PHYSICAL REVIEW D, VOLUME 63, 063507
Waveforms for gravitational radiation from cosmic string loops
Bruce Allen*
Department of Physics, University of Wisconsin–Milwaukee, P.O. Box 413, Milwaukee, Wisconsin 53201
Adrian C. Ottewill†
Department of Mathematical Physics, University College Dublin, Belfield, Dublin 4, Ireland
共Received 20 October 2000; published 20 February 2001兲
We obtain general formulas for the plus- and cross-polarized waveforms of gravitational radiation emitted
by a cosmic string loop in a transverse, traceless 共synchronous, harmonic兲 gauge. These equations are then
specialized to the case of piecewise linear loops, and it is shown that the general waveform for such a loop is
a piecewise linear function. We give several simple examples of the waveforms from such loops. We also
discuss the relation between the gravitational radiation by a smooth loop and by a piecewise linear approximation to it.
DOI: 10.1103/PhysRevD.63.063507
PACS number共s兲: 98.80.Cq, 04.30.Db, 11.27.⫹d
I. INTRODUCTION
Cosmic strings are one dimensional topological defects
that may have formed if the vacuum underwent a phase transition at a very early time breaking a local U(1) symmetry
关1–4兴. The resulting network of strings is of cosmological
interest if the strings have a large enough mass per unit
length, ␮ . If G ␮ /c 2 ⬃10⫺6 , where G is Newton’s constant
and c is the speed of light 共i.e. ␮ ⬃1022 g/cm), then cosmic
strings may be massive enough to have provided the density
perturbations necessary to produce the large scale structure
we observe in the Universe today and could explain the pattern of anisotropies observed in the cosmic microwave background 关5兴.
The main constraints on ␮ come from observational
bounds on the amount of gravitational background radiation
emitted by cosmic string loops 共关4,6,7兴 and references
therein兲. A loop of cosmic string is formed when two sections of a long string 共a string with length greater than the
horizon length兲 meet and intercommute. Once formed, loops
begin to oscillate under their own tension, undergoing a process of self-intersection 共fragmentation兲 and eventually creating a family of non-self-intersecting oscillating loops. The
gravitational radiation emitted by each loop as it oscillates
contributes to the total background gravitational radiation.
In a pair of papers, we introduced and tested a new
method for calculating the rates at which energy and momentum are radiated by cosmic strings 关8,9兴. Our investigation
found that many of the published radiation rates were numerically inaccurate 共typically too low by a factor of two兲.
Remarkably, we also found a lower bound 共in the center-ofmass frame兲 for the rate of gravitational radiation from a
cosmic string loop 关10兴. Our method involved the use of
piecewise linear cosmic strings. In this paper we wish to
provide greater insight into the behavior of such loops and,
in particular, how they approximate smooth loops by exam-
*Email address: [email protected]
†
Email address: [email protected]
0556-2821/2001/63共6兲/063507共8兲/$15.00
ining the waveforms of the gravitational waveforms of such
loops.
It has long been known 关6,11兴 that the first generation of
ground-based interferometric gravitational-wave detectors
关for example, the Laser Interferometric Gravitational Wave
Observatory 共LIGO-I兲兴 will not be able to detect the
gravitational-wave stochastic background produced by a network of cosmic strings in the Universe. The amplitude of this
background is too weak to be detectable, except by a future
generation of more advanced instruments. However, a recent
paper by Damour and Vilenkin 关12兴 has shown that the nonGaussian bursts of radiation produced by cusps on the closest
loops of strings would be a detectable LIGO-I source. While
the specific examples studied here do not include these types
of cusps, the general method developed can be applied to
such loops.
Our space-time conventions follow those of Misner,
Thorne and Wheeler 关13兴 so that ␩ ␮ ␯ ⫽diag(⫺1,1,1,1) ␮ ␯ .
We also set ប⫽c⫽1, but we leave G explicit.
II. GENERAL THEORY
In the center-of-mass frame, a cosmic string loop is specified by the 3-vector position x(t, ␴ ) of the string as a function of two variables: time t and a space-like parameter ␴
that runs from 0 to L. 共The total energy of the loop is ␮ L.)
When the gravitational back-reaction is neglected 共a good
approximation if G ␮ 2 Ⰶ1), the string loop satisfies equations of motion whose most general solution in the centerof-mass frame is
x共 t, ␴ 兲 ⫽ 21 关 a共 u 兲 ⫹b共 v 兲兴 ,
共2.1兲
where
u⫽t⫹ ␴ ,
v ⫽t⫺ ␴ .
