-
Rybak, I. Yu and Avgoustidis, Anastasios and Martins, C.J.A.P.
(2017) Semi-analytic calculation of cosmic microwave background
anisotropies from wiggly and superconducting cosmic strings.
Physical Review D, 96 (10). 103535/1-103535/20. ISSN 2470-0029
Access from the University of Nottingham repository:
http://eprints.nottingham.ac.uk/48326/1/wiggly_PRDaccepted.pdf
Copyright and reuse:
The Nottingham ePrints service makes this work by researchers of
the University of Nottingham available open access under the
following conditions.
This article is made available under the University of
Nottingham End User licence and may be reused according to the
conditions of the licence. For more details see:
http://eprints.nottingham.ac.uk/end_user_agreement.pdf
A note on versions:
The version presented here may differ from the published version
or from the version of record. If you wish to cite this item you
are advised to consult the publisher’s version. Please see the
repository url above for details on accessing the published version
and note that access may require a subscription.
For more information, please contact
[email protected]
mailto:[email protected]
-
Semi-analytic calculation of cosmic microwave background
anisotropies from wigglyand superconducting cosmic strings
I. Yu. Rybak,1, 2, 3, ∗ A. Avgoustidis,4, † and C. J. A. P.
Martins1, 2, ‡
1Centro de Astrof́ısica da Universidade do Porto, Rua das
Estrelas, 4150-762 Porto, Portugal2Instituto de Astrof́ısica e
Ciências do Espaço, CAUP, Rua das Estrelas, 4150-762 Porto,
Portugal
3Faculdade de Ciências, Universidade do Porto, Rua do Campo
Alegre 687, 4169-007 Porto, Portugal4School of Physics and
Astronomy, University of Nottingham,
University Park, Nottingham NG7 2RD, England(Dated: 4 September
2017)
We study how the presence of world-sheet currents affects the
evolution of cosmic string networks,and their impact on predictions
for the cosmic microwave background (CMB) anisotropies generatedby
these networks. We provide a general description of string networks
with currents and explicitlyinvestigate in detail two physically
motivated examples: wiggly and superconducting cosmic
stringnetworks. By using a modified version of the CMBact code, we
show quantitatively how the relevantnetwork parameters in both of
these cases influence the predicted CMB signal. Our analysis
suggeststhat previous studies have overestimated the amplitude of
the anisotropies for wiggly strings. Forsuperconducting strings the
amplitude of the anisotropies depends on parameters which
presentlyare not well known—but which can be measured in future
high resolution numerical simulations.
I. INTRODUCTION
As demonstrated in [1], symmetry breaking processesin early
universe scenarios can lead to the formationof topologically stable
line-like concentrations of energy,known as cosmic strings (for
general reviews see [2, 3]).These one-dimensional objects evolve
and interact witheach other, forming a cosmic string network.
Depend-ing on their origin, strings can have significantly
differ-ent properties and observational signatures. Examples
oftheoretically well-motivated scenarios where the presenceof
cosmic strings is expected include brane inflation [4–7],
supersymmetric grand unified theories with hybridinflation [8–13]
and many others [14–17]. In most cases,cosmic strings are stable
and survive to the present era,acting as fossils for these models.
Hence, quantitativebounds placed on string networks can lead to
strong con-straints on the underlying early universe model.
One difficulty is precisely that different models can pro-duce
strings with different properties, with varying ob-servational
predictions for the corresponding string net-works. Hence, in order
to achieve reliable observationalconstraints on the underlying
early universe models fromcosmic string network phenomenology, one
needs to de-velop an accurate description of cosmic string
networkevolution, taking into account the distinctive features
ofdifferent types of cosmic strings. One way to accomplishthis task
is through numerical simulations [18, 19]. Thisapproach provides
reliable results, but is currently lim-ited by computer
capabilities, especially when one tries toinclude non-trivial
cosmic string features like world-sheetcurrents. At present,
multi-tension cosmic string net-works and strings with currents are
very time-consuming
∗ [email protected]†
[email protected]‡
[email protected]
to model, and cannot be simulated with both high-resolution and
sufficiently large dynamic range (see [20]for a recent field theory
simulation of pq-stings). Fur-ther, numerical simulations have to
be repeated for dif-ferent values of cosmological and string
parameters andare thus not particularly flexible for parameter
determi-nation through direct confrontation with
observationaldata.
There is an alternative – and largely complementary–
semi-analytic approach for the description of cosmicstring network
evolution based on the velocity-dependentone-scale (VOS) model [21,
22]. In this treatment itis much easier to add non-trivial features
for cosmicstrings [23–30] allowing evolution over large
dynamicalranges that cannot be achieved by numerical
simulations.However, semi-analytic descriptions involve free
parame-ters, which can only be reliably calibrated by comparisonto
simulations. As a result, a combination of such ana-lytic
descriptions and numerical simulations is at presentthe best
approach for studying the evolution of cosmicstring networks with
non-trivial properties.
Among different methods for detecting observationalsignals from
cosmic string networks, cosmic microwavebackground (CMB)
anisotropies offer one of the mostsensitive and robust probes [31].
Current results ob-tained using cosmic string network simulations
[32, 33]and calibrated semi-analytic descriptions [34] yield
verysimilar constraints for simple global cosmic strings,
thecurrent limit on the string tension being at the level ofGµ
-
2
(which is common in supersymmetric models [36, 37]),by trapped
vector fluxes on non-Abelian stings [38], andother specific
mechanisms (for example symmetry break-ing of an accidental
symmetry in SU(2) stings [39]). Froma phenomenological point of
view, the presence of suchcurrents gives rise to effective,
macroscopic properties onthe string. For example, small-scale
structure (wiggles)on strings can be described by a specific type
of cur-rent [40–42].
In what follows we will show quantitatively how thepresence of
currents on the string worldsheet can affectobservational
predictions for the string CMB signal, pay-ing particular attention
to the special cases of wigglystrings and superconducting strings.
In the case of wig-gly strings this generalises and extends the
work of [43],where string wiggles were taken into account through
aconstant free parameter α. In our approach we can con-struct the
most general model for wiggly strings, leadingto a full description
of wiggles, including their evolutionand their effect on the string
equations of motion. Forsuperconducting strings, some relevant
model parametersare less well known (due to the lack of numerical
simu-lations of these models) but we are also able to providea full
description. In both cases, our results will enablea more detailed
and robust comparison to observations,which we leave for future
work.
II. STRING MODEL WITH CURRENTS
In order to obtain an effective two-dimensional La-grangian of a
string-like object from a four-dimensionalfield theory, one usually
follows the procedure of [35].This coarse-graining approach
unavoidably involves theloss of some of the features of the
original four-dimensional description; in particular, it cannot
describekey properties of superconducting strings like current
sat-uration and supersonic wiggle propagation. As a result,there is
only a phenomenological approach to reproduceproperties of the
original four-dimensional model [44, 45].On the other hand, one is
often interested in averagedequations of motion and these can be
the same for differ-ent Lagrangians (for an explicit example see
[25]). Thus,focusing on deriving the exact form of the Lagrangian
isnot necessarily the most productive route to obtainingaccurate
string network evolution.
Bearing in mind the subtleties described above, we con-sider the
general form of a two-dimensional Lagrangianinvolving an arbitrary
function of a string current. First,note that a current on a
two-dimensional space can berepresented as a derivative of a scalar
field ϕ,
Ja = ϕ,a, (1)
where ,a =∂∂σa , with σ
a the coordinates on the stringworldsheet (Latin indexes a, b
run over 0, 1).
We can thus build three possible terms “living” onthe
worldsheet, out of which the Lagrangian will be con-
structed
[1]: ϕ,aϕ,bγab = κ,
[2]: εacεbdγabγcd = γ,
[3]: εacεbdγabϕ,cϕ,d = ∆,
(2)
where γab = gµνxµ,ax
ν,b is the induced metric on the string
worldsheet (gµν being the background space-time metricwith Greek
indexes µ, ν corresponding to 4-dimensionalspace-time coordinates),
γ is the determinant of the in-duced metric and εab is the
Levi-Civita symbol in twodimensions.
The term ∆ is motivated by the Dirac-Born-Infeld(DBI) action for
cosmic strings (relevant studies can befound in [46], [47] and
[26]).
Taking into account the three possible terms in Eq. (2)we can
write down the general form of the action general-ising the
Nambu-Goto action to the case of a string withcurrent
S = −µ0∫f(κ, γ,∆)
√−γd2σ, (3)
where µ0 is a constant of dimensions [Energy]2 deter-
mined by the symmetry breaking scale giving rise tostring
formation.
