc(t) is an oscillating function which is normalized to an
upper turning point of unity.
If the oscillon oscillates approximately harmonically we can write
Aosc(t) 'χ
1 + χ+
cos(ωnlt)
1 + χ(4.51)
where ωnl is the time-dependent frequency of the oscillon and χ is a dimen-
sionless constant which accounts for a possible non-zero center of oscillation.
Now assume that we are interested in calculating the radiation rate of
the oscillon at t = 0. Since the oscillon radiation rate depends only on the
instantaneous properties of the oscillon, writing ωnl(t) = ωnl(0) in Eq. 4.51
will still yield the correct radiation rate at t = 0. Combining Eqs. 4.50 and
4.51 and taking the unitary cosine transform of Ac we have
F(ω) =
√2
π
∫ ∞0
cos(ωt)Ac(t)dt (4.52)
=
√2
π
∫ ∞0
cos(ωt) cos(ωnl(0))A(t)
1 + χdt,
where in the second step we’ve multiplied A(t) by Aosc(t) in Eq. 4.51 and
dropped all terms except the one proportional to cos(ωnlt)A(t) which will cre-
ate a finite width peak centered at the oscillon frequency (all others are irrel-
evant and do not contribute to the radiation rate). We can rewrite Eq. 4.52
as
F(ω) =1√2π
∫ ∞0
[cos(ω+t) + cos(ω−t)]A(t)
1 + χdt, (4.53)
where ω+ ≡ ωnl + ω and ω− ≡ ωnl − ω. Since each of the two cosine terms
above will contribute an identical peak (one centered at +ωnl and the other at
55
−ωnl) and both will contribute identically to the radiation rate, we can simply
drop the one centered on −ωnl and multiply by two, yielding
F(ω) =
√2
π
∫ ∞0
cos([ωnl − ω]t)A(t)
1 + χdt. (4.54)
Now, since the behavior of an oscillon’s amplitude A(t) is to tend to a finite
value (even if it dies before reaching this value, it still behaves as if tending to
a finite value), denoted A∞, as t→∞, we define A∆ ≡ A(t)− A∞ and write
F(ω) =
√2
π
∫ ∞0
cos([ωnl − ω]t)(A∆ + A∞)
1 + χdt (4.55)
=
√2
π
∫ ∞0
cos([ωnl − ω]t)A∆
1 + χdt,
where we’ve dropped the transform of the A∞ term since it will produce a
delta function centered at ωnl which will not in anyway affect F(ω) in the
region of interest (the radiation zone).
To proceed, we use the fact (found by performing successive integrations
by parts) that, for a function f(x) which tends to zero as x→∞,
∫ ∞0
cos(sx)f(x)dx =∞∑n=1
(−1)nf (2n−1)(0)
s2n, (4.56)
where f (m) is the mth derivative of the function f . We apply Eq. 4.56 to
Eq. 4.55 and note that, for a long-lived oscillon, the flatness of A(t) implies
that, for s in the tail, we need only keep the first term, yielding
F(ω) ' −√
2
π
A
1 + χ
1
(ωnl − ω)2, (4.57)
where we’ve switched the evaluations at t = 0 to a general time t, since the
above calculation can be applied to any physical time. Eq. 4.57 essentially
56
states that when a peak associated with a decaying function has a very small
width, the tail goes like (ωnl − ω)−2.
It will be useful to note that F(ω) in Eq. 4.55 will be a peak whose width,
denoted ΓA, scales as
ΓA ∼(
A∆
1 + χ
)−1d
dt
(A∆
1 + χ
) ∣∣∣∣t=t′
=A
A− A∞, (4.58)
where we’ve used that χ is a constant, A∆ ≡ A− A∞, and A∞ is a constant.
4.4 Analysis of Oscillon Stability as a Func-
tion of Time
Recall from the first chapter that the three conditions representing an oscillon
before decay, at the point of decay, and after decay, respectively, are:
ωgap > Γlin; (4.59)
ωgap = Γlin;
ωgap < Γlin.
In this section, we seek to investigate the concept of oscillon stability in more
depth. In doing so, we will obtain a more general formulation of Eqs. 4.59
which will provide a precise measure of the oscillon’s stability when ωgap 6= Γlin.
