Constraints on cosmic strings using data from the first
Advanced LIGO observing run
B. P. Abbott
etal.
*
(LIGO Scientific Collaboration and Virgo Collaboration)
(Received 4 December 2017; published 8 May 2018)
Cosmic strings are topological defects which can be formed in grand unified theory scale phase
transitions in the early universe. They are also predicted to form in the context of string theory. The main
mechanism for a network of Nambu-Goto cosmic strings to lose energy is through the production of loops
and the subsequent emission of gravitational waves, thus offering an experimental signature for the
existence of cosmic strings. Here we report on the analysis conducted to specifically search for
gravitational-wave bursts from cosmic string loops in the data of Advanced LIGO 2015-2016 observing
run (O1). No evidence of such signals was found in the data, and as a result we set upper limits on the
cosmic string parameters for three recent loop distribution models. In this paper, we initially derive
constraints on the string tension
G
μ
and the intercommutation probability, using not only the burst analysis
performed on the O1 data set but also results from the previously published LIGO stochastic O1 analysis,
pulsar timing arrays, cosmic microwave background and big-bang nucleosynthesis experiments. We show
that these data sets are complementary in that they probe gravitational waves produced by cosmic string
loops during very different epochs. Finally, we show that the data sets exclude large parts of the parameter
space of the three loop distribution models we consider.
DOI:
10.1103/PhysRevD.97.102002
I. INTRODUCTION
The recent observation of gravitational waves
[1]
(GWs)
has started a new era in astronomy
[2,3]
. In the coming
years Advanced LIGO
[4]
and Advanced Virgo
[5]
will be
targeting a wide variety of GW sources
[6]
. Some of these
potential sources could yield new physics and information
about the Universe at its earliest moments. This would be
the case for the observation of GWs from cosmic strings,
which are one-dimensional topological defects, formed
after a spontaneous symmetry phase transition character-
ized by a vacuum manifold with noncontractible loops.
Cosmic strings were first introduced by Kibble
[7]
(for a
review see for instance
[8
–
10]
). They can be generically
produced in the context of grand unified theories
[11]
.
Linear-type topological defects of different forms should
leave a variety of observational signatures, opening up a
fascinating window to fundamental physics at very high
energy scales. In particular, they should lens distant
galaxies
[12
–
14]
, produce high energy cosmic rays
[15]
,
lead to anisotropies in the cosmic microwave background
[16,17]
, and produce GWs
[18,19]
.
A network of cosmic strings is primarily characterized
by the string tension
G
μ
(
c
¼
1
), where
G
is Newton
’
s
constant and
μ
the mass per unit length. The existence of
cosmic strings can be tested using the cosmic microwave
background (CMB) measurements. Confronting experi-
mental CMB data with numerical simulations of cosmic
string networks
[20
–
23]
, the string tension is constrained to
be smaller than a few
10
−
7
.
Cosmic superstrings are coherent macroscopic states of
fundamental superstrings (F-strings) and also D-branes
extended in one macroscopic direction (D-strings). They
are predicted in superstring inspired inflationary models
with spacetime-wrapping D-branes
[24,25]
. For cosmic
superstrings, one must introduce another parameter to
account for the fact that they interact probabilistically. In
[26]
, it is suggested that this intercommutation probability
p
must take values between
10
−
1
and 1 for D-strings and
between
10
−
3
and 1 for F-strings. In this paper, we will
refer to both topological strings and superstrings as
“
strings,
”
and parametrize them by
p
and
G
μ
.
Cosmic string parameters can also be accessed through
GWs. Indeed, the dynamics of the network is driven by the
formation of loops and the emission of GWs. In particular,
cusps and kinks propagating on string loops are expected to
produce powerful bursts of GWs. The superposition of
these bursts gives rise to a stochastic background which can
be probed over a large range of frequencies by different
observations. Historically, the big-bang nucleosynthesis
(BBN) data provided the first constraints on cosmic strings
[27]
. It was then surpassed by CMB bounds
[28]
to then be
surpassed more recently by pulsar timing bounds
[29]
.In
this paper, we report on the search for GW burst signals
*
Full author list given at the end of the article.