共2.2兲
Here a(u)⬅a(u⫹L) and b( v )⬅b( v ⫹L) are a pair of periodic functions, satisfying the gauge condition 兩 a⬘ (u) 兩
⫽ 兩 b⬘ ( v ) 兩 ⫽1, where the prime denotes differentiation with
respect to the function’s argument. Because the functions a
63 063507-1
©2001 The American Physical Society
BRUCE ALLEN AND ADRIAN C. OTTEWILL
PHYSICAL REVIEW D 63 063507
and b are periodic in their arguments, the string loop is periodic in time. The period of the loop is L/2 since
冉
x t⫹
h ␮ ␯ 共 t,x兲 ⫽4G
冊
冋
L
L
1
, ␴ ⫹ ⫽ 关 a共 t⫹ ␴ ⫹L 兲 ⫹b共 t⫺ ␴ 兲兴
2
2
2
Far from the string loop center-of-mass the dominant behavior is that of an outgoing spherical wave given by
With our choice of coordinates and gauge, the energymomentum tensor T ␮ ␯ for the string loop is given by
冠
冡
1
⫻ ␦ t⫺ 共 u⫹ v 兲 ␦ (3) „y⫺x共 u, v 兲 …,
2
where G
␮␯
⬁
h ␮ ␯ 共 t,x兲 ⫽4G
冋
du d v G ␮ ␯ 共 u, v 兲
0⭐u⫺ v ⭐2L
共2.4兲
d 3y
册
where r⫽ 兩 x 兩 and ⍀̂⫽x/r is a unit vector pointing away from
the source. Inserting Eq. 共2.10兲 into Eq. 共2.12兲, we find the
field far from a cosmic string loop is
共2.5兲
⬁
8G ␮
h ␮ ␯ 共 t,x兲 ⫽
L
with x ⫽„t,x(t, ␴ )…. In terms of a and b,
G 0i ⫽ 41 关 a i⬘ ⫹b i⬘ 兴 ,
冕
共2.12兲
␮
G 00⫽ 21 ,
兺
n⫽⫺⬁
e i ␻ n (t⫺r)
r
1
⫻ T̃ ␮ ␯ 共 ␻ n ,y兲 ⫺ ␩ ␮ ␯ T̃ ␭ ␭ 共 ␻ n ,y兲 e i ␻ n ⍀̂•y,
2
is defined by
G ␮ ␯ 共 u, v 兲 ⫽ ⳵ u x ␮ ⳵ v x ␯ ⫹ ⳵ v x ␮ ⳵ u x ␯ ,
册
共2.11兲
共2.3兲
冕冕
d 3y
1
⫻ T̃ ␮ ␯ 共 ␻ n ,y兲 ⫺ ␩ ␮ ␯ T̃ ␭ ␭ 共 ␻ n ,y兲 e i ␻ n (t⫺ 兩 x⫺y兩 ) .
2
1
⫽ 关 a共 t⫹ ␴ 兲 ⫹b共 t⫺ ␴ 兲兴 ⫽x共 t, ␴ 兲 .
2
T ␮ ␯ 共 t,y兲 ⫽ ␮
兺 冕 兩 x⫺y兩
n⫽⫺⬁
⬁
兺
n⫽⫺⬁
冋
G i j ⫽ 41 关 a i⬘ b ⬘j ⫹a ⬘j b i⬘ 兴 ,
共2.6兲
e i ␻ n (t⫺r)
r
冕 冕
L
L
du
0
dv
0
1
⫻ G̃ ␮ ␯ 共 u, v 兲 ⫺ ␩ ␮ ␯ G̃ ␭ ␭ 共 u, v 兲
2
册
⫻e ⫺i ␻ n [(u⫹v)/2⫺⍀̂•x(u, v )] .
and the trace is
G ␭ ␭ ⫽ 21 关 ⫺1⫹a⬘ •b⬘ 兴 .
共2.7兲
The n⫽0 term in this sum corresponds to the static field
Alternatively we may introduce the four-vectors A ␮ (u)
⫽„u,a(u)… and B ␮ ( v )⫽„v ,b( v )… so that
G ␮ ␯ ⫽ 41 共 A ⬘ ␮ B ⬘ ␯ ⫹B ⬘ ␮ A ⬘ ␯ 兲 .
h ␮static
␯ 共 t,x 兲 ⫽
兺
e i ␻ n t T̃ ␮ ␯ 共 ␻ n ,y兲 ,
共2.9兲
where ␻ n ⫽4 ␲ n/L and
T̃ ␮ ␯ 共 ␻ n ,y兲 ⫽
2
L
冕
2␮
⫽
L
L/2
0
⫽
冕 冕
L
dv
du
0
0
册
2GM
2G ␮ L
␦ ␮␯ ,
共 ␩ ␮ ␯ ⫹2 t̂ ␮ t̂ ␯ 兲 ⫽
r
r
共2.15兲
h ␮rad␯ ⫽h ␮ ␯ ⫺h ␮static
␯ .
共2.16兲
We may rewrite Eq. 共2.13兲 as
L
du
0
L
as appropriately to an object with mass M as may be seen by
comparison with the Schwarzschild metric in isotropic coordinates 共see, for example, Eq. 共31.22兲 of Ref. 关13兴兲. We denote the radiative part of the field by
dt e ⫺i ␻ n t T ␮ ␯ 共 t,y兲
L
冕 冕
1
⫻ G ␮ ␯ 共 u, v 兲 ⫺ ␩ ␮ ␯ G ␭ ␭ 共 u, v 兲 , 共2.14兲
2
⬁
n⫽⫺⬁
8G ␮
rL
冋
共2.8兲
The gauge conditions are satisfied if and only if A ⬘ ␮ (u) and
B ⬘ ␮ ( v ) are null vectors.