We will then see that an expression for the frequency of the superimposed
oscillation seen in Fig. 4.9, which is clearly related to stability, will naturally
emerge.
In deriving the equations governing the radiation rate and lifetime in the
previous sections, we made the simplifying assumption that the oscillons under
57
study are long-lived. Mathematically, this assumption is employed in approx-
imating the series expansion in Eq. 4.56 by its first term, yielding Eq. 4.57.
The more stable the oscillon, the smaller a given term in the series expansion
will be relative to the term before it.
Let νn denote the magnitude of the nth term in the series of Eq. 4.56.
In this section, instead of assuming that νn νn+1, we will compute the
fractional difference between two adjacent terms and take the result to be a
natural measure of the stability of the oscillon.
Define the (dimensionless) stability function Σ from two adjacent terms in
the series of Eq. 4.56 as
Σ ≡ νn − νn+1
νn. (4.60)
When the oscillon is highly stable, νn νn+1, and Σ → 1; conversely, as
νn+1 → νn (causing the series in Eq. 4.56 to fail to converge) then Σ→ 0. In
section 4.8 (see Eq. 4.91) it is shown that, for any value of n,
Σ = 1−(
Γnl
ωgap
)2
. (4.61)
Eq. 4.61 is refered to as the stability function; as would be expected, it involves
the ratio between ωgap and Γnl. Using Eq. 4.41 for Γnl, we can plot Eq. 4.61
for the longest-lived oscillon (using Eqs. 4.44 and 4.45). This is shown in the
top graph of Fig. 4.15. The extreme closeness of Σ to unity for most of the
oscillon’s life, when compared to Eq. 4.60, verifies that we are quite justified
in assuming νn νn+1.
The minimum stability allowed at a given time, Σmin, is attained when Γnl
is at its maximum, namely, when Γnl = Γlin,
Σmin = 1−(
Γlin
ωgap
)2
. (4.62)
58
Now note that the conditions in Eq. 4.59 can be written in terms of Σmin as
Σmin > 0; (4.63)
Σmin = 0;
Σmin < 0,
respectively. In the bottom graph of Fig. 4.15, we plot Σmin for the longest
lived oscillon (again, using Eqs. 4.44 and 4.45 and with Γlin given by Eq. 4.5).
It is clear that one can interpret the stability Σ as a measure of the radia-
tion rate of the oscillon: the radiation rate decreases in time, so the stability
increases. On the other hand, Σmin measures the resistance of the oscillon
against spontaneous decay: as the oscillon evolves and approaches the line of
existence, the buffer against instability of this kind decreases.
We will now write the conditions in Eqs. 4.59 and 4.63 in a third and final
form. First, observe that we can write Σmin as
Σmin =
(ωmod
ωgap
)2
, (4.64)
where
ωmod ≡√ω2
gap − Γ2lin. (4.65)
In terms of ωmod, the conditions in Eq. 4.63 become
ωmod ∈ <, 6= 0; (4.66)
ωmod = 0;
ωmod ∈ =,
for Σmin > 0, Σmin = 0, and Σmin < 0, respectively. In other words, if we
59
0 1000 2000 3000 4000 5000 6000 70001x10-81x10-71x10-6
0.000010.0001
0.001
0.01
0.1
1 - (
Stab
ility
Func
tion)
0 1000 2000 3000 4000 5000 6000 7000
0 1000 2000 3000 4000 5000 6000 7000Time
0
0.05
0.1
0.15
0.2
0.25
0.3
Min
. Sta
bility
Fun
ctio
n
0 1000 2000 3000 4000 5000 6000 7000
Figure 4.15: The top graph shows the theoretical calculation of (1− Σ) vs.time and the bottom shows Σmin, both for the oscillon with (Ar, Rr) '(.92, 3.25). The stability measured by Σ is clearly related to the radiationrate of the oscillon: this kind of stability becomes greater in time, since theradiation rate decreases. On the other hand, the stability measured by Σmin
is related to the resistance of the oscillon to decay: as time evolves and theoscillon moves closer to the line of existence, the buffer against instabilitydiminishes.
60
consider the quantity B(t) ≡ eiωmodt, then the real part of B(t) before the
decay, at the decay point, and after the decay are
B(t) = cos(√
ω2gap − Γ2
lint)
; (4.67)
B(t) = 1;
B(t) = e±Γlint,
for Σmin > 0, Σmin = 0, and Σmin < 0, respectively, where the last condition
follows since, after the decay is initiated, ωgap tends to zero, making ωmod tend
to ±iΓlin.