PHYSICAL REVIEW D
97,
102002 (2018)
2470-0010
=
2018
=
97(10)
=
102002(20)
102002-1
© 2018 American Physical Society
produced by cosmic string cusps and kinks using Advanced
LIGO data collected between September 12, 2015 06
∶
00
UTC and January 19, 2016 17
∶
00 UTC
[30]
, offering a total
of
T
obs
¼
4 163 421
s(
∼
48
.
2
days) of coincident data
between the two LIGO detectors. Moreover, combining
the result from the stochastic GW background search
previously published in
[31]
, we test and constrain cosmic
string models. While the LIGO O1 burst limit remains
weak, the stochastic bound now surpasses the BBN bound
for the first time and is competitive with the CMB bound
across much of the parameter space.
We will place constraints on the most up-to-date string
loop distributions. In particular, we select three analytic
cosmic string models (
M
¼f
1
;
2
;
3
g
)
[8,32
–
35]
for the
number density of string loops, developed in part from
numerical simulations of Nambu-Goto string networks
(zero thickness strings with intercommutation probability
equal to unity), in a Friedman-Lemaître-Robertson-Walker
geometry. These models are more fully described in Sec.
II
where their fundamental differences are also discussed.
Section
III
presents an overview of the experimental data
sets which are used to constrain the cosmic string param-
eters. Finally, the resulting limits are discussed in Sec.
IV
.
II. COSMIC STRING MODELS
We constrain three different models of cosmic strings
indexed by
M
. Common to all these models is the
assumption that the width of the strings is negligible
compared to the size of the horizon, so that the string
dynamics is given by the Nambu-Goto action. A further
input is the strings intercommutation probability
p
.For
field theory strings, and in particular
U
ð
1
Þ
Abelian-Higgs
strings in the Bogomol
’
nyi
–
Prasad
–
Sommerfield limit
[8]
,
intercommutation occurs with effective unit probability
[36,37]
,
p
¼
1
. That is, when two superhorizon (infinite)
strings intersect, they always swap partners; and if a string
intersects itself, it therefore chops off a (subhorizon) loop.
The latter can also result from string-string intersections at
two points, leading to the formation of two new infinite
strings and a loop.
Cosmic string loops oscillate periodically in time, emit-
ting GWs.
1
A loop of invariant length
l
has period
T
¼
l
=
2
and corresponding fundamental frequency
ω
¼
4
π
=
l
.Asa
result it radiates GWs with frequencies which are multiples
of
ω
, and decays in a lifetime
τ
¼
l
=
γ
d
where
[18,40,41]
γ
d
≡
Γ
G
μ
with
Γ
≃
50
:
ð
1
Þ
If a loop contains kinks
[41
–
43]
(discontinuities on the
tangent vector of a string) and cusps (points where the
string instantaneously reaches the speed of light), these
source bursts of beamed GWs
[44
–
46]
. The incoherent
superposition of these bursts give rise to a stationary and
nearly Gaussian stochastic GW background. Occasionally,
sharp and high-amplitude bursts of GWs stand above this
stochastic GW background.
The three models considered here differ in the loop
distribution
n
ð
l
;t
Þ
d
l
, namely the number density of
cosmic string loops of invariant length between
l
and
l
þ
d
l
at cosmic time
t
. To determine the consequences of
these differences on their GW signal, we work in units of
cosmic time
t
and introduce the dimensionless variables
γ
≡
l
=t
and
F
ð
γ
;t
Þ
≡
n
ð
l
;t
Þ
×
t
4
:
ð
2
Þ
We will often refer to
γ
as the relative size of loops and
F
as
simply the loop distribution. All GWs observed today are
formed when the string network is in its
scaling regime
,
namely a self-similar, attractor solution in which all the
typical length scales in the problem are proportional to
cosmic time.