As a consequence of the time periodicity of the loop the
stress tensor can be expressed as a Fourier series
T ␮ ␯ 共 t,y兲 ⫽
共2.13兲
dv e
⫺i ␻ n (u⫹ v )/2
⬁
0
h ␮ ␯ 共 t,x兲 ⫽
⫻G ␮ ␯ 共 u, v 兲 ␦ (3) „y⫺x共 u, v 兲 …. 共2.10兲
The retarded solution for the linear metric perturbation
due to this source in harmonic gauge is 关14兴
兺 e ⫺i ␻ k
n⫽⫺⬁
n ␮x
␮ (n)
e
␮ ␯ 共 ⍀̂ 兲
共2.17兲
where k ␮ ⫽(1,⍀̂) is a null vector in the direction of propagation and
063507-2
WAVEFORMS FOR GRAVITATIONAL RADIATION FROM . . .
e(n) ␮ ␯ ⫽
冕 冕
8GM
rL 2
L
L
du
0
冋
␧ i⫽
dv
0
1
⫻ G ␮ ␯ 共 u, v 兲 ⫺ ␩ ␮ ␯ G ␭ ␭ 共 u, v 兲
2
⫻e i ␻ n /2[k ␮ A
册
␮ (u)⫹k B ␮ ( v )]
␮
1
L
冕
L
0
GM
兵 关 I 0 J 0 ⫹I•J兴 ⍀ i ⫹2 关 I 0 J i ⫹J 0 I i 兴 其
r
e0⬘ ␮ ⫽0.
共2.18兲
du A ⬘ ␮ 共 u 兲 e i ␻ n k ␮ A
␮ (u)/2
共2.25兲
The spatial components are given by
e⬘i j ⫽
共2.19兲
2GM
兵 关 I i J j ⫹I j J i 兴 ⫹ ␦ i j 关 I 0 J 0 ⫺I•J兴
r
⫹⍀ i ⍀ j 关 I 0 J 0 ⫹I•J兴 ⫹I 0 关 J i ⍀ j ⫹⍀ i J j 兴
⫹J 0 关 I i ⍀ j ⫹⍀ i I j 兴 其 ,
and
J (n) ␮ ⫽
1
L
冕
L
0
d v B ⬘␮共 v 兲 e i␻nk␮B
␮ ( v )/2
.
共2.24b兲
then
are polarization tensors. From Eq. 共2.8兲, it is clear that the
polarization tensors may be written in terms of the fundamental integrals
I (n) ␮ ⫽
PHYSICAL REVIEW D 63 063507
共2.20兲
共2.26兲
and these satisfy the gauge conditions
e␮⬘ ␮ ⫽e⬘i i ⫽0
共2.27兲
e⬘j i ⍀ i ⫽0.
共2.28兲
In terms of these integrals,
and
2GM
e00⫽
关 I 0 J 0 ⫹I•J兴
r
共2.21a兲
e0i ⫽
2GM
关 I 0 J i ⫹J 0 I i 兴
r
共2.21b兲
ei j ⫽
2GM
兵 关 I i J j ⫹J i I j 兴
r
⫹ ␦ i j 关 I 0 J 0 ⫺I•J兴 其 ,
L
0
du k ␮ A ⬘ ␮ 共 u 兲 e ⫺i ␻ n k ␯ A
2i
⫽
␻
冕
L
0
and
0
冊
0 ,
0
共2.29兲
e ⫹⫽
2GM
关 I 1 ⬘ J 1 ⬘ ⫺I 2 ⬘ J 2 ⬘ 兴
r
共2.30a兲
e ⫻⫽
2GM
关 I 1 ⬘ J 2 ⬘ ⫹I 2 ⬘ J 1 ⬘ 兴
r
共2.30b兲
define two modes of linear polarization.
In terms of the original basis we can write
2GM
共 cos 2 ␺ A ⫹ ⫹sin 2 ␺ A ⫻ 兲
r
共2.31a兲
2GM
共 ⫺sin 2 ␺ A ⫹ ⫹cos 2 ␺ A ⫻ 兲
r
共2.31b兲
e ⫹⫽
共2.23兲
and
If we make the choice
␧ 0⫽
⫺e ⫹
0
where
共2.22兲
which is a consequence of periodicity, and the corresponding
equation for B ␮ . The harmonic gauge condition does not
determine the gauge completely and we are left with the
freedom to make transformations of the form
e␮⬘ ␯ ⫽e␮ ␯ ⫹k ␮ ␧ ␯ ⫹k ␯ ␧ ␮ .
e⫻
and
␯ (u)/2
d
␯
du e ⫺i ␻ n k ␯ A (u)/2 ⫽0,
du
冉
e⫹
e⬘i ⬘ j ⬘ ⫽ e ⫻
0
共2.21c兲
where we have dropped the superscript n for clarity.
