Axions from String Decay

C. Hagmanna, S. Changb, and P. Sikivieb

a Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA 94550

b University of Florida, Physics Department, Gainesville, FL 32611

We have studied numerically the evolution and decay of axion strings. These global defects decay
mainly by axion emission and thus contribute to the cosmological axion energy density. The relative
importance of this source relative to misalignment production of axions depends on the spectrum.
Radiation spectra for various string loop configurations are presented. They support the contention
that the string decay contribution is of the same order of magnitude as the contribution from
misalignment.

1. INTRODUCTION -
with core size 1
(
= f )
a and energy per
unit length
Axion strings arise in the early universe
when the global U(1) symmetry breaks
PQ 2
 f ln(L/
a ) (2)
spontaneously at the scale f a ~ 1012 GeV. A
Brownian network of strings is formed initially where L is the long distance cutoff, i.e. the
with O(1) string per horizon volume. As the interstring distance. The logarithmic factor in
horizon expands, wiggles on the strings will Equation 2 distinguishes a global from a gauge
start oscillating and radiate axions. Strings will (local) string, whose energy density decays
also intercommute and form closed loops, exponentially outside of its core. For axion
which collapse and convert into axions. t f a
Simulations of gauge string networks show strings near QCD time, ln ( ) 70
QCD and
that a scaling solution [1,2] is reached after a most of the string energy resides outside of the
few Hubble times with about one long string core.
per horizon size and a population of decaying The number of axions emitted by strings
loops. This process continues until QCD time, during the string epoch (1 GeV < T < 1012 GeV)
when the axion acquires a small mass. As a can in principle be calculated from the
 v2 10 Aug 1998 consequence, domain walls form between radiation spectrum of the evolving string
strings. The wall-string system is short-lived network. Analytic techniques are not well
because wall tension pulls the strings together, developed and one must rely on numerical
followed by mutual annihilation into free simulations. A major difficulty is the
axions. If there is an inflationary period after enormous loop to core size ratio of realistic
PQ symmetry breaking with a reheat axion strings. Currently, the largest computer
temperature < f simulations have ln (L / ) 7 . Yet, it is
a , strings get diluted and do
not contribute to the cosmological axion valuable to study the spectrum as a function of
density. ln (L/ ) in order to deduce a possible trend.
The dynamics of axion strings is governed This is especially important for closed loops as
by the classical Lagrangian density they dissipate most of the string network
energy.
1  1 2 2
L =  - ( - f ) (1) In the past, there has been considerable
2 4 a controversy over the correct radiation
spectrum. One group [3,4,5,6,7] (scenario A)
where = exp (ia / f )
a is a complex scalar argues that the spectrum is strongly peaked at
field whose phase is the axion field. A simple wavelengths of order the loop size. This
solution of (1) is a static, straight global string corresponds to an under-damped decay with


about 5-10 oscillations per loop half-life when configurations are set up on large (~107 points)
ln (L / ) 70 . A second group [8,9,10] Cartesian grids, and then time-evolved using
(scenario B) argues that the loops decay the finite-difference equations derived from
without oscillations, in a time of order the Eq. (1). A FFT spectrum analysis of the kinetic
initial size divided by the speed of light. The and gradient energies during the collapse
radiation spectrum is dE / dk ~1/ k (which is yields N (t)
ax .
also the spectrum of the static string field) with
cutoffs at the inverse loop size and core 2.1. Circular loops
thickness. Of course, it is possible that neither
scenario A or B is correct. Because of azimuthal symmetry, circular loops
The cosmological axion number density at can be studied in r-z space. By mirror
time t is related to the radiation spectrum symmetry, the problem can be further reduced
through to one quarter-plane.
The static axion field far from the string core
t
1 t
d  (t) is (with f a 1)
a
n (t) ~ (3)
t3 / 2 3 / 2
t t (t)
PQ
where 1
a (r, z) (r, z) (6)
2