The arbitrary choice of the function f(κ, γ,∆) canbreak
reparametrisation invariance of the generalizedaction of Eq. (3).
In order to preserve invarianceof the action under
reparametrizations, the last twoterms 2 should be connected in the
following wayf(κ,∆/γ). Hereinafter for the sake of simplicity the
func-tion f(κ,∆/γ) in equations will be denoted just as f .
Assuming that cosmic strings are moving in a
flatFriedmann-Lemaitre-Robertson-Walker (FLRW) back-ground with
metric ds2 = a(τ)
(dτ2 − dl2
), we can build
the stress-energy tensor from the action (3)
Tµν(y) =µ0√−g
∫d2σ√−γδ(4)(y − x(σ))(
Ũ ũµũν − T̃ ṽµṽν − Φ(ũµṽν + ṽµũν)),
(4)
where ũµ =√�ẋµ
(−γ)1/4 and ṽµ = x
′µ√�(−γ)1/4 are orthonormal
timelike and spacelike vectors respectively (ũµũµ = 1,
ṽµṽµ = −1), � =√
x′ 2
1−ẋ2 and
Ũ = f − 2∂f∂γ∆γ + 2γ
00 ∂f∂κ ϕ̇
2 + 2γ11 ∂f∂∆ϕ′ 2, (5)
T̃ = f − 2∂f∂γ∆γ + 2γ
11 ∂f∂κϕ
′ 2 + 2γ00 ∂f∂∆ ϕ̇2, (6)
Φ = 2√−γ
(∂f∂κ −
∂f∂∆
)ϕ′ϕ̇ ; (7)
here and henceforth dots and primes respectively denotetime and
space derivatives.
It is important to note that for this modification of
theLagrangian, the stress-energy tensor (4) has non-diagonalterms
induced by the presence of the current. Let usobtain the equations
of motion for the action (3) using
-
3
the definitions of Ũ in (5), T̃ in (6) and Φ in (7).
Variationof the action (3) with respect to xµ and ϕ gives
∂τ (�Ũ) +ȧa�(ẋ2(Ũ + T̃ ) + Ũ − T̃
)= ∂σΦ, (8)
ẍ�Ũ + ẋ� ȧa(1− ẋ2
) (Ũ + T̃
)=
= ∂σ
(T̃� x′)
+ x′(
2 ȧaΦ + Φ̇)
+ 2Φẋ′, (9)
∂τ
((∂f∂κ +
∂f∂∆
)�ϕ̇)
= ∂σ
((∂f∂κ +
∂f∂∆
)ϕ′
�
). (10)
where we have chosen a parametrisation satisfying thetransverse
temporal conditions ẋ · x′ = 0 and x0 = τ .
As can be seen from the equations of motion (8)and (9), string
dynamics does not depend explicitly onthe form of the current
contribution f(κ,∆/γ). The dy-
namics of the string is defined completely by Ũ , T̃ andΦ,
which can be associated to mass per unit length andstring tension.
Indeed, it is only the dynamics of ϕ it-self – equation (10) – that
explicitly depends on ∂f/∂κand ∂f/∂∆. This provides us an
alternative approachto studying string dynamics effectively,
without an ex-plicit connection between an effective
Nambu-Goto-likeaction and the original field theory model. One can
in-stead study the behaviour of Ũ , T̃ and Φ in the
originalfour-dimensional model in the framework of field theory(as
it was done for example in [48–52]) and then insert
the dynamics of Ũ , T̃ and Φ in the equations of motion(8) and
(9).
Additionally, we also note that one can easily general-ize the
equations of motion (8)-(10) to include any num-ber of uncoupled
scalar fields, associated to correspond-ing currents. In this case,
we can simply rewrite thevariables κ and ∆ as
κi = γabϕ,ai ϕ
,bi , ∆i = ε
acεbdγabϕi,cϕi,d , (11)
where the index i runs over the number of fields. Thereis no
summation over i; if a sum over this index is to betaken it will be
written explicitly.
Definitions (5), (6) and (7) in the case of multiple cur-rents
generalise to
Ũ = f + 2∑i
(γ00 ∂f∂κi ϕ̇i
2 + γ11 ∂f∂∆iϕ′ 2i −
∂f∂γ
∆iγ
), (12)
T̃ = f + 2∑i
(γ11 ∂f∂κiϕ
′ 2i + γ
00 ∂f∂∆i
ϕ̇i2 − ∂f∂γ
∆iγ
), (13)
Φ = 2√−γ∑i
(∂f∂κi− ∂f∂∆i
)ϕ′iϕ̇i. (14)
With definitions (12)-(14) the form of the stress-energytensor
(4) and the equations of motion (8)-(9) stay un-changed. On the
other hand, the equation of motion forthe scalar field (10) is
substituted by the set of equations
∂τ
((∂f∂κi
+ ∂f∂∆i
)�ϕ̇i
)= ∂σ
((∂f∂κi
+ ∂f∂∆i
)ϕ′i�
). (15)
We see, therefore, that if we extend the action (3) toinclude
additional scalar fields ϕi, the structure of theequations of
motion together with the form of the general
stress-energy tensor remains unchanged; we only need toadd a new
index i to κ and ∆. This fact will be usefulin our considerations
below. For now, let us diagonalisethe stress-energy tensor (4) and
define the mass per unitlength and tension for these strings with
currents, follow-ing references [40, 49]
Tµν uν = Uδµν u
ν , (16)
Tµν vν = Tδµν v
ν . (17)
The new orthonormal timelike uµ and spacelike vµ vec-tors are
eigenvectors of the stress-energy tensor (4) withcorresponding
eigenvalues U (mass per unit length) andT (tension). These
eigenvalues are related to the original
Ũ , T̃ and Φ in (5-7) by
U = µ0/2(Ũ + T̃ + ∆
), (18)
T = µ0/2(Ũ + T̃ −∆
), (19)
while the eigenvectors can be expressed in terms of theoriginal
ũµ and ṽµ as
uµ = aũµ +√a2 − 1ṽµ, (20)
vµ =√a2 − 1ũµ + aṽµ, (21)
with
a =1
2
[1 +
Ũ − T̃∆
]
and
∆ =
√(Ũ − T̃ )2 − 4Φ2 .
The passage from the equations of motion of a sin-gle string
segment to an effective description of a wholenetwork of strings is
done through an averaging proce-dure [21] leading to the VOS model
for cosmic strings.Following this approach, we begin by dotting
equation (9)with vectors ẋ and x′ and using the property of
ourparametrization ẋ · x′ = 0 to obtain
ẋ · ẍ�Ũ + ẋ2� ȧa(1− ẋ2
) (Ũ + T̃
)=
= T̃� ẋ · x′′ − 2Φẍ · x′, (22)
x′ · ẍ�Ũ − x′ · x′′ T̃� + 2Φx′′ · ẋ =
= x′2(
2 ȧaΦ + Φ̇ +T̃ ′
� −T̃�2 �′). (23)
Using the expression �′
� =x′·x′′x′2 −
x′·ẍ1−ẋ2 we can elimi-
nate the terms proportional to �′ and x′ · ẍ obtaining
theequation
ẋ · ẍ�Ũ + ẋ2� ȧa
(1− ẋ2
) (Ũ + T̃
)− T̃�ẋ · x′′ =
= 2Φ1− ẋ2
Ũ − T̃
(T̃ ′ + �
(2ȧ
aΦ + Φ̇− 2Φx
′′ · ẋx′2
)).
(24)
-
4
We now introduce the macroscopic variables
E = µ0a∫Ũ�dσ , (25)
E0 = µ0a∫�dσ , (26)
v2 =〈ẋ2〉, (27)
where 〈...〉 =∫... �dσ∫�dσ
denotes the (energy-weighted) av-
eraging operation. These macroscopic quantities are,
re-spectively, the total energy, the ’bare’ energy (withoutthe
contribution form the current) and the Root-MeanSquared (RMS)
velocity. Using these definitions we pro-ceed to average equations
(24) and (8) finding
Ė + ȧaE(υ2 (1 +W )−W
)= 〈Φ′/�〉E0, (28)
υ̇ + υ ȧa(1− υ2
)(1 +W )− (1− υ2)k(υ)Rc =
=〈
2 ΦŨ
1−ẋ21−T̃ /Ũ
(T̃ ′
� + 2ȧaΦ + Φ̇− 2Φ
x′′·ẋx′2
)〉. (29)
Here, we have defined W =〈T̃ /Ũ
〉and introduced the
average comoving radius of curvature of strings in thenetwork,
Rc, and the curvature parameter, k(υ), satisfy-
ing
〈ẋ� ·(
x′
�
)′〉= k(υ)Rc υ(1 − υ
2). For ordinary cosmic
strings, an accurate ansatz for the curvature parameteras a
function of velocity
k(υ) =2√
2
π(1− v2)(1 + 2
√2v3)
1− 8v6
1 + 8v6, (30)
has been derived in [22]. We assume that this functionstays
valid for strings with currents as well.