Therefore, we can conclude that ωmod is a special frequency associated
with the decay of the oscillon whose value decreases in time and, at a certain
point, becomes imaginary, signaling the oscillon’s final demise with timescale
on the order of the linear decay width (Eq. 4.67). In fact, as mentioned
previously, such a phenomenon is commonly observed in, for example, Fig. 4.9.
Specifically, there exists a modulation oscillation whose frequency tends to
decrease as time progresses, until, at a certain point, the oscillon decays with
width ∼ Γlin.
In Fig. 4.16 we plot the period
Tdecay ≡2π
ωmod
=2π√
ω2gap − Γ2
lin
, (4.68)
along with the numerically measured period of the superimposed oscillation.
As shown, Tdecay quite accurately reproduces this frequency.
In conclusion, we now have three separate (yet equivalent) formulations
of the condition for oscillon decay. The first says that the nonlinear and
linear peaks must significantly overlap (Eqs. 4.59). The second says that the
61
0 1000 2000 3000 4000 5000 6000 7000Time
0
100
200
300
400
500
Deca
y O
scilla
tion
Perio
ds
0 1000 2000 3000 4000 5000 6000 7000
0
100
200
300
400
500
Figure 4.16: The solid curve is the theoretical calculation of Tdecay for(Ar, Rr) ' (.92, 3.25). The dashed curve is the numerically measured periodof the superimposed oscillation, showing very good agreement.
measure of oscillon stability must fall to zero (Eqs. 4.63). The last states that
the modulation frequency ωmod must become imaginary (Eqs. 4.66).
4.5 The Four Oscillon Timescales
We will now review the four timescales associated with oscillons encountered
in our theory. They are:
Trelax = −E − E∞E
=1
γE[E − E∞]g= Γ−1
E (4.69)
Tdecay =2π√
ω2mod − Γ2
lin
(4.70)
Tlinear ∼ ωmassR2 (4.71)
Tosc =2π
ωnl
. (4.72)
62
The first, Trelax, is the relaxation time of the oscillon and is typically the longest
timescale present. This is the timescale over which the oscillon experiences
significant change. Its net value expresses the inverse rate of energy radiation,
thus being largest where the oscillon radiates the least, as can be seen from
the flatness of curves such as those in Fig. 4.11 and 4.12. This is linked to the
lifetime by,
Tlife =1
γEg
1
[ED − E∞]g=
1
gTrelax
∣∣∣∣E=ED
(4.73)
∼ Trelax
∣∣∣∣E=ED
.
where we’ve used Eq. 4.40 and the fact that the dynamical exponents are
typically of order unity (see Fig. 4.6).
The second timescale, Tdecay, is the period of the superimposed oscillations
seen in the oscillon as a result of its motion towards the line of existence. The
third is the decay time of an object in the linear theory [8]. The fourth, and
shortest, timescale is the oscillation period of the oscillon.
The theoretical values of the four timescales are plotted in Fig. 4.17 vs.
time for the longest-lived oscillon. Note how, together, they span many orders
of magnitude. It is the presence of these four widely different timescales in
one single system that makes oscillons such intriguing objects to study.
4.6 Linear Radiation Distribution in d = 2
We begin by expanding the Gaussian in eigenfunctions (zeroth-order bessel
function of the first kind) of the two-dimensional, spherically symmetric Klein-
Gordon equation:
A(t)e−ρ2
R2 =
∫ ∞0
b(k)J0(kρ)dk. (4.74)
63
0 1000 2000 3000 4000 5000 6000 7000Time
1
10
100
1000
10000
100000
Tim
e Sc
ales
0 1000 2000 3000 4000 5000 6000 7000
1
10
100
1000
10000
100000
Figure 4.17: The four oscillon time scales plotted over time. From top tobottom we have plotted Trelax, Tdecay, Tlinear, and Tosc. It can clearly be seenhere that an oscillon is an object governed by multiple timescales spanningmany orders of magnitude.