1 dE d dE
= (t) (t) d
(4) in the infinite volume limit, where is the
(t) d d solid angle subtended by the loop. We use as
initial configuration the outcome of a
and dE
d is the radiated axion energy relaxation routine starting with Equation 6
spectrum. A useful quantity is outside the core and

dE dk
N (t) = (t) . () 0.58( )exp(
i ) (7)
ax (5)
dk k


In scenario B, N ax is predicted to stay constant 2500
since both axion radiation and static string
field have the same spectrum. Conversely, one 2000
expects N L
ax to increase by ln( / ) in scenario n
itio 1500
A. The factor final initial
r = N / s
ax Nax , by which N ax op
increases during the decay of a string loop re 1000
determines the string decay contribution to the oC
axion cosmological energy density. The string 500
decay contribution is r times the contribution
from misalignment. In scenario A, r is of the 0
order ln (L/ ) , whereas in scenario B, r is of 0 1000 2000 3000 4000 5000
order unity. Tim e


Figure 1: Core position of collapsing loop versus time for
= 0.001 (dotted line) and = 0.004 (solid line). The
2. LOOP SIMULATIONS lattice size is 40964096.

We have performed simulations of various
loop geometries: (1) circular loops initially at within the core. Here, is the distance to the
rest, (2) noncircular loops with angular string, and is the winding angle. The
momentum, and (3) string-antistring pairs relaxation and the subsequent dynamical
with angular momentum. The initial evolution are done with reflective boundary


conditions. A step size dt = 0.2 was used for = c J (k r) sin (k z)
z mn 0 m n
the time evolution and the total energy was mn
conserved to better than 1 %. In general the = c J (k r) cos (k z)
r mn 1 m n (8)
loops collapsed at nearly the speed of light mn

without a rebound. For a small range of = c J (k r) cos (k z)
mn 0 m n
parameters, 80 < R / mn
0 < 190, where R0 is the
initial loop radius, we noticed a small bounce with the boundary conditions J (k R )
m = 0
as shown in Figure 1. 1 max
There is a substantial Lorentz contraction of and sin (k Z )
n = 0
max . The dispersion
the string core as it collapses (see Figure 2). A relationship is given by 2 2
= k + k .
lattice effect became evident when the reduced mn m n
core size becomes comparable to the lattice Figure 3 shows the power spectrum
spacing. This lattice effect consists of a dE / d (ln k ) displayed in ln k bins of
"scraping" of the string core on the underlying width ln k = .
0 5 at t = 0 and after the collapse
grid, during which the kinetic energy of the at t = 3000. At both times, the spectrum
string gets dissipated into high frequency exhibits an almost flat plateau, consistent with
axion radiation. We always choose small a dE / dk 1/ k spectrum. The high frequency
enough to avoid this phenomenon. cutoff of the spectrum is increased however
after the collapse and is associated with the
Lorentz contraction of the core.
The evolution of N ax (E / k )
mn mn during
mn
the loop collapse was studied for various
values of R /
0 . We observe a marked
decrease of N ax by ~ 20 % during the collapse
roughly independent of R /
0 .





10000


8000


) 6000
k
(ln /dE 4000
d


2000


0
Figure 2: Intensity plot of potential energy (contours 1 2 3 4 5 6 7 8 9 10 11
represent constant potential energy) in vicinity of string ln k
core for R = 2400, = 0.001 at t = 2600. The Lorentz factor is
0

about 4. The arrows represent the axion field.

Figure 3: Energy spectrum of collapsing loop for R =
0

2400 and = 0.001. The white (black) histogram represents
A spectrum analysis of the fields was the spectrum at t = 0 (t = 3000). The increased high
performed by expanding the gradient and frequency cutoff of the final spectrum is due to the Lorentz
kinetic energies as contraction of the core.