Following the procedure of [21, 22], we rewrite the aver-aged
equations of motion (28-29) in terms of more conve-nient
macroscopic variables: the comoving characteristiclength Lc and the
comoving correlation length ξc, whichare related to the energies in
(25-26) by the followingexpressions
E =µ0V
a2L2c
and
E0 =µ0V
a2ξ2c,
where V is the volume over which the averaging has
beenperformed. In addition, we employ the VOS model ap-proximation
that the average radius of curvature of cos-mic strings in the
network is equal to the correlationlength, i.e. Rc ≈ ξc. Assuming
further that the av-eraged macroscopic quantities can be split
as
〈ΦŨ T̃
〉=
〈Φ〉〈Ũ〉〈
T̃〉
we obtain the following system of equations
2L̇c =ȧaLc
(υ2 (1 +W )−W + 1
)−√
1−υ2Q,sÛ
, (31)
υ̇ + υ ȧa(1− υ2
)(1 +W )− (1− υ2)k(υ)ξc =
= 2QÛ
1−υ21−W
(√1− υ2 T̂,s + Q̇+ 2Q
(ȧa −
k(υ)υξc
)), (32)
where 〈Φ〉 = Q,〈
Φ̇〉
= Q̇, Û =〈Ũ〉
, T̂ =〈T̃〉
, the
correlation and characteristic lengths are related by ξc =
Lc√Û and a new derivative variable ,s =
∂∂s has been
introduced, corresponding to the parametrization ds =√x′2dσ.
Equations (31-32) are the averaged macroscopic equa-tions
describing a network of cosmic strings with a cur-rent. It is
apparent that scaling solutions (Lc = εcτ, υ =const when a ∝ τn
with εc and n constants [21, 22]) existif the averaged quantities
Û , T̂ and Q are appropriatelyrestricted. In particular we see
from (31)-(32) that scal-ing behaviour – typical for ordinary
string networks –can arise when Û , T̂ , Q = const, while T,s and
Q,s ∼ 1/τ .Additionally, the requirement of a well-defined εc
impliesthe condition
υ2 <2 + n(W − 1)n(W + 1)
. (33)
Equation (33) relates the rms string velocity υ to the
ratio W =〈T̃ /Ũ
〉for a given expansion rate (charac-
terisd by n) for a cosmic string network with currents.These
general relations will be useful when we considerthe special case
of a wiggly string network.
We now concentrate on how these modifications caninfluence
predictions for the CMB anisotropy from cos-mic strings. We follow
the approach of [43, 53, 54].Rather than working with the full
network of cosmicstrings, we consider a number of straight string
segmentsin Minkowski space that decay according to the evolu-tion
of strings in an expanding FLRW metric, and havevelocities and
lengths determined by the VOS model.
We start from the Fourier transform of the stress-energy tensor
(4) of a single straight string segment onwhich the contribution
from string currents has been av-eraged as above
Θµν = µ0
∫ ξ0τ/2−ξ0τ/2
[Û�ẊµẊν − T̂ X
′µX ′ν
�−
−Q(ẊµX ′ν + ẊνX ′µ
)]eik·Xdσ,
(34)
where the vector Xµ = xµ0 + σX′µ + τẊµ represents the
straight, stick-like solution for a string moving with ve-locity
υ (so that ẊµẊµ = 1−υ2) and with worldsheet co-ordinates σ and τ
in the transverse temporal gauge. Thecomoving length of a string
segment at conformal time τis ξ0τ , where ξ0 will be determined
from the macroscopicevolution equations (31-32). Variables Û , T̂
and Q areconstants for the straight string, as follows from the
equa-tions of motion (8-10). The four-vector xµ0 = (1,x0) isa
random location for a single string segment, while X ′µ
and Ẋµ are randomly oriented and satisfy the transverse
-
5
condition X ′µẊµ = 0. We can choose these vectors1 as
Ẋµ =
1υ(cos θ cosφ cosψ − sinφ sinψ)υ(cos θ sinφ cosψ + cosφ sinψ)−υ
sin θ cosψ
, (35)
X ′ µ =
0sin θ cosφsin θ sinφcos θ
. (36)Without loss of generality we can choose the wave vec-
tor along the third axis k = kk̂3 and integrating over σwe
obtain the following expressions
Θ00 =µ0Û√1−v2
sin(kX3ξ0τ/2)kX3/2
cos(k · x0 + kX3vτ), (37)
Θij = Θ00
[v2ẊiẊj − T̂ /Û(1− v2)X ′iX ′j −
− vQ/Û(ẊiX
′j + ẊjX
′i
)], (38)
where the indices i, j run over the 3-dimensional
spatialcoordinates.
The scalar, vector and tensor components can be de-fined as
ΘS = (2Θ33 −Θ11 −Θ22) /2, (39)
ΘV = Θ13, (40)
ΘT = Θ12. (41)
Substituting (37) and (38) in (39)-(41), we obtain thescalar,
vector and tensor contributions for a straightstring segment with
stress-energy tensor (4), (34)
2ΘS
Θ00=
[v2(3Ẋ3Ẋ3 − 1)− 6vQ/ÛX ′3Ẋ3−
−(1− v2)T̂ /Û(3X ′3X ′3 − 1)],
(42)
ΘV
Θ00=
[v2Ẋ1Ẋ3 − T̂ /Û(1− v2)X ′1X ′3−
−vQ/Û(X ′1Ẋ3 + Ẋ1X
′3
)],
(43)
ΘT
Θ00=
[v2Ẋ1Ẋ2 − T̂ /Û(1− v2)X ′1X ′2−
−vQ/Û(X ′1Ẋ2 + Ẋ1X
′2
)].
(44)
Following the prescription of reference [55], we canthen
calculate the unequal time two-point correlators byaveraging over
locations, string orientations and velocityorientations of the
string segment
〈ΘI(k, τ1)Θ
J(k, τ2)〉
=2µ20F(τ1, τ2, ξ0)
16π3
∫ 2π0
dφ
∫ π0
sin θdθ
∫ 2π0
dψ
∫ 2π0
dχΘI(k, τ1)ΘJ(k, τ2) . (45)
Here, the indices I and J correspond to the scalar,
vector,tensor and “00” components. The function F(τ1, τ2,
ξ0)describes the string decay rate. It is chosen to havethe same
form as for ordinary (without currents) cosmic
1 Note that although we work in the transverse temporal gauge
wehave chosen the normalization X′2 = 1. This may seem to
beinconsistent as X′2 = �2(1 − Ẋ2) and � is evolving according
toequation (8). However, we are implicitly taking this effect
intoaccount by having the limits of the integral (34)
time-dependentthrough the time evolution of ξ0. This evolves
according to themacroscopic equation (31), which has been derived
by averagingequations (24) and (8).
strings [43]
F(τ1, τ2, ξ0) =1
(ξ0Max(τ1, τ2))3 , (46)
but here ξ0 is determined by modified VOS equations (28-29). The
phase χ = k ·x0 arises from varying over stringlocations x0 (refer
to equation (37)), which we integrateover.
We can write the general form of the correlators as〈ΘI(k,
τ1)Θ
J(k, τ2)〉
=µ20F(τ1,τ2,ξ0)k2(1−υ2) B
I−J(τ1, τ2). (47)
If we are only interested in the approximation kτ <
1(superhorizon scales), we can expand BI−J keeping only
-
6
terms that are up to k2. In this case the non-zero corre-lators
are the following
B00−00(τ1, τ2) ≈ Û2ξ20k2τ1τ2, (48)BS−S ≈ 15
B00−00
Û2(τ1, τ2)× (49)(
Û2υ4 + T̂ Ûυ2(1− υ2) + T̂ 2(1− υ2)2 + 3υ2Q2),
BV−V ≈ 13BS−S , (50)
BT−T ≈ 13BS−S . (51)
In the Appendix we give exact expressions for the equaltime
two-point correlators BI−J(τ) and provide semi-analytic expressions
for the unequal time two-point cor-relators valid for all (i.e.
from subhorizon through tosuperhorizon) modes k.