The harmonic gauge condition requires that the polarization tensors satisfy k ␮ e␮ ␯ ⫽ 21 k ␯ e␮ ␮ . This is easily verified
by noting that I 0 ⫽⍀̂•I and J 0 ⫽⍀̂•J. These equations follow from the identity
冕
If we perform a spatial rotation to coordinates (x ⬘ ,y ⬘ ,z ⬘ )
where ⍀̂ points along the z ⬘ -axis, then we can write
GM
关 I 0 J 0 ⫹I•J兴
r
e ⫻⫽
共2.24a兲
with
063507-3
BRUCE ALLEN AND ADRIAN C. OTTEWILL
PHYSICAL REVIEW D 63 063507
A ⫹ ⫽ 兵 共 cos2 ␪ cos2 ␾ ⫺sin2 ␾ 兲 I 1 J 1
tween the ith and (i⫹1)th kink. The kink at u⫽u N a is the
same as the first kink at u⫽u 0 ⫽0 but, even though u 0 and
u N a are at the same position on the loop, u 0 ⫽0 while u N a
⫽1. The loop is extended to all values of u by periodicity
共with period 1兲. We denote ai ⫽a(u i ), and the constant unit
vector tangent to the ith segment by ai⬘ . Then we have
⫹ 共 cos2 ␪ sin2 ␾ ⫺cos2 ␾ 兲 I 2 J 2 ⫹sin2 ␪ I 3 J 3
⫹ 共 1⫹cos2 ␪ 兲 cos ␾ sin ␾ 共 I 2 J 1 ⫹I 1 J 2 兲
⫺sin ␪ cos ␪ cos ␾ 共 I 3 J 1 ⫹I 1 J 3 兲
⫺sin ␪ cos ␪ sin ␾ 共 I 2 J 3 ⫹I 3 J 2 兲 其
共2.32a兲
A ⫻ ⫽ 兵 ⫺2 cos ␪ sin ␾ cos ␾ I 1 J 1
a共 u 兲 ⫽ai ⫹ai⬘ 共 u⫺u i 兲
ai⫹1 ⫽ai ⫹a⬘i 共 u i⫹1 ⫺u i 兲 .
⫹cos ␪ 共 2 cos2 ␾ ⫺1 兲共 I 2 J 1 ⫹I 1 J 2 兲
⫹sin ␪ sin ␾ 共 I 3 J 1 ⫹I 1 J 3 兲
共2.32b兲
I(n) ⫽
兺
(n)
(n)
*其 .
⫹e ⫺i4 ␲ n(t⫺r)/L e⫹/⫻
兵 e i4 ␲ n(t⫺r)/L e⫹/⫻
i
⫽
2␲n
共2.33兲
Recall that h rad is obtained from the full metric perturbation
h by dropping the n⫽0 term, which corresponds to the static
共non-radiative兲 part of the field.
The power emitted to infinity per solid angle may be written as
r2
dP
⫽ lim
具 h h ␣␤ ,r 典
d⍀ r→⬁ 32␲ G ␣␤ ,t
GM 2
⫽
2␲
1
2 ␲ in
N a ⫺1
兺
i⫽0
a⬘i
1⫺⍀̂•a⬘i
⫻ 兵 e ⫺2 ␲ in(u i⫹1 ⫺⍀̂•ai⫹1 ) ⫺e ⫺2 ␲ in(u i ⫺⍀̂•ai ) 其
⬁
n⫽1
共3.2兲
We have corresponding definitions for the b-loop, and we
follow the convention of Ref. 关8兴 by labeling the kinks by the
index j.
It is now elementary to calculate that, for n⫽0,
where ␪ , ␾ and ␺ are the Euler angles defining the orientation of the frame (x ⬘ ,y ⬘ ,z ⬘ ) relative to the original frame
共our conventions follow those of Ref. 关15兴兲. The corresponding linearly polarized waveforms are then defined by
rad
h ⫹/⫻
共 t,x兲 ⫽
N a ⫺1
兺
i⫽0
再
ai⬘
1⫺⍀̂•a⬘i
⫺
⬘
ai⫺1
⬘
1⫺⍀̂•ai⫺1
冎
⫻e ⫺2 ␲ in(u i ⫺⍀̂•ai ) ,
共3.3兲
with a similar equation for J. If we insert these expressions
into Eq. 共2.32兲 and then into Eq. 共2.33兲 the sum over n for
rad
h R/L
consists of terms of the form
⬁
共2.34兲
1
兺 2 cos 2 ␲ n 关 2 共 t⫺r 兲 ⫹ 共 u i ⫺⍀̂•ai 兲 ⫹ 共 v j ⫺⍀̂•bj 兲兴
n⫽1 n
共3.4兲
⬁
兺
n⫽1
共3.1兲
and for consistency,
⫹2 cos ␪ sin ␾ cos ␾ I 2 J 2
⫺sin ␪ cos ␾ 共 I 2 J 3 ⫹I 3 J 2 兲 其
u苸 关 u i ,u i⫹1 兴 ,
for
(n) 2
(n) 2
␻ 2n 兵 具 兩 A ⫹
兩 典⫹具兩A⫻
兩 典其.
which may be performed exactly using the identity
共2.35兲
⬁
III. EXAMPLES
For convenience we shall now set the length of the loop
L⫽1, and take ␺ ⫽0.