64
Using the orthogonality relation
∫ ∞0
ρJ0(kρ)J0(k′ρ)dρ =1
kδ(k − k′) (4.75)
we can invert Eq. 4.74 to yield
b(k) = kA(t)
∫ ∞0
J0(kρ)e−ρ2
R2 ρdρ. (4.76)
In [20] it is shown that integration gives
b(k) =R2A
2keR
2k2/4. (4.77)
Note that this function has the same form as the corresponding one in d = 3.
Hence the results in Eqs. 4.6 and 4.7 need not be recomputed. To obtain Γlin
in d = 2 one could, in principle, solve the equation of motion as was done
in d = 3; however, for our purposes it will be sufficient to determine this
parameter numerically (by numerically integrating the linear Klein-Gordon
equation in d = 2 with a Gaussian initial condition), yielding
1
2Γlin =
1
Tlinear
' 0.848
ωmassR2' 0.6
R2. (4.78)
Eq. 4.78 simply gives the (inverse) decay timescale of a linear field in two
spatial dimensions which is initiated to a Gaussian of radius R.
4.7 Overlap Function
Consider the overlap function Ω(ω) in Eq. 4.17,
Ω(ω) = αF(ω)b(ω) ≡ F(ω)b(ω). (4.79)
65
Since Ω(ω) has dimension of amplitude per unit frequency (see Eq. 4.16), as
does F(ω), it follows that b(ω) is dimensionless. Therefore, one can interpret
b(ω) as a dimensionless coupling factor that modulates the distribution F(ω).
If the oscillon couples weakly to a certain mode, b(ω) will be small at that
value of ω.
Now, b(ω), which is proportional to the linear radiation distribution, will
be a peak whose frequency of maximum coupling ωmax is found by
db
dω
∣∣∣∣ω=ωmax
= 0. (4.80)
Consider now that the largest-amplitude radiation wave that can be created
by a driving force of frequency ω > ωmass will be produced by driving at the
frequency ωmax. However, the largest-amplitude radiation wave that can be
created by a given driving force will have an amplitude on the order of the
driving force itself (never larger). Therefore, at frequencies near ωmax, the
amplitude of the radiation wave Ω will be roughly equal to the amplitude of
the “driving force” F . In other words, Ω(ωmax) ' F(ωmax), implying that
b(ωmax) ' 1 from Eq. 4.79. This leads to
α ' 1
b(ωmax), (4.81)
which, when combined with Eqs. 4.17 and 4.57 yields
Ω(ω) ' −√
2
π
A
1 + χ
1
(ωnl − ω)2
b(ω)
b(ωmax). (4.82)
For example, in the system given by Eq. 4.1, where b(ω) is given by Eq. 4.6,
ωmax =√
2 + 2/R2. This leads to α = e1/2R/√
2 and, when combined with
66
Eq. 4.17, gives
Ω(ω) = − R√π
A
1 + χ
(ω2 − 2)12
(ωnl − ω)2e
12−R
2
4.(ω2−2). (4.83)
This function is plotted in the inset of Fig. 4.5 for typical values of the param-
eters in the system given by Eq. 4.1 in d = 3. Eq. 4.83 gives the distribution
(amplitude per unit frequency) of the radiation wave emitted by the oscillon
in the vicinity of a time t. Its integral with respect to frequency gives the total
amplitude of the radiation wave emitted by the oscillon, denoted A.
Given an expression for Ω(ω), δ and ωrad can then be given by
Ω′(ωrad) = 0; (4.84)
1
2δ =
[∫(ω − ωrad)2 Ω(ω)dω∫
Ω(ω)dω
] 12
,
where the prime denotes a derivative with respect to ω.