Having computed the correlators (47), let us now as-sume that
the cosmic string network under considerationhas reached a scaling
regime. We can then assume thatξ0, υ together with Û , T̂ and Q do
not depend on τ andσ. To obtain an analytic estimate of the
string-inducedCMB anisotropy, let us consider the string network
evolv-ing in the matter domination epoch (n = 2). For thiscase we
can use the following solution of the linearisedEinstein-Boltzmann
equations [53, 56]
δT
T= −1
2
∫ τfτi
dτḣijninj ,
ḣij = ḣSij + ḣ
Vij + ḣ
Tij ,
(52)
ḣSij = −ρ∑k
eik·x∫ τ
0
dτ ′(1
3δij
(τ ′
τ
)6(ΘTr + 2ΘS)− kikj
(τ ′
τ
)4ΘS
),
(53)
ḣVij =∑k
eik·x(V̇ikj + V̇jki
),
V̇i = ρ
∫ τ0
dτ ′(τ ′
τ
)ΘVi ,
(54)
ḣTij = ρ
∫ τ0
dτ ′k3τ ′4F (kτ ′, kτ)ΘTij ,
F (kτ ′, kτ) = G1(kτ′)Ġ2(kτ)−G2(kτ ′)Ġ1(kτ),
(55)
where ρ = 16πG, G1(kτ) =cos(kτ)(kτ)2 +
cos(kτ)(kτ)3 , G2(kτ) =
cos(kτ)(kτ)3 +
sin(kτ)(kτ)2 ,
δTT are the CMB temperature fluctua-
tions, ni is a unit vector defining the direction of CMB
photons, and ΘTr is the trace of the Fourier
transformedstress-energy tensor.
We can now compute the angular power spectrum Clof the CMB
anisotropy using the expression [53]:
CSl =1
2π
∫ ∞0
k2dk〈∫ τ00
dτ
(1
3ḣ1 + ḣ2
d2
d(k∆τ)2
)jl(k∆τ)
〉2,
(56)
CVl =2
π
∫ ∞0
k2dkl(l + 1)〈∫ τ00
dτḣVd
d(k∆τ)(jl(k∆τ)/(k∆τ))
〉2,
(57)
CTl =1
2π
∫ ∞0
k2dk(l + 2)!
(l − 2)!〈∫ τ00
dτ
(k∆τ)2ḣT jl(k∆τ)
〉2,
(58)
where ∆τ = τ0 − τ (with τ0 the value of conformal timetoday),
jl(k∆τ) are spherical Bessel functions, and ḣ1,
ḣ2 are defined as
ḣ1(τ) = −ρ∫dτ ′(τ ′
τ
)6(ΘTr(τ ′) + 2ΘS(τ ′)), (59)
ḣ2(τ) = −ρ∫dτ ′(τ ′
τ
)4ΘS(τ ′). (60)
We proceed by making a further approximation on thecorrelators
(47). The dominant contribution to the two-point correlator is when
τ1 → τ2 (see for example [55]),which allows us to approximate (47)
as
〈ΘI(k, τ1)Θ
J(k, τ2)〉
=µ20F(τ1, τ2, ξ0)k2(1− υ2)
×
BI−J(τ1)δ(τ1 − τ2),(61)
where δ(τ1 − τ2) is Dirac delta function and BI−J(τ1) =BI−J(τ1,
τ1).
By using this form of the correlators (61) one canrewrite
equations (56), (57) and (58) as
-
7
CSl =κ2
2π
∫∞0k2dk
∫ τ00dτ1∫ τ0
0dτ2∫ τ1
0dτ ′1
f(τ ′1,ε)k2(1−v2)
τ ′81τ41 τ
42Fsc(τ
′1), (62)
CVl =2κ2
π
∫∞0k2dkl(l + 1)
∫ τ00dτ1∫ τ0
0dτ2
j′l(kτ1)kτ1
j′l(kτ2)kτ2
∫ τ10dτ ′1
τ ′81 f(τ′1,ξ0)
k2(1−v2) BV−V (τ ′1), (63)
CTl =κ2
2π
∫∞0k2dk (l+2)!(l−2)!
∫ τ00dτ1∫ τ0
0dτ2
jl(kτ1)(kτ1)2
jl(kτ2)(kτ2)2
∫ τ10dτ ′1k
6τ ′81 F (τ′1, τ1)F (τ
′2, τ2)
f(τ ′1,ξ0)k2(1−v2)B
T−T (τ ′1), (64)
where
Fsc =19jl(kτ1)jl(kτ2)
τ ′41τ21 τ
22
(BTr−Tr(τ ′1) + 4B
Tr−S(τ ′1) + 4BS−S(τ ′1)
)+
+ 13
(j′′l (kτ1)jl(kτ2)
τ ′ 2
τ22+ j′′l (kτ2)jl(kτ1)
τ ′ 2
τ21
) (BTr−S(τ ′1) + 2B
S−S(τ ′1))
+ j′′l (kτ1)j′′l (kτ2)B
S−S(τ ′1), (65)
with trace components: BTr−Tr(τ ′1) =[1 + v2 − T̃ /Ũ(1− v2)
]2B00−00(τ ′1), (66)
BTr−S(τ ′1) =[1 + v2 − T̃ /Ũ(1− v2)
]B00−S(τ ′1). (67)
In the final form of equations (62) and (65) we haveexpressed
the contribution from the “00” componentin terms of the trace
component “Tr” using the rela-tions (66) and (67), which can be
derived from (37)and (38). It should be stressed that in obtaining
equa-tions (62), (63) and (64) we have only used the approx-imation
(61). We have thus succeeded to derive fullsemi-analytic
expressions for the scalar, vector and tensorcontributions to the
angular powerspectrum from cosmicstrings with arbitrary currents,
valid in matter domina-tion and under the approximation (61).
In the superhorizon limit kτ < 1 considered above,
thetwo-point correlators have the simple form (48)-(51) andwe can
factor out from the integrals (62)-(64) the keyquantities
characterising the cosmic string network: υ,ξ0, Û , T̂ and Q. This
allows us to establish a direct con-nection between cosmic string
network parameters andthe string contribution to CMB anisotropies,
valid onsuperhorizon scales. For the vector (63) and tensor
(64)contributions it is easy to see that
CV,Tl ∼ (Gµ0)2×
Û2υ4 + T̂ Ûυ2(1− υ2) + T̂ 2(1− υ2)2 + 3υ2Q2
ξ0(1− v2),
(68)
which agrees with the result of [55] in the limit Q = 0,U = αµ0
and T = µ0/α.
The treatment of the scalar mode (62) is more sub-tle. We will
estimate it to leading order, using the fol-lowing asymptotic form
of the spherical Bessel functionjl(x) ∼ xl, valid when 0 < x 500
we can take the leading term of (65) asj′′l (kτ1)j
′′l (kτ2)B
S−S(τ ′1). It follows that, in this approx-imation, the scalar
contribution will be the same as theabove approximate expressions
for the vector and tensor
components
CS1
-
8
or equivalently
Û T̂ =1 ,
Û = µ, T̂ =1/µ, Q = 0 ,(71)
where µ is a dimensionless parameter quantifying theamount of
wiggles on the string, and µ = 1 correspondsto the usual Nambu-Goto
string (the same parameter wasdenoted as α in [43]).
Applying the equation of state (70) to the averagedequations of
motion (31) and (32) we obtain
2dLcdτ =ȧaLc
[1 + υ2 − 1−υ
2
µ2
], (72)
dυdτ =
(1− υ2
) [ k(υ)Lcµ5/2
− ȧaυ(
1 + 1µ2
)]. (73)
Note that the comoving correlation length is connected
tocomoving characteristic length by the following relationξc =
õLc.
We can now include an energy loss term F (υ, µ) onthe right-hand
side of equation (72) and assume scalingbehaviour of the network Lc
= ετ (while ξc = ξ0τ). Notethat in this case the previously
obtained constraint (33)has the form
v2 <2/n− 1 + 1/µ2
1 + 1/µ2, (74)
where, in the scaling regime, µ is a constant.The expression
(74) means that the rms string velocity
has an upper limit, determined by the expansion rate nand amount
of wiggles µ on the string; this is illustratedin figure 1.
Figure 1. The constraint (74) on the square of the rms
veloc-ity, v2, depending on the expansion rate n and the amount
ofwiggles µ.
It is important to note that this restriction was ob-tained just
by using the equation of state for wiggly cos-mic strings (70) in
our general equation (31). This means
that any Lagrangian suitable for wiggly string description(i.e.
any choice of f(κ,∆/γ) satisfying (70) for the equa-tion of state)
cannot change this relation. Moreover, it isvalid for any energy
loss function F (v, µ). Thus, any wig-gly cosmic string network
with any energy loss functionof the form F (v, µ) must satisfy the
constraint (74).