1.1 10 7 emission of gravitational radiation and the
loop diameter shrinks with time. The power is
1.0 10 7 2
P = G  , where  is the energy per unit
length, G is the gravitational constant, and
9.0 10 6 is a constant ~ 50 - 100 which depends on
xaN , . Evidently, the power is independent of
8.0 10 6 loop size, and the loop undergoes
4
1/( G ) 10 oscillations in its lifetime. The
7.0 10 6
radiation power spectrum was numerically
6.0 10 6 determined [15,16] to be 4 / 3
P 1/ k
k .
0 2000 4000 6000 8000 Axion strings are much more short-lived
Time
Figure 4: N than gauge strings and radiate axions
ax of a circular loop as a function of time for efficiently due to the strong topological
R = 2400, and = 0.004 (solid line), = 0.001 (dashed line),
0
and = 0.00025 (dotted line). coupling between string and field. No closed
loop solutions are known however.
2.2. Rotating loops According to scenario A [7,15], the radiated
power is 2
P 50 fa , independent of loop size,
There exists a family of nonintersecting and the loop shrinks linearly with time. The
("Kibble-Turok") [11,12,13,14] loops, which expected number of oscillations per loop half-
have been studied in the context of gauge life is ln (L / ) / 50 4 for ln (L / ) 70 . In
strings. They are solutions of the Nambu-Goto addition, N ax should increase by a factor
equations of motion and all have non-zero ln (L/ ) . In scenario B on the other hand, N
angular momentum. ax
Intercommuting (self-intersection with should remain approximately constant.
reconnection) causes the loop sizes to shrink, We performed numerous simulations of
and hence the average energy of radiated rotating loops on a 3D (2563) lattice with
periodic boundary conditions. Standard
axions to increase and hence N ax to decrease. Fourier techniques were used for the spectrum
Intercommuting favors scenario B for these analysis, and N was computed as a function
reasons. We picked the Kibble-Turok ax
of time using the dispersion relationship
configuration as an initial condition to avoid
intercommuting as much as possible, thus
giving scenario A the best possible chance to = 2(3 - cos k - cos k - cos k )
mnp m n p . (10)
get realized. 1.2 10 5
A common loop parameterization is given by
1.1 10 5

x
( , t) = R (
2 {1 - )
sin + 1
-
3
sin +
-
sin +
3 }
1.0 10 5
y
( , t) = R2 {- (1-)
cos - 1
-
3
cos -
-
cos
cos +
3 }
xa 9.0 104
z
( ,t) = R N
2 {- 2 1
( - )
cos -
-
sin
cos + }
(9) 8.0 10 4
where  = (  t) / R , and ( ,
0 2 R) is the 7.0 10 4
length along the loop. For a significant subset
[12,13,14] of the free parameters (
)
1
,
0 , 6.0 10 4
(
- , ) the loop never self-intersects. A 0 50 100 150 200 250 300 350 400
Time
noteworthy feature is the periodic appearance Figure 5: Nax of non-intersecting ("Kibble-Turok") loops
of cusps, where the string velocity as a function of time for = 0.01, = 0, and = 0.2 (solid
momentarily reaches the speed of light. The line), = 0.1 (dashed line), and = 0.0625 (dotted line). The
motion of a gauge string is damped by lattice size is 2563 and R = 72.


6000 system to form, which decays by emission of
Etot axions and eventually annihilates.
5000


4000
E
y kin
rge 3000
nE
2000 Egrad

1000
Epot
0 0 50 100 150 200 250 300 350 400
Time



Figure 6: Energy of non-intersecting loop. Shown are
total, gradient, kinetic, and potential energy as a function of
time for = 0.01, = 0, and = 0.0625.


Figure 5 shows N ax for various core sizes
and constant , . The behavior is very
similar to that of a non-rotating circular loop
with a reduction of ~ 25 %. Figure 6 depicts the
energy of the collapsing loop. Clearly, the total
energy is well conserved. A few percent of the
loop energy is dissipated as massive radiation,
shown here as Epot .
2.3. String-antistring pairs

Lastly, we studied rotating, parallel string-
antistring pairs, which can be thought of as
cross-sections of loops with large eccentricity.
The 20482 lattice was initialized with the
Abrikosov ansatz [17]

(x, y) =
(x - x , y - y ) (x - x , y - y )
1 1 1 2 2 2 (11)


where ( x , y ), (x , y )
1 1 2 2 are the locations of
string and antistring respectively. The fields
were relaxed with periodic boundary
conditions and the cores held fixed. The time
derivative (x, y) was obtained by a finite
difference over a small time step. The
attractive force per unit length between strings

(
( )
) = V
F
2 /

(12)
Figure 7: Collapsing string-antistring pair. The initial
rotational velocity of the cores is 0.4 and = 0.005. The
is competing against the centrifugal force. In snapshots are for t = 40, 800, 1440.
general, one expects a bound relativistic


Figure 7 shows the decay of a string-antistring Acknowledgements
system for an initial rotational velocity of 0.4.
For the parameter range (3 < ln ( / ) , This work was performed under the
v
rot 0.6, ~ 500), the system collapsed auspices of the U.S. Department of Energy
immediately in a time of order . under Contract No. W-7405-Eng-48 at LLNL
and under DE-FG05-86ER40272 at the
University of Florida. We thank Livermore
The spectrum and N ax were computed as Computing for granting us generous access to
well and Figure 8 shows the evolution for their facilities.
several cases.

8 103
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