A. Piecewise linear loops
These are the loops for which the functions a(u) and b( v )
are piecewise linear functions. The functions a(u) and b( v )
may be pictured as a pair of closed loops which consist of
joined straight segments. The segments join together at kinks
where a⬘ (u) and b⬘ ( v ) are discontinuous.
Following the notation of Ref. 关8兴 we take the a- and
b-loops to have N a and N b linear segments, respectively. The
coordinate u on the a-loop is chosen to take the value zero at
one of the kinks and increases along the loop. The kinks are
labeled by the index i, where i⫽0,1, . . . ,N a ⫺1. The value
of u at the ith kink is denoted by u i and without loss of
generality we set u 0 ⫽0.The segments on the loop are also
labeled by i, with the ith segment being the one lying be-
1
兺 2 cos 2 ␲ nx⫽ ␲ 2
n⫽1 n
冉
x 2 ⫺x⫹
1
6
冊
x苸 关 0,1兲 . 共3.5兲
This function is extended to other values by periodicity, for
example, for x苸 关 1,2) we merely replace x by x⫺1 in Eq.
共3.5兲. Such transformations leave the coefficient of x 2 unchanged and can only change the coefficient of x by a multiple of 2. As a result, when the sum in Eq. 共3.3兲 is performed
for the coefficient of x 2 the sum telescopes and gives zero.
Thus, the wave form of a piecewise linear loop will be a
piecewise linear function. In addition, considering the coefficient of x all slopes of the waveform must be a multiple of
some fundamental slope. The slope only changes when a
共4-dimensional兲 kink crosses the past light cone of the observer at (t,x). These properties are illustrated in the examples below.
B. Garfinkle-Vachaspati loops
As our first set of loops we study the loops considered by
Garfinkle and Vachaspati 关16兴. The vectors a(u) and b( v ) lie
063507-4
WAVEFORMS FOR GRAVITATIONAL RADIATION FROM . . .
in a plane and make a constant angle 2 ␣ with each other
where ␣ 苸(0,␲ /2). To be specific, we may take a(u) and
b( v ) to be given by
a共 u 兲 ⫽
b共 v 兲 ⫽
再
再
冉 冊
冉 冊
u 共 cos ␣ i⫹sin ␣ j兲 ,
1
u苸 0, ,
2
共 1⫺u 兲共 cos ␣ i⫹sin ␣ j兲 ,
1
u苸 ,1 ,
2
冉 冊
冉 冊
PHYSICAL REVIEW D 63 063507
and correspondingly
(n)
A⫹
⫽
共 1⫺ v 兲共 cos ␣ i⫺sin ␣ j兲 ,
v苸
1
,1 .
2
„1⫺sin2 ␪ cos2 共 ␾ ⫺ ␣ 兲 …„1⫺sin2 ␪ cos2 共 ␾ ⫹ ␣ 兲 …
⫻
共3.6a兲
(n)
⫽
A⫻
共3.6b兲
„1⫺sin2 ␪ cos2 共 ␾ ⫺ ␣ 兲 …„1⫺sin2 ␪ cos2 共 ␾ ⫹ ␣ 兲 …
J(n) ⫽
⫻
e i ␲ n„1⫺sin ␪ cos( ␾ ⫺ ␣ )…⫺1
i ␲ n„1⫺sin2 ␪ cos2 共 ␾ ⫺ ␣ 兲 …
e i ␲ n„1⫺sin ␪ cos( ␾ ⫹ ␣ )…⫺1
i ␲ n„1⫺sin ␪ cos 共 ␾ ⫹ ␣ 兲 …
2
h ⫹⫽
2
共 cos ␣ i⫹sin ␣ j兲
共 cos ␣ i⫺sin ␣ j兲
共3.7b兲
1
共 e i ␲ n„1⫺sin ␪ cos( ␾ ⫺ ␣ )…⫺1 兲
n ␲2
2
⫻共 e i ␲ n„1⫺sin ␪ cos( ␾ ⫹ ␣ )…⫺1 兲 .