To get a feeling for the size of the maximum networkvelocity in
(74) we consider two limiting cases: stringswithout wiggles (µ = 1)
and highly wiggly strings (µ →∞):
v2 < 1/n (µ = 1), (75)
v2 < 2/n− 1 (µ→∞). (76)
As seen from Eq. (75), for strings without wiggles onlyvery fast
expansion rates n can cause a significant re-striction to the
string network velocity, while for highlywiggled strings the limit
(76) provides a severe constrainteven when n = 2 (matter domination
era). For wigglystrings with µ = 1.5 in the matter domination era
(n = 2)the velocity is limited as v2 < 0.3, which is close to
thevalues of rms velocities from field theory simulations [19]in
the matter domination era.
Let us now study the full description of the wigglycosmic string
network model [24, 57] described by theaction
S = µ0
∫ω√−γd2σ, (77)
where ω =√
1− κ.The derivation of the averaged equations of motion for
this model can be found in [24, 57]. We will use the finalsystem
of equations in the following form (where we haveomitted the term
responsible for scale dependence)
2dLcdτ =ȧaLc
[1 + υ2 − 1−υ
2
µ2
]+ cfaυ√µ , (78)
dυdτ =
(1− υ2
) [k
Lcµ5/2− ȧaυ
(1 + 1µ2
)], (79)
1µdµdτ =
υL√µ
[k(
1− 1µ2)− c(fa − fo − S)
]−
− ȧa(
1− 1µ2), (80)
where the three functions fa(µ), f0(µ) and S(µ) quantifyenergy
loss/transfer:
2(dξdt
)only big loops
= cf0(µ)υ, (81)
2(dLdt
)all loops
= cfa(µ)υLξ , (82)
2(dξdt
)energy transfer
= cS(µ)υ . (83)
Here, c is a constant “loop chopping” parameter (see be-low), t
=
∫adτ , and ξ = ξca, L = Lca are the physical
(rather than comoving) lengthscales corresponding to ξcand
Lc.
The term f0(µ) accounts for the energy loss due to theformation
of big loops. Here, “big” means that they areformed by
intersections of strings separated by distances
-
9
of order the correlation length ξ or by self-intersections atthe
scale of the radius of curvature R ≈ ξ. The functionfa(µ) describes
the energy loss caused by all types ofloops, and the difference
fa(µ) − f0(µ) corresponds tothe energy loss by small loops only,
which is driven bythe presence of wiggles.
In order to reproduce correctly the original model with-out
wiggles, we can use the energy loss/transfer functionsas discussed
in [57]
f0(µ) = 1, (84)
fa(µ) = 1 + η(
1− 1√µ), (85)
S(µ) = D(1− 1µ2 ), (86)
where D and η are constants.
Thus, the evolution of a wiggly string network isdescribed by
the system of ordinary differential equa-tions (78)-(80), which, in
view of equations (84)-(86) in-cludes three free constant
parameters c, D and η [57]:
• c is the “loop chopping efficiency” parameter quan-tifying how
much energy the network looses due to theproduction of ordinary
loops;
• η is a parameter describing the energy loss enhance-ment due
to the creation of small loops caused by thepresence of
wiggles;
• D is a parameter quantifying the amount of energytransferred
from large to small scales.
By making various different choices of parametersc, η and D we
can explore the effects of the energyloss/transfer mechanisms
described above on the evolu-tion of the string network. Note that
c has been mea-sured in Abelian-Higgs and Goto-Nambu simulations
tobe c = 0.23±0.04 [58, 59], but there are no such measure-ments
for the other two parameters. Let us study howthese
phenomenological quantities can change the pre-diction for the CMB
anisotropy caused by wiggly cosmicstring networks.
In order to investigate in detail the effects of stringwiggles
on the predicted CMB anisotropies from cos-mic string networks, we
implement the wiggly VOSmodel (78)-(80) into the CMBact code [43].
The originalcode was developed so as to take into account the
pres-ence of string wiggles in the computation of the
string-induced CMB anisotropy. However, in the original CM-Bact
package, wiggles were modelled by a single (con-stant)
phenomenological parameter α = µ modifying theeffective mass per
unit length and string tension at thelevel of the stress-energy
tensor (87). In other words,within the approximations of the
original CMBact code,the amount of wiggles was not a dynamical
parameterand did not influence the equations of motion, while
fromthe wiggly VOS model we have just discussed it is clearthat
these effects must, in general, be present. Here, weimplement the
full description of wiggly strings in CM-Bact. Using the equation
of state for wiggly strings (70)
we first rewrite the stress-energy tensor (4) as
Tµν(y) =µ0√−g
∫d2σ(
�µẋµẋν − x′µx′ν
�µ
)δ(4)(y − x(σ)) ,
(87)
where µ is the amount of wiggles, which is now dynami-cal,
satisfying equation (80). The size of string segmentsis set to be
equal2 to the correlation length ξ0τ . Wealso change the VOS
equations of motion in CMBactto the full system (78)-(80) and
implement the stress-energy components (42)-(44). With these
modifications,we achieve a full treatment of wiggly cosmic string
net-works in CMBact.
In figure 2 we show our results for network evolutionand in
figure 3 the corresponding CMB anisotropies com-puted in our
modified version of CMBact. In both figureswe also show the
corresponding results of the originalCMBact code [43] for
comparison. Regarding figure 2,we note thet the accuracy of CMBact
is comparativelyworse at low redshifts; this explains why the
effects ofthe matter to acceleration transition seemingly
becomevisible around reshifts of a few, while the onset of
accel-eration occurs below z = 1. This point is not crucial forour
analysis, since our goal is to make a comparaive studyof the
effects of the additional degrees of freedom on thestrings.
Moreover, these low redshifts have a relativelysmall effect on the
overall CMB signal. Nevertheless, thisis an issue which should be
adressed if this code is to beused for quantitative comparisons
with current or forth-coming CMB data.
We have chosen to vary parameter D, keeping η fixed,which allows
us to cover a wide range of µ values. Fix-ing D and increasing η is
equivalent to decreasing theamount of wiggles and an effective
change of c, which isalready covered from our variation of D with
fixed η. Itis also important to note that in order to have an
attrac-tor scaling solution when a ∝ τn the following conditionmust
be satisfied
η >D(1− 1/µ2
)1− 1/√µ
. (88)
Physically, this means that in order to achieve a scal-ing
solution, small scale structure should be able to loseenergy
(controlled by parameter η) faster than it receivesthe energy from
large scales (controlled by parameterD). When the condition (88) is
violated, energy accumu-lates at small scales and there is no
stable scaling regime
2 For an even more realistic model we could consider the
stringssegments to have a range of sizes and speeds picked from
appro-priate distributions as in [34], but here we want to focus on
theeffects of string wiggles only and compare to the results of
theoriginal CMBact code, which also takes all segments to have
thesame size and speed.
-
10
z
υ
zz
μ
η=3, c=0.23, 10*D=
5.50 6.00 6.50 6.75 7.00 7.25 7.50 7.75 8.00
α=2
0.00
Lc/τ
Figure 2. Evolution of the rms velocity υ, comoving
characteristic length Lc and amount of wiggles µ as a function of
redshiftz for wiggly cosmic string networks with different values
of the parameter D, obtained by a modified version of the
CMBactcode [43]. The horizontal dashed red and blue lines
correspond to the usual (without wiggles; µ = 1) scaling regimes
forradiation (red shaded area) and matter domination (blue shaded
area) epochs respectively. Note that the horizontal (redshift)axis
is depicted in a linear scale in the redshift range 0 < z < 1
and in a logarithmic scale for z > 1.
for these wiggly cosmic strings. In practice, the condi-tion
(88) is used as a guide for estimating the range ofvariation of
D.
Figure 3 shows how the full treatment of wiggly cos-mic string
networks affects the prediction for the string-induced CMB
anisotropy. Note that the CMB contribu-tion is generally smaller
than for ordinary cosmic strings(i.e. without wiggles, µ = 1). This
is mainly due toa reduction in the rms string velocity υ (see
figure 2)when the amount of wiggles µ increases. In view of
theobserved changes to the usual CMB predictions for cos-mic
strings, we argue that to achieve accurate resultsfor wiggly cosmic
strings, one should study them in theframework of the complete
wiggly model (78)-(80) andthe modified version of CMBact developed
here. Thisgenerally leads to a weakening of the CMB-derived
con-straint on the string tension µ0 (but note that there is
also a region in parameter space – for large D – wherethe
correlation length can actually become smaller thanfor ordinary
strings, see figure 2).