4.8 Derivation of Stability Function
From Eq. 4.24 it can be shown by differentiating that,
X ∝ (X −X∞)a (4.85)
where a ≡ 1 + gρX
. By further differentiating, it can be shown that
X(m)
X(n)∝
(X
X −X∞
)m−n
= Γm−nX , (4.86)
67
where X(m) denotes the mth derivative of X with respect to time. Now, from
Eq. 4.60,
Σ ≡ νn − νn+1
νn= 1− νn+1
νn. (4.87)
From Eq. 4.56 we have,
νnνn+1
=s2
f (2n+1)/f (2n−1)=
(ω − ωnl)2
A(2n+1)/A(2n−1), (4.88)
where we’ve used Eq. 4.57 and the fact that χ is a constant. Combining
Eq. 4.88 with Eq. 4.86 (in the case that X = A) we have
νnνn+1
∝(ω − ωnl
Γnl
)2
= β
(ω − ωnl
Γnl
)2
, (4.89)
where β is a proportionality constant and we’ve used that Γnl = ΓA. Combin-
ing Eq. 4.89 with Eq. 4.87 we have
Σ = 1− 1
β
(Γnl
ω − ωnl
)2
. (4.90)
Eq. 4.90 is a function of ω; to determine the appropriate value of ω we note
that, if Γnl were to attain its maximum value of Γlin and if Γlin = ωgap, the
oscillon would decay; therefore, Σ = 0 when Γnl = Γlin = ωgap. This implies
that β(ω − ωnl)2 = ω2
gap. Substitution of this into Eq. 4.90 yields the desired
result of
Σ = 1−(
Γnl
ωgap
)2
. (4.91)
The quantity Σ in Eq. 4.91 measures the stability of the oscillon by comparing
the decay width Γnl with the frequency gap ωgap. For a very small radiation
rate Γnl ωgap and so Σ ' 1. On the other hand, when Γnl = ωgap then
Σ = 0.
68
Chapter 5
The Oscillon Profile
In this chapter I will summarize some preliminary work that I have done in
trying to derive an equation for the spatial profile of the oscillon. It begins with
a simple separability assumption. This then leads to a differential equation for
the spatial profile of the oscillon at the top of its oscillation cycle. An empirical
relationship between two parameters in this equation is then shown to provide
a possible explanation for the phenomenon of sudden oscillon death.
5.1 Assumption of Partial Separability of Space
and Time
Suppose we have a time periodic field configuration which is oscillating about
the vacuum and suppose that at some point in the oscillation φ(r, t) = 0 for
all r (i.e., the upper turning point of the oscillation). Also, suppose that, in
the vicinity of the upper turning point, the field satisfies
φ(r, t) ' α(t)f(β(t)r
)(5.1)
69
where α(t) and β(t) are time-dependent functions and where f(·) is the spatial
profile of the field normalized so that f(0) = 1. It is easy to show that since
φ(r, t) = 0 for all r (at turning point) we must have α = β = 0 at the upper
turning point. Using this one can show that
φ = α(t)
[(α
α
)f(r) +
(β
β
)rf ′(r)
](5.2)
at the upper turning point. Given the equation of motion of a spherically
symmetric, scalar field with a symmetric double well potential
φ = φrr +d− 1
rφr − 2φ+ 3φ2 − φ3, (5.3)
then Eq. 5.2 implies that, in the vicinity of the upper turning point we have
φrr = −d− 1
rφr + χ2rφr + Λ2φ− 3φ2 + φ3 (5.4)
where
χ2 ≡ β
β, Λ2 ≡ 2 +
α
α(5.5)
evaluated at the upper turning point. In the next section we integrate Eq. 5.4
and compare the result to the numerical profiles obtained from simulation. At
this point there are three undetermined constants required to solve Eq. 5.4: χ
and Λ and φ(r = 0). Henceforth we will take φ(r = 0) and Λ to be free param-
eters (determined by numerics) and will let χ be determined by the shooting
method (i.e., the requirement that φ → 0 as r → ∞). Roughly speaking
φ(r = 0) represents the amplitude of the configuration and Λ determines the
inverse spatial extent.
70
0 2 4 6 8 10 12 14r
0
0.2
0.4
0.6
0.8
1
1.2
Osc
illon
Pro
file
0 2 4 6 8 10 12 14
0
0.2
0.4
0.6
0.8
1
1.2
Figure 5.1: Plotted are the profiles of the longest lived oscillon in the SWDPtaken at three different times during the oscillon’s life. The parameters usedare (φ(r = 0) = 1.075,Λ = 1.33), (φ(r = 0) = .96,Λ = 1.27) and (φ(r = 0) =.84,Λ = 1.2). Dashed curves are numerical and solid curves are theoretical.
5.2 Results
It seems that for any oscillon profile at the turning point (and at any such
point during the oscillon’s life) we can find values of φ(r = 0) and Λ which give
excellent agreement between numerics and Eq. 5.4. Fig. 5.1 shows the spatial
profiles of the longest lived oscillon in the SDWP taken at three different times
during the oscillon’s life (but always at the turning point). These numerical
profiles (dashed) are compared with solutions of Eq. 5.4 (solid curves) with
parameters φ(r = 0) and Λ chosen to give the best fit in each case.