2
The models considered here were developed (in part)
using numerical simulations of Nambu-Goto strings, for
which
p
¼
1
. As mentioned above, cosmic superstrings
intercommute with probability
p<
1
. The effect of a
reduced intercommutation probability on the loop distri-
bution has been studied in
[47]
. Following this reference
we take
3
F
p<
1
¼
F
=p
, leading to an increased density of
strings
[48]
and to an enhancement of various observational
signatures.
A. Model
M
= 1: Original large loop distribution
The first model we consider is the oldest, developed in
[8,32]
. It assumes that, in the scaling regime, all loops
chopped off the infinite string network are formed with the
same
relative size, which we denote by
α
. At time
t
, the
distribution of loops of length
l
to
l
þ
d
l
contains loops
chopped off the infinite string network at earlier times, and
diluted by the expansion of the Universe and by the
emission of GWs. Assuming that loops do not self-intersect
once formed, and taking into account that the length of
a loop decays at the rate
d
l
=dt
¼
−
γ
d
, the scaling loop
distribution (for
γ
≤
α
) in the radiation era is given by
[8]
F
ð
1
Þ
rad
ð
γ
Þ¼
C
rad
ð
γ
þ
γ
d
Þ
5
=
2
Θ
ð
α
−
γ
Þ
;
ð
3
Þ
where
Θ
is the Heaviside function, and the superscript
(1) stands for model
M
¼
1
. Some of these loops formed in
the radiation era can survive into the matter era, meaning
1
Superhorizon cosmic strings also emit GWs, due to their
small-scale structure
[19,38,39]
.
2
Scaling breaks down for a short time in the transition between
the radiation and matter eras, and similarly in the transition to
dark energy domination.
3
In
[47]
the exponent of the power-law behavior was found
to be slightly different, namely 0.6. Since our goal here is to
highlight the effect of
p<
1
, we used a simple dependence of
1
=p
as many others in the literature have done.
B. P. ABBOTT
et al.
PHYS. REV. D
97,
102002 (2018)
102002-2
that in the matter era the loop distribution has two
components. Those loops surviving from the radiation
era have distribution
F
ð
1
Þ
;a
mat
ð
γ
;t
Þ¼
C
rad
ð
γ
þ
γ
d
Þ
5
=
2
t
eq
t
1
=
2
Θ
ð
−
γ
þ
β
ð
t
ÞÞ
;
ð
4
Þ
with
t
eq
the time of the radiation to matter transition, and
where the lower bound,
β
ð
t
Þ
, is the length in scaling units,
of the last loops formed in the radiation era at time
t
eq
:
β
ð
t
Þ¼
α
t
eq
t
−
γ
d
1
−
t
eq
t
:
ð
5
Þ
The loops formed in the matter era itself have a distribution
F
ð
1
Þ
;b
mat
ð
γ
;t
Þ¼
C
mat
ð
γ
þ
γ
d
Þ
2
Θ
ð
α
−
γ
Þ
Θ
ð
γ
−
β
ð
t
ÞÞ
:
ð
6
Þ
The normalization constants
C
rad
and
C
mat
cannot be
determined from analytical arguments, but rather are fixed
by matching with numerical simulations of Nambu-Goto
strings. Following
[8,32]
, we set them to
C
rad
≃
1
.
6
;C
mat
≃
0
.
48
:
ð
7
Þ
Furthermore we shall assume that
α
≃
0
.
1
. The loop
distribution in the matter era is thus given by the sum of
distributions in Eqs.
(4)
and
(6)
.
The loop distribution
F
ð
1
Þ
is plotted in Fig.
1
for different
redshift values and fixing
G
μ
at
10
−
8
. A discontinuity,
visible for low redshift values, results from the radiation-
matter transition which is modeled by Heaviside functions.