Note that both the evolution and CMB results fromour wiggly VOS
model are somewhat closer in compari-son to results from
Abelian-Higgs simulations (and sim-ilarly ordinary VOS results are
closer to Nambu-Gotosimulations). It is then tempting to speculate
that wig-gles play a dynamical role analogous to that of the
aver-aged field fluctuations that appear in Abelian-Higgs
fieldtheory simulations (as opposed to effective
Nambu-Gotosimulations). This hypothesis may be investigated
bydirect comparisons of Abelian-Higgs and Goto-Nambusimulations
with suitably high resolutions and dynamicranges.
To end this section, let us return to the wigglymodel but this
time without referring to the specific La-
-
11
η=3, c=0.23, 10*D=
5.50 6.00 6.50 6.75 7.00 7.25 7.50 7.75 8.00
α=2
0.00
Figure 3. CMB anisotropy for wiggly cosmic string networks
obtained by a modified version of the CMBact code [43]. Thepanels
show scalar, vector and tensor contributions (top to bottom) of the
BB, TT , TE and EE modes (left to right). (Notethere is no BB
contribution from scalar modes.) These have been computed for
different values of D with fixed η. The CMBactresult from [43] with
α = 2 is shown by the black dashed line for comparison.
grangian (77). We wish to study the scaling regime forwiggly
strings but leaving the amount of wiggles µ as afree parameter that
we can tune. Instead of varying pa-rameter D, as it was done above,
we can vary µ. Thisapproach does not require an assumption on the
energytransfer function (86); we only need to define how energyloss
depends on the amount of wiggles (85). Let us nowestimate how the
rms velocity υ and comoving charac-teristic length Lc are related
to the parameters c, η andµ in the scaling regime. We insert the
scaling solutionLc = ετ , υ =const to equations (72), (73) to
obtain thealgebraic equations
ε(
2− n[1 + υ2 − 1−υ
2
µ2
])= cfa(µ)υ√µ , (89)
k(υ)εµ5/2
= nυ(
1 + 1µ2
), (90)
where we have included energy loss function fa(µ) givenby
(85).
Despite the reduction of the equations of motion toalgebraic
equations (89) and (90) in the scaling regime,it is still not
possible to solve them analytically, mainlydue to the complicated
form of the momentum parame-ter (30). To study how the amount of
wiggles affects themacroscopic parameters υ (rms string velocities)
and ε(comoving correlation length in units of conformal time)
in the scaling regime we solve the system (89)-(90) nu-merically
for different expansion rates n. The results areshown in figure 4.
It is seen that the rms velocity υ, asanticipated from the
restriction (74), decreases with thegrowth of the amount of wiggles
µ. This is also in agree-ment with our results for the rms velocity
evolution (seefigure 2) in the dynamical wiggly model for a
realistic ex-pansion history. The situation for ε is more
interesting.The correlation length does not increase
monotonicallywith the amount of wiggles but has a maximum aroundµ =
1.5− 1.9. This is also in agreement to our full treat-ment in
figure 2 where we modelled string wiggles byvarying parameter D and
took a realistic expansion his-tory.
Since we have computed the velocity υ and correlationlength ξ0τ
=
√µετ in the scaling regime, we can use
equations (68) and (69) to estimate how the contributionto the
CMB anisotropy from cosmic strings depends onthe amount of string
wiggles. For wiggly cosmic stringsthe angular power spectrum Cl has
the following depen-dence (which coincides with the result in
[55])
Cl ∼ (Gµ0)2µ4υ4 + µ2υ2(1− υ2) + (1− υ2)2
µ2ξ0(1− v2), (91)
where scalar, vector and tensor components depend on
-
12
Figure 4. Dependence of the scaling values of the rms velocity,
v, and the comoving correlation length divided by conformaltime, ε,
on the amount of wiggles µ for different expansion rates n.
Figure 5. Comparison between the behaviour of the string-induced
angular power spectrum Cl for different amount ofwiggles in our
analytic approximation (solid lines) and thenumerical computation
using our modified CMBact code (cir-cles). The dependence on µ has
been estimated analyticallyusing equations (69), (68) together with
the equations for thescaling regime of the network (89)-(90). Using
the value ofµ in the matter domination era and Cl’s for scalar
(green),vector (blue) and tensor (red) components at l = 700
(wherethe sum peaks), we have obtained the Cl−µ dependence fromthe
CMBact code.
string parameters in the same way.We can now compare the
dependence in equation (91)
with our numerical results using our modified CMBactcode. By
choosing the µ value for the matter dominationera and looking at
the peak (l ≈ 700) of the sum of thescalar, vector and tensor
contributions we plot them incomparison to the analytic estimate
from (68) and (69).This comparison is shown in figure III. For our
approx-imate estimate it is seen that after a fast decrease of
Cl’s with growing amount of wiggles µ, the value of Clreaches a
plateau. A similar behaviour is seen for vec-tor, tensor and scalar
components obtained from the fulltreatment using our modified
CMBact code, even thoughthe agreement is somewhat weaker for the
scalar contri-bution. These results reaffirm the approximations
usedto estimate the analytic dependence of Cl on the stringnetwork
characteristics.
IV. SUPERCONDUCTING MODEL(CHIRAL CASE)
Another special case of current-carrying cosmic stringsof
notable physical interest is the case of superconductingcosmic
strings. This type of strings has been studiedthoroughly in the
framework of field-theory [35, 48–51,60, 61]. In all these cases
the stress-energy tensor on thestring worldsheet has the following
form:
T ab =
(A+B −CC A−B
). (92)
where A arises from the field responsible for the stringcore
formation, while B and C represent additional con-tributions due to
coupling with external fields (dynamicsof currents). The
stress-energy tensor (92) is written for
the worldsheet metric ηab =
(1 00 −1
)on a 4-dimensional
Minskowski spacetime background with � = 1.Consider now the
two-dimensional stress-energy tensor
for the action (3), which reads
T ab =
(µ0Ũ −µ0 Φ�µ0�Φ µ0T̃
). (93)
There is an obvious correspondence between the stress-energy
tensors (92) and (93); they are in agreement if we
-
13
demand the chiral condition [25, 26]
κ→ 0 , (94)
which also means
∆→ 0 ; (95)
here κ and ∆ are defined by Eq. (2). These imply that
Ũ = 1 + Φ, T̃ = 1 − Φ. In Minkowski space (� = 1)we see that A
= µ0, B = µ0Φ and C = µ0Φ, so wehave the condition B = C. In order
to avoid this sit-uation and be able to reproduce a stress-energy
tensorof the form (92) within the Nambu-Goto approximation,we need
to use at least two scalar fields. It has alreadybeen demonstrated
that adding any number of additionalfields (11) together with the
definitions (12)-(14) keepsthe evolution equations (8)-(9)
unchanged, replacing thescalar field equation (10) by the set of
equations (15).In effect, introducing additional fields makes C and
Bdifferent in Minkowski space.
Indeed, when we add extra scalar fields we obtain astress-energy
tensor in the form of (92) with the corre-
spondence3 A = µ0, B = 2µ0γ00(∂f∂κ −
∂f∂∆
)∑i ϕ̇i
2 =
µ0Ψ and C = µ0Φ, where Φ is given by equation (14)and we have
assumed a Minkowski background. Thesecorrespond to
Û = 1 + 〈Ψ〉 , T̂ = 1− 〈Ψ〉 , Q = 〈Φ〉 . (96)
Thus, this multiple worldsheet field approach pro-vides enough
flexibility to reproduce the field-theoreticalstress-energy tensor
variables in (92) within the Nambu-Goto approximation.
Let us now consider the equations of motion for chi-ral
currents. We will apply our averaging procedure tothe system of
equations (15) for the currents, similarly towhat we already did
for first two equations (31) and (32)for the correlation length and
string velocity. First of all,we note that in order to have the
appropriate Nambu-Goto limit for the action (3) when ϕ = 0 we need
to have
f(κ,∆) −−−→κ→0
1 and additionally ∂f(κ,∆)∂κ −−−→κ→0 const (as
well as ∂f(κ,∆)∂∆ −−−→∆→0 const). These conditions allow usto
make simplifications, similar to what was done in refer-ence [26],
and consider the case of conserved microscopiccharges for each
field
�ϕ̇i = φi = const , (97)
ϕ′i = ψi = const , (98)
which leads to the additional condition �′ = 0. Further-more, in
this case we can define Ψ and Φ as
Ψ =
(∂f
∂κ+∂f
∂∆
)∑i
φ2ia2x′2
, (99)
3 Here we used an assumption that all multipliers ∂f∂∆i
are equal
as well as all ∂f∂κi
are equal (94).