共3.7a兲
共3.8a兲
cos ␪ sin2 ␾
It is then straightforward to calculate that, for n⫽0,
I(n) ⫽
1
共 e i ␲ n„1⫺sin ␪ cos( ␾ ⫺ ␣ )…⫺1 兲
n 2␲ 2
⫻ 共 e i ␲ n„1⫺sin ␪ cos( ␾ ⫹ ␣ )…⫺1 兲
1
v 苸 0, ,
2
v共 cos ␣ i⫺sin ␣ j兲 ,
sin2 ␪ cos共 ␾ ⫹ ␣ 兲 cos共 ␾ ⫺ ␣ 兲 ⫺cos 2 ␾
共3.8b兲
As described above, the sum over n in Eq. 共2.33兲 may be
performed explicitly to yield a piecewise linear function. For
example, ␾ 苸 关 0,␲ /2), h ⫹ is given explicitly by
sin2 ␪ cos共 ␾ ⫹ ␣ 兲 cos共 ␾ ⫺ ␣ 兲 ⫺cos 2 ␾
2GM
r „1⫺sin2 ␪ cos2 共 ␾ ⫺ ␣ 兲 …„1⫺sin2 ␪ cos2 共 ␾ ⫹ ␣ 兲 …
„1⫺sin ␪ cos共 ␾ ⫺ ␣ 兲 …„1⫺sin ␪ cos共 ␾ ⫹ ␣ 兲 …,
⫻
¦
1
0⭐ 共 t⫺r 兲 ⬍ sin ␪ „cos共 ␾ ⫺ ␣ 兲 ⫹cos共 ␾ ⫹ ␣ 兲 …
4
⫺8 共 t⫺r 兲 ⫹„1⫹sin ␪ cos共 ␾ ⫹ ␣ 兲 …„1⫹sin ␪ cos共 ␾ ⫺ ␣ 兲 …,
1
1
sin ␪ „cos共 ␾ ⫺ ␣ 兲 ⫹cos共 ␾ ⫹ ␣ 兲 …⭐ 共 t⫺r 兲 ⬍ „1⫹sin ␪ cos共 ␾ ⫹ ␣ 兲 …
4
4
⫺„1⫺sin ␪ cos共 ␾ ⫺ ␣ 兲 …„1⫹sin ␪ cos共 ␾ ⫹ ␣ 兲 …,
冉
共3.9兲
1
1
„1⫹sin ␪ cos共 ␾ ⫹ ␣ 兲 …⭐ 共 t⫺r 兲 ⬍ „1⫹sin ␪ cos共 ␾ ⫺ ␣ 兲 …
4
4
8 t⫺r⫺
冊
1
⫹„1⫺sin ␪ cos共 ␾ ⫺ ␣ 兲 …„1⫺sin ␪ cos共 ␾ ⫹ ␣ 兲 …,
2
1
1
„1⫹sin ␪ cos共 ␾ ⫺ ␣ 兲 …⭐ 共 t⫺r 兲 ⬍ ,
4
2
and the waveforms are periodic in t with period 21 . The intervals are ordered in the given way for our choice of ␾
苸 关 0,␲ /2). h ⫻ is obtained simply by replacing the prefactor by one appropriate to A ⫻ , as is clear from Eq. 共3.8兲. To obtain
the waveforms for other angles we may note that the transformation ␾ → ␾ ⫹ ␲ is equivalent to changing the sign of (t
⫺r), while the transformation ␾ → ␲ ⫺ ␾ is equivalent to changing the sign of (t⫺r) and the sign in front of the sin 2 ␾ term
in the prefactor in h ⫻ .
Note that the apparent singularity in the waveforms in the plane of the loop ( ␪ ⫽ ␲ /2) at ␾ ⫽⫾ ␣ and ␾ ⫽ ␲ ⫾ ␣ is spurious.
This may be seen by noting that the waveform is bounded by the two constant sections of the piecewise linear curve which
063507-5
BRUCE ALLEN AND ADRIAN C. OTTEWILL
PHYSICAL REVIEW D 63 063507
take on a value which tends to zero in this limit. In fact, the numerator of the prefactor also vanishes in this limit, which
ensures that the amplitude tends to zero at these points and hence that even the time derivatives 共which determine the power兲
are finite. Along the axis ␪ ⫽0, Eq. 共3.9兲 reduces to
h ⫹/⫻ 共 ␪ ⫽0 兲 ⫽⫺
2GM
cos 2 ␾ /sin 2 ␾
r
Waveforms for various angles are plotted in Fig. 1 for the
case of ␣ ⫽ ␲ /4, corresponding to two lines at right angles.
This is the configuration which radiates minimum gravitational radiation for this class of loops, P⫽64 ln 2G␮2
⬇44.3614G ␮ 2 .
C. Plane-line loops
As our next set of examples we study the set of loops in
which a(u) lies along the z-axis and b( v ) is always in the
x-y plane. This class of loops was studied by us in Ref. 关17兴,
where we gave an analytic result for the power lost in gravitational radiation by such loops. Explicitly a(u) is given by
a共 u 兲 ⫽
再
冉 冊
冉 冊
uk
1
u苸 0,
2
共 1⫺u 兲 k
1
u苸 ,1 .
2
共3.11兲
再冉
⫺8 共 t⫺r 兲 ⫹1,
8 t⫺r⫺
0⭐t⫺r⬍
冊
1
4
共3.10兲
1
1
1
⭐t⫺r⬍ .
⫹1,
2
4
2
Also J (n)
3 ⫽0, so we have
A ⫹ ⫽⫺sin ␪ cos ␪ 共 cos ␾ J 1 ⫹sin ␾ J 2 兲 I 3
共3.13a兲
A ⫻ ⫽sin ␪ 共 sin ␾ J 1 ⫺cos ␾ J 2 兲 I 3 .
共3.13b兲
It follows immediately that the waveforms vanish along the
z-axis.