71
5.3 An Empirical Relationship between Λ and
χ
Over the course of the lifetime of the oscillon (but always at the turning points)
the values of φ(r = 0), Λ, and χ will change with time to reflect the changing
amplitude and radius of the oscillon. It is found empirically that, as these
parameters evolve, the relation
χΛ = const ≡ qω2m (5.6)
is satisfied quite precisely, where q is a dimensionless number. Specifically,
the values used in Fig. 5.1 can be reproduced to within < 1% by letting
q ' .283. Fig. 5.2 show the value of q which produces a best fit between
theoretical and numerical profiles as a function of time for the longest-lived
oscillon in the SWDP. As can be seen, q(t) is essentially constant during the
stable phase of the oscillon’s life, confirming Eq. 5.6. In addition to reproducing
the results in Fig. 5.1, Eq. 5.6 also reduces Eq. 5.4 to only one free parameter
(which we take to be φ(r = 0)). We can then solve Eq. 5.4 for any value of
φ(r = 0), compute its energy and radius, then plot the energy and radius as
functions of φ(r = 0). The results of this are shown in Figs. 5.3 and 5.4 for
energy and radius, respectively. It is interesting to note that the energy curve
possesses a minimum which agrees very well with the actual decay energy of
the oscillon. We can therefore say that the oscillon, over the course of its
life, “travels” along this curve (from right to left) and decays when it reaches
the minimum (with the solution of Eq. 5.4 at each point yielding the spatial
profile). Unfortunately, attempts at deriving Eq. 5.6 have not been met with
success; given its predictive power, the derivation of this relation (or something
72
0 1000 2000 3000 4000 5000 6000 7000time
0
0.1
0.2
0.3
0.4
0.5
q
0 1000 2000 3000 4000 5000 6000 7000
0
0.1
0.2
0.3
0.4
0.5
Figure 5.2: The values of q which produce the best fit between theoreticaland numerical profiles as a function of time for longest-lived oscillon in theSWDP. As can be seen, q(t) is essentially constant during the stable phase ofthe oscillon’s life, confirming Eq. 5.6.
close to it) would therefore seem to be a good direction for future research.
73
0 0.5 1 1.5 2Core Amplitude
0
20
40
60
80
Ener
gy
0 0.5 1 1.5 2
0
20
40
60
80
Figure 5.3: Energy as a function of φ(r = 0) after substituting Eq. 5.6 intoEq. 5.4. The curve possesses a minimum at E ' 41.5. The oscillon “travels”along the curve (from right to left) and decays when it reaches the minimum.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6Core Amplitude
0
2
4
6
8
10
Radi
us
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
0
2
4
6
8
10
Figure 5.4: Radius as a function of φ(r = 0) after substituting Eq. 5.6 intoEq. 5.4. The curve possesses a minimum at R ' 2.7.
74
Chapter 6
Conclusions and Future
Directions
In this chapter I will summarize what has been presented and will suggest
some possible future directions that this research should take. In short, I
have presented some simple models which permit an estimation of the basic
properties of oscillons such as amplitude, radius, energy, radiation rate, decay
energy and lifetime. Although these methods are more heuristic in nature
than the standard systematic methods found in the literature, they do allow
a rough estimation of the more complicated properties of oscillons (such as
lifetime, large oscillation amplitude, etc.) which other methods (such as the
so-called ε expansion method) fail to provide.
With regard to the calculation of radiation rate, it seems necessary to find
a more systematic method for determining the oscillons coupling to radiation
modes of a given frequency. I have done my best to estimate this, but a more
rigorous treatment is necessary. One of the fundamental questions associated
with oscillons is how one determines the minimum energy configuration. Al-
though I have proposed a simple method by which this could be estimated,
75
a more rigorous treatment is necessary; at present, the only reliable way to
obtain the attractor configuration is to measure it in d = 2 and to make a
guess in d = 3. For this reason I have avoided examining other field potentials
in favor of trying to develop more sophisticated techniques for characterizing
simple potentials, such as the symmetric double well potential.
76
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