Φ =
(∂f
∂κ+∂f
∂∆
)∑i
φiψia2x′2
, (100)
and (94) gives us ∑i
φ2i =∑i
ψ2i . (101)
Expressions (99) and (100) tell us that if we use thecondition
of conserved microscopic charges (97)-(98) wehave two variables Ψ
and Φ which evolve in the same wayand differ only by a
multiplicative constant β:
Ψβ = Φ, (102)
where β =∑i φiψi∑i φ
2i
. Together with (101), this implies
that 0 < β < 1.
By direct differentiation of equation (99) we obtain
thefollowing evolution equations for the field Ψ (clearly, thesame
equations are also obeyed by Φ)
Ψ̇ + 2 ȧaΨ = 2Ψẋ·x′′x′ 2 , (103)
Ψ′ + 2Ψx′·x′′x′ 2 = 0. (104)
Following the approach of [26], we average the equa-tions of
motion (103), (104) and substitute the equationof state (96) into
equations (31) and (32). This leads tothe VOS model for
superconducting chiral strings, tak-ing into account energy and
charge losses (for details onthese loss terms see [26])
dLcdτ =
ȧaLc
υ2+Q1+Q + υ
(Qsβ
(1+Q)3/2+ c2
), (105)
dυdτ =
1−υ21+Q
[k(υ)
Lc√
1+Q
(1−Q(1 + 2sβk(υ) )
)− 2 ȧaυ
],(106)
dQdτ = 2Q
(k(υ)υ
Lc√
1+Q− ȧa
)+
cυ(1−√
1+Q)Lc
√1 +Q .(107)
We have used the assumption〈
Ψ′
�(1+Ψ)
〉= −s υRc
2Q1+Q [26]
and that the correlation and characteristic lengths arerelated
by ξc = Lc
√1 +Q.
Therefore, our general analysis of chiral current depen-dence in
the action (3) including the addition of extraworldsheet fields has
not introduced significant changesin the macroscopic equations
describing superconductingchiral cosmic string networks, as
compared to the resultsin [26] (the only difference is that the
constant s hasnow been changed to βs). Note also that the final
resultdoes not depend explicitly on the precise form of the
La-grangian; the important physics can be encoded in theequations
of state of the strings, in agreement with ourprevious
discussion.
The evolution of string networks described by equa-tions
(105)-(107) was carefully studied in [26]. It was
-
14
shown that these networks have generalized scaling solu-tions4
only if the following relation is satisfied
n =2k(υ)− cW̃c+ k(υ)
, (108)
where W̃ =√
1+Qs−11+Q−1s
, with Qs a constant corresponding
to the scaling value of the function Q. As we can see
fromequation (108), the expansion rate n for scaling solutions(with
constant charge) cannot be larger than n ≤ 2. Themaximal value of n
is reached when c = 0, while for c =0.23 (which we use here) we
have the condition n
-
15
s=1.0, c=0.23, Q0=
0.00 1.00 1.20 1.21 1.22
1.23 1.24 1.25 1.26
0.00
1.27 1.30
10*Q0=4.42+10-2x
z
υ
zz
Q
Lc/τ
Figure 7. Evolution of the rms velocity υ, comoving
characteristic length Lc and charge Q depending on redshift z
forsuperconducting (chiral) cosmic string networks with different
initial conditions Q0 for the string charge, obtained by a
modifiedversion of the CMBact code [43]. The horizontal dashed red
and blue lines correspond to the usual (without charge, Q =
0)scaling regimes for radiation (red shaded area) and matter
domination (blue shaded area) eras respectively. Note that
thehorizontal (redshift) axis is depicted in a linear scale in the
redshift range 0 < z < 1 and in a logarithmic scale for z
> 1.
choose from our numerical results in figure 8. However,it is
clear that the analytic approach and numerical com-putation are in
qualitative agreement. In particular theangular power spectrum Cl
decreases as we increase thecharge Q on the string.
V. CONCLUSIONS
There are many well-motivated scenarios in Early Uni-verse
Physics that can leave behind relic defects in theform of cosmic
strings. These relics can be utilised as“fossils” for cosmological
research, helping us to obtain abetter understanding of the
physical processes that tookplace in the early universe. By
developing an accurate de-scription of the evolution of cosmic
string networks andusing it to calculate quantitative predictions
of string-
induced observational signals, we can obtain strong con-straints
on theoretical models leading to a better under-standing of Early
Universe Physics. Here, we presented adetailed study of the
evolution of cosmic strings with cur-rents and demonstrated how the
presence of worldsheetcurrents affects the predictions for the CMB
anisotropyproduced by cosmic string networks.
In section II we considered the action (3) describingstrings
with an arbitrary dependence on worldsheet cur-rents. We have
described how to average the microscopicequations of motion for
this model to obtain macroscopicevolution equations (without energy
loss) for the stringnetwork (72)-(73). These describe the time
evolutionof the rms string velocity υ and characteristic lengthL,
and depend only on three parameters Û , T̂ and Qdefining the
string equation of state. These same pa-rameters, together with the
network quantities L and
-
16
s=1.0, c=0.23, Q0=
0.00 1.00 1.20 1.21 1.22
1.23 1.24 1.25 1.26
0.00
1.27 1.30
10*Q0=4.42+10-2x
Figure 8. CMB anisotropy results for superconducting (chiral)
cosmic sting networks obtained by our modified version ofCMBact
[43]. The panels show the scalar, vector and tensor contributions
(top to bottom) to the BB, TT , TE and EEpowerspectra (left to
right). The calculations are done for different initial conditions
of the charge Q0.
Figure 9. Scaling values of rms velocity, v, and comoving
correlation length divided by conformal time, ε, depending on
thecharge Q, for different expansion rates n.
υ, appear directly in the string stress-energy tensor (4)which
seeds the string-induced CMB anisotropy. Thisprovides a direct
connection between modelling stringevolution and computing CMB
anisotropies from cosmicstring networks, which has allowed us to
obtain simpleanalytic estimates for the dependence of the string
an-
gular power spectrum Cl on macroscopic network pa-rameters
(68)-(69). For a more complete semi-analytictreatment of the CMB
anisotropy for strings with cur-rents, we have adapted the
methodology of [34] and haveprovided coefficients for the relevant
integrals in the Ap-pendix. In sections III, IV we considered two
specific
-
17
Figure 10. Behaviour of Cl for different string charge Q,
ob-tained from the analytical approximation.
cases of strings with currents: wiggly and superconduct-ing
cosmic strings respectively. In each case we computedthe CMB signal
numerically using appropriately modifiedversions of CMBact.
For wiggly string networks (section III) we studied thespecific
case when the parameter κ in (2) only carriesa time dependence, κ =
κ(τ). We studied network dy-namics using an effective action for
wiggly cosmic strings,and introduced the averaged macroscopic
equations CM-Bact, allowing us to compute CMB anisotropies
fromthese strings. CMBact has already built in the optionto study
wiggly strings, but this was done through a sin-gle constant
parameter. Here, for the first time, we wereable to take into
account the time evolution of wigglesand their influence on the
macroscopic equations of mo-tion for the string network. This full
treatment broughtimportant changes in modelling wiggly cosmic
string net-works. From figure (3) we see that wiggly strings
canproduce a lower signal in CMB anisotropy than ordinarystrings
(when the other parameters are fixed), which hadnot been
appreciated before our work. We have also com-pared our analytic
estimation (91) to our numerical re-sults from the CMBact code. The
comparison shows thatthe main trend for Cl (decreasing of Cl as µ
increases,for multiple moments 1
-
18
by an FCT Research Professorship, contract
referenceIF/00064/2012, funded by FCT/MCTES (Portugal) andPOPH/FSE
(EC). We would like to thank Patrick Peter,Tom Charnock, José
Pedro Vieira and colleagues fromP.S. for fruitful discussions and
help.
APPENDIX
As shown in [55] the integral (45) can be expanded inthe
following way
〈ΘI(k, τ1)Θ
J(k, τ2)〉
=f(τ1, τ2, ξ0)µ
20
k2(1− υ2)×
6∑i=1
AIJi [Ii(x−, ρ)− Ii(x+, ρ)] ,(114)
where I, J correspond to the “00”, scalar, vector andtensor
components of the stress-energy tensor and theform of the six
integrals Ii are as given in [55].