In Ref. 关17兴 we proved that the minimum gravitational
radiation emitted by any loop in this class is given by taking
the b-loop to be a circle:
b共 v 兲 ⫽
1
„cos共 2 ␲ v 兲 i⫹sin共 2 ␲ v 兲 j….
2␲
共3.14兲
The power emitted in gravitational radiation by this loop is
P⫽16
冕
2 ␲ 共 1⫺cos x 兲
0
x
dx G ␮ 2 ⬇39.0025 G ␮ 2 .
共3.15兲
It follows that
I ⫽
(n)
e i ␲ n(1⫺cos ␪ ) ⫺1
i ␲ nsin2 ␪
k.
共3.12兲
J(n) may be determined explicitly as
i i(n⫹1)( ␾ ⫺ ␲ )
2 J n⫹1 共 nsin ␪ 兲
J (n)
1 ⫽ 关e
2
␲
⫺e i(n⫺1)( ␾ ⫺ 2 ) J n⫺1 共 nsin ␪ 兲兴
共3.16a兲
1 i(n⫹1)( ␾ ⫺ ␲ )
2 J n⫹1 共 nsin ␪ 兲
J (n)
2 ⫽ 关e
2
␲
⫹e i(n⫺1)( ␾ ⫺ 2 ) J n⫺1 共 nsin ␪ 兲兴 .
共3.16b兲
This gives the equivalent forms
(n)
⫽⫺
A⫹
sin„␲ n sin2 共 ␪ /2兲 …cos ␪
关 J n⫹1 共 n sin ␪ 兲
sin ␪
⫹J n⫺1 共 nsin ␪ 兲兴
FIG. 1. Plus-polarized waveforms for the Garfinkle-Vachaspati
loops with ␣ ⫽ ␲ /4. The solid line corresponds to the wave travelling up the z-axis ( ␪ ⫽0). The dotted line corresponds to a direction
at elevation ␪ ⫽ ␲ /3 along ␾ ⫽0. The dashed line corresponds to a
wave travelling in the plane of the loop ␪ ⫽ ␲ /2 at angle ␾ ⫽ ␲ /5.
The cross-polarized waveforms differ only in that their amplitude
has a different dependece on ␪ and ␾ .
063507-6
⫽⫺2
⫻
1 i ␲ nsin2 ( ␪ /2)⫹in( ␾ ⫺ ␲ /2)
e
␲n
sin„␲ nsin2 共 ␪ /2兲 …cos共 ␪ 兲
sin2 ␪
1 in ␾ ⫺in( ␲ /2)cos ␪
e
␲n
J n 共 nsin ␪ 兲
共3.17兲
WAVEFORMS FOR GRAVITATIONAL RADIATION FROM . . .
PHYSICAL REVIEW D 63 063507
FIG. 2. Plus-polarized waveforms for the circle-line loop. Plotted is h ⫹ /sin ␪ for ␪ ⫽ ␲ /2 共solid line兲, ␪ ⫽ ␲ /4 共dashed line兲 and
␪ ⫽ ␲ /20 共dotted line兲. The choice of scaling is chosen on the basis
of the asymptotic form Eq. 共3.19兲.
FIG. 4. The solid lines are the plus-polarized waveforms for the
hexagon-line loop with ␪ ⫽ ␲ /4 and with ␾ ⫽0 共left through兲 and
␾ ⫽5 ␲ /6 共right through兲. The dotted lines are the corresponding
waveforms for the circle-line loop.
and
In the plane of the b-loop h ⫹ vanishes so that the wave
becomes linearly polarized. On the other hand, as we approach the axis ␪ ⫽0 the fundamental mode (n⫽1 term兲
dominates and we have
(n)
⫽isin„␲ nsin2 共 ␪ /2兲 …sin ␪ 关 J n⫹1 共 nsin ␪ 兲
A⫻
⫺J n⫺1 共 n sin ␪ 兲兴
⫽2i
⫻
1 i ␲ n sin2 ( ␪ /2)⫹in( ␾ ⫺ ␲ /2)
e
␲n
h ⫹ ⬃⫺
sin„␲ nsin2 共 ␪ /2兲 …
J ⬘n 共 nsin ␪ 兲
sin ␪
1 in ␾ ⫺in( ␲ /2)cos ␪
.
e
␲n
GM
sin ␪ sin„4 ␲ 共 t⫺r 兲 ⫹ ␾ …
r
共3.19兲
and
h ⫻⬃
共3.18兲
GM
sin ␪ cos„4 ␲ 共 t⫺r 兲 ⫹ ␾ ….
r
共3.20兲
The corresponding waveforms for various choices of ␪ are
plotted in Figs. 2 and 3. 共As the system simply rotates cylindrically, with time the choice of ␾ is irrelevant, corresponding simply to a shift in t⫺r.)
Thus the wave approaches circular polarization but its amplitude vanishes as sin ␪.