For
t<t
eq
, the loop distribution is entirely determined by
Eq.
(3)
and is time independent.
B. Model
M
= 2: Large loop Nambu-Goto
distribution of Blanco-Pillado
etal.
Rather than postulating that all loops are formed with a
given size
α
t
at time
t
as in model 1, the loop production
function can be determined from numerical simulations.
This approach was taken in
[33]
, determining the rate of
production of loops of size
l
and momentum
⃗
p
at time
t
.
Armed with this information,
n
ð
l
;t
Þ
is determined ana-
lytically as in model 1 with the additional assumption that
the momentum dependence of the loop production function
is weak so that it can be integrated out.
In the radiation era, the scaling distribution reads
F
ð
2
Þ
rad
ð
γ
Þ¼
0
.
18
ð
γ
þ
γ
d
Þ
5
=
2
Θ
ð
0
.
1
−
γ
Þ
;
ð
8
Þ
where the superscript (2) stands for model 2. In the matter
era, analogously to above, there are two contributions.
The loops left over from the radiation era can be deduced
from above, whereas loops formed in the matter era have
distribution
F
ð
2
Þ
;b
mat
ð
γ
;t
Þ¼
0
.
27
−
0
.
45
γ
0
.
31
ð
γ
þ
γ
d
Þ
2
Θ
ð
0
.
18
−
γ
Þ
Θ
ð
γ
−
β
ð
t
ÞÞ
;
ð
9
Þ
where
β
ð
t
Þ
is given in Eq.
(5)
with
α
¼
0
.
1
.
The loop distribution of model 2 is plotted in Fig.
1
.
Notice that in the radiation era, the distributions in models 1
and 2 take the same functional form, though their nor-
malization differs by a factor of order 10. In the matter era,
the functional form is slightly different and the normali-
zation is smaller by a factor of order 2. The authors of
[33]
attribute this reduction in the number of loops to two
effects: (i) only about 10% of the power is radiated into
large loops
—
indeed, most of it is lost directly into smaller
loops which radiate away very quickly; (ii) most of the
energy leaving the network goes into loop kinetic energy
which is lost to redshifting.
C. Model
M
= 3: Large loop Nambu-Goto
distribution of Ringeval
etal.
This analytical model was presented in
[34]
, and is based
in part on the numerical simulations of
[35]
.
As opposed to model 2, here the (different) numerical
simulation is not used to determine the loop production
function at time
t
, but rather the distribution of non-self-
intersecting loops at time
t
. The analytical modeling also
differs from that of model 2 in that an extra ingredient is
12
−
10
10
−
10
8
−
10
6
−
10
4
−
10
2
−
10
1
γ
Loop size
1
4
10
8
10
12
10
16
10
20
10
22
10
Loop distribution
Model 1
-2
z=10
2
z=10
z=1
4
z=10
6
z=10
12
−
10
10
−
10
8
−
10
6
−
10
4
−
10
2
−
10
1
γ
Loop size
1
4
10
8
10
12
10
16
10
20
10
22
10
Loop distribution
Model 2
-2
z=10
2
z=10
z=1
4
z=10
6
z=10
12
−
10
10
−
10
8
−
10
6
−
10
4
−
10
2
−
10
1
γ
Loop size
1
4
10
8
10
12
10
16
10
20
10
22
10
Loop distribution
Model 3
-2
z=10
2
z=10
z=1
4
z=10
6
z=10
FIG. 1. Loop size distributions predicted by three models:
M
¼
1
, 2, 3. For each model, the loop distribution,
F
ð
γ
;t
ð
z
ÞÞ
, is plotted for
different redshift values and fixing
G
μ
at
10
−
8
.
CONSTRAINTS ON COSMIC STRINGS USING DATA FROM
...
PHYS. REV. D
97,
102002 (2018)
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