The coefficients AIJi , together with the full expressionsfor
the analytic equal time correlators BIJ , are listedbelow (where,
in this Appendix, we use the definitionsρ = k|τ1 − τ2|υ, x1,2 =
kξ0τ1,2, x± = (x1 ± x2)/2):
A00−001 = 2Û2,
A00−00i = 0,
(i = 2, .., 6)
A00−S1 = Û(T̂ + (2Û − T̂ )v2),
A00−S2 = −3Û(T̂ (1− v2) + Ûv2
)A00−S3 = 0
A00−S4 = −3Û2v2
A00−S5 = 3Û2v2
A00−S6 = 0
AS−S1 =−27Û2v4 + ρ2(T̂ + (2Û − T̂ )v2)2
2ρ2
AS−S2 =3(
9Û2v4 + ρ2(T̂ 2(1− v2)2 − Û2v4
))2ρ2
AS−S3 = −9
2
((Ûv2 + T̂ (1− v2)
)2− 4v2Q2
)
AS−S4 =3Ûv2
(9Ûv2 − ρ2(T̂ (1− v2) + 2Ûv2)
)ρ2
AS−S5 = −3Ûv2
(9Ûv2 − ρ2(T̂ (1− v2) + 2Ûv2)
)ρ2
AS−S6 = 9v2(Û2v2 + T̂ Û(1− v2)− 2Q2)
)AV−V1 =
3Û2v4 + ρ2v2Q2
ρ2
AV−V2 = −3Û2v4
ρ2
AV−V3 =(Ûv2 + T̂ (1− v2)
)2− 4v2Q2
AV−V4 = −(6/ρ2 − 1
)Û2v4 − v2Q2
AV−V5 =(6/ρ2 − 1
)Û2v4 + v2Q2
AV−V6 = −2v2(Û2v2 + T̂ Û(1− v2)− 2Q2
)AT−T1 =
ρ2T̂ 2(1− v2
)2 − 3Û2v44ρ2
AT−T2 =3Û2v4 − ρ2
(T̂ 2(1− v2
)2 − Û2v4)4ρ2
AT−T3 = −1
4
(Ûv2 + T̂ (1− v2)
)2+ v2Q2
AT−T4 =v2(
3Û2v2 + ρ2(T̂ Û(1− v2) + 2Q2
))2ρ2
AT−T5 = −v2(
3Û2v2 + ρ2(T̂ Û(1− v2) + 2Q2
))2ρ2
AT−T6 =v2
2
(Û2v2 + T̂ Û(1− v2)− 2Q2
)
B00−00(τ) = 2Û2(cos(x)− 1 + xSi(x)),
B00−S = 12x
(Û(2T̂ + v2(Û − 2T̂ ))(x cos(x) + 3 sin(x) + x(xSi(x)− 4))
),
BS−S = 116x3
([8T̂ Ûv2(1− v2)(x2 − 18) + 8T̂ 2(1− v2)2(x2 − 18) + Û2v4(11x2
− 54) + 288v2Q2
]x cos(x) +
+ x3[32(
3v2Q2 − Û2v4 − T̂ Ûv2(1− v2)− T̂ 2(1− v2)2))
+(
11Û2v4 + 8T̂ Ûv2(1− v2) + 8T̂ 2(1− v2)2)xSi(x)
]−
http://polozov.cf/
-
19
− 3 sin(x)[8T̂ Ûv2(1− v2)(x2 − 6) + 8T̂ 2(1− v2)2(x2 − 6)−
Û2v4(18 + z2) + 96v2Q2
],
BV−V = 124x3
(3x cos(x)
[16T̂ (1− v2)(T̂ − (T̂ − Û)v2) + Û2v4(6 + z2) + 4v2(x2 −
8)Q2
]+
x3[16T̂ (1− v2)(T̂ − (T̂ − Û)v2)− 32v2Q2 + 3v2x(Û2v2 +
4Q2)Si(x)
]−
3 sin(x)[16T̂ (1− v2)(T̂ − (T̂ − Û)v2) + Û2v4(6− x2) + 4v2(x2
− 8)Q2
]),
BT−T = 196x3
(3x cos(x)
[(3Û2v4 + 8T̂ Ûv2(1− v2) + 8T̂ 2(1− v2)2)(x2 − 2) + 16v2(2 +
x2)Q2
]+
+ x3[64T̂ (1− v2)(v2(T̂ − Û)− T̂ )− 64v2Q2 + 3x(3Û2v4 + 8T̂
Ûv2(1− v2) + 8T̂ 2(1− v2)2 + 16v2Q2)Si(x)
]+
+ 3 sin(x)[Û2v4(6− 5x2) + 8T̂ Ûv2(1− v2)(2 + x2) + 8T̂ 2(1−
v2)2(2 + x2) + 16v2(x2 − 2)Q2
]).
[1] T. W. B. Kibble, J.Phys. A 9, 1387 (1976).[2] A. Vilenkin
and E. P. S. Shellard, Cosmic Strings and
Other Topological Defects (Cambridge University Press,Cambridge,
2000).
[3] M. B. Hindmarsh and T. W. B. Kibble, Rept.Prog.Phys.58, 477
(1995), arXiv:hep-ph/9411342 [astro-ph.CO].
[4] E. J. Copeland, R. C. Myers, and J. Polchinski, JHEP2004,
013 (2004), arXiv:hep-th/0312067 [hep-th].
[5] S. Sarangi and S.-H. H. Tye, Phys.Lett.B 536, 185(2002),
arXiv:hep-th/0204074 [hep-th].
[6] H. Firouzjahi and S.-H. H. Tye, JCAP 0503, 009
(2005),arXiv:hep-th/0501099 [astro-ph.CO].
[7] N. T. Jones, H. Stoica, and S.-H. H. Tye, Phys.Lett. B,6
(2003), arXiv:hep-th/0303269 [astro-ph.CO].
[8] R. Jeannerot, J. Rocher, and M. Sakellariadou,Phys.Rev.D 68,
103514 (2003), arXiv:hep-ph/0308134[hep-th].
[9] Y. Cui, S. Martin, D. E. Morrissey, and J. Wells,Phys.Rev.D
77, 043528 (2008), arXiv:arXiv:0709.0950v2[hep-th].
[10] R. Jeannerot and M. Postma, JHEP 0412, 043
(2004),arXiv:hep-ph/0411260 [astro-ph.CO].
[11] A. Achucarro, A. Celi, M. Esole, J. Van den Bergh, andV. P.
A., JHEP 0601, 102 (2006), arXiv:hep-th/0511001[astro-ph.CO].
[12] A.-C. Davis and M. Majumdar, Phys.Lett. B, 257
(1999),arXiv:hep-ph/9904392 [astro-ph.CO].
[13] E. Allys, JCAP 1604, 009 (2016),arXiv:arXiv:1505.07888
[astro-ph.CO].
[14] M. Koehn and M. Trodden, Phys.Lett. B, 498
(2016),arXiv:arXiv:1512.09138 [astro-ph.CO].
[15] G. Ballesteros, R. J., A. Ringwald, and C. Tamarit,JCAP
1708, 001 (2017), arXiv:1610.01639 [astro-ph.CO].
[16] G. Lazarides, I. N. R. Peddie, and A. Vamvasakis,Phys.Rev.
D, 043518 (2008), arXiv:arXiv:0804.3661[astro-ph.CO].
[17] D. F. Chernoff and S.-H. H. Tye, Int.J.Mod.Phys.
D24,1530010 (2015), arXiv:arXiv:1412.0579 [astro-ph.CO].
[18] J. J. Blanco-Pillado, K. D. Olum, and B. Shlaer,
Phys.Rev. D, 083514 (2011),
arXiv:arXiv:1101.5173[astro-ph.CO].
[19] M. Hindmarsh, J. Lizarraga, J. Urrestilla, D. Dave-rio, and
M. Kunz, Phys.Rev. D96, 023525 (2017),arXiv:1703.06696
[hep-ph].
[20] J. Lizarraga and J. Urrestilla, JCAP 1604, 053
(2016),arXiv:arXiv:1602.08014 [astro-ph.CO].
[21] C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev.D54,
2535 (1996), arXiv:hep-ph/9602271 [hep-ph].
[22] C. J. A. P. Martins and E. P. S. Shellard, Phys. Rev.D65,
043514 (2002), arXiv:hep-ph/0003298 [hep-ph].
[23] C. Martins, Astrophys.Space Sci. 261, 311 (1998).[24] C.
Martins, E. Shellard, and J. Vieira, Phys.Rev. D90,
043518 (2014), arXiv:1405.7722 [hep-ph].[25] C. Martins and E.
Shellard, Phys.Rev. D, 7155 (1998),
arXiv:hep-ph/9804378 [astro-ph.CO].[26] M. F. Oliveira, A.
Avgoustidis, and C. J. A. P. Martins,
Phys.Rev. D85, 083515 (2012), arXiv:1201.5064 [hep-ph].
[27] A. Avgoustidis and E. P. S. Shellard, Phys.Rev. D78,103510
(2008), arXiv:0705.3395 [hep-ph].
[