As in Ref. 关17兴 we may also consider the case where the
b-loop forms a regular N-sided polygon. In Figs. 4 and 5 we
FIG. 3. Cross-polarized waveforms for the circle-line loop. Plotted is h ⫻ /sin ␪ for ␪ ⫽ ␲ /2 共solid line兲, ␪ ⫽ ␲ /4 共dashed line兲 and
␪ ⫽ ␲ /20 共dotted line兲. The choice of scaling is chosen on the basis
of the asymptotic form Eq. 共3.20兲.
FIG. 5. The solid lines are the cross-polarized waveforms for the
hexagon-line loop with ␪ ⫽ ␲ /4 and with ␾ ⫽0 共left peak兲 and ␾
⫽5 ␲ /6 共right peak兲. The dotted lines are the corresponding waveforms for the circle-line loop.
063507-7
BRUCE ALLEN AND ADRIAN C. OTTEWILL
PHYSICAL REVIEW D 63 063507
compare the waveform for the circle with that for a
regular hexagon for which P⫽(40 ln 5⫺32 ln 2)G␮2
⬇42.1968G ␮ 2 . As mentioned above a change in ␾ for the
circle-line loop corresponds simply to a shift in t; however,
this is no longer the case for the polygon for which the waveform will only repeat every 2 ␲ /N. Hence in Figs. 4 and 5
we include hexagon-line waveforms for both ␾ ⫽0 and ␾
⫽5 ␲ /6 共this choice was made simply to disentangle the two
graphs as far as possible兲. It is remarkable that even for such
a crude approximation to the circle as a hexagon, the waveform of the hexagon-line loop provides remarkably good
piecewise linear approximations to the circle-line waveforms.
IV. CONCLUSION
Given the remarkable agreement of the waveforms it is of
interest to compare the instantaneous power defined by
P ⫹/⫻ ⫽
GM 2
2␲
⬁
兺
n⫽1
(n) 2
␻ 2n 兩 A ⫹/⫻
兩
共4.1兲
FIG. 6. Comparison of the instantaneous power in pluspolarized waves for the 24-sided polygon-line loop 共dotted line兲 and
for the circle-line loop 共solid line兲 with ␪ ⫽ ␲ /4 and ␾ ⫽0.
in the different polarizations. While this quantity is not
gauge invariant its time average is and gives the total power
radiated in each polarization. By comparing the function for
the polygon-line loops with the circle-line loop we can certainly see that their time averages agree well. As the waveform for a piecewise linear loop is a piecewise linear func-
tion, the instantaneous power, which is the square of its
derivative, will be piecewise constant. For example, in Fig. 6
we compare the instantaneous power in the plus-polarization
between the circle-line loop and a regular 24-sided polygonline loop. The very close agreement between the two curves
provides further evidence for the validity of the piecewise
linear approximation of string loops used by 关8兴.
关1兴 T.W.B. Kibble, J. Phys. A 9, 1387 共1976兲; T.W.B. Kibble, G.
Lazarides, and Q. Shafi, Phys. Rev. D 26, 435 共1982兲.
关2兴 Y.B. Zel’dovich, Mon. Not. R. Astron. Soc. 192, 663 共1980兲.
关3兴 A. Vilenkin, Phys. Rev. D 24, 2082 共1981兲; Phys. Rep. 121,
263 共1985兲.
关4兴 E.P.S. Shellard and A. Vilenkin, Cosmic Strings and Other
Topological Defects 共Cambridge University Press, Cambridge,
England, 1994兲.
关5兴 B. Allen, R.R. Caldwell, E.P.S. Shellard, A. Stebbins, and S.
Veeraraghavan, Phys. Rev. Lett. 77, 3061 共1996兲.
关6兴 B. Allen and R.R. Caldwell, Phys. Rev. D 45, 3447 共1992兲.
关7兴 R.R. Caldwell, in Proceedings of the Fifth Canadian General
Relativity and Gravitation Conference, 1993, edited by R.
McLenaghan and R. Mann 共World Scientific, New York,
1993兲.
关8兴 B. Allen and P. Casper, Phys. Rev. D 50, 2496 共1994兲.
关9兴 B. Allen, P. Casper, and A. Ottewill, Phys. Rev. D 51, 1546
共1995兲.
关10兴 B. Allen and P. Casper, Phys. Rev. D 52, 4337 共1995兲.
关11兴 B. Allen, in Proceedings of the Les Houches School on Astrophysical Sources of Gravitational Radiation, edited by J.A.
Marck and J.P. Lasota 共Cambridge University Press, Cambridge, England, 1997兲.
关12兴 T. Damour and A. Vilenkin, Phys. Rev. Lett. 85, 3761 共2000兲.
关13兴 C.W. Misner, K.S. Thorne, and J.A. Wheeler, Gravitation
共Freeman, San Francisco, 1973兲.
关14兴 S. Weinberg, Gravitation and Cosmology 共John Wiley, New
York, 1972兲.
关15兴 G. Arfken, Mathematical Methods for Physicists, third edition
共Academic, Orlando, 1985兲.
关16兴 D. Garfinkle and T. Vachaspati, Phys. Rev. D 36, 2229 共1987兲.
关17兴 B. Allen, P. Casper, and A. Ottewill, Phys. Rev. D 50, 3703
共1994兲.
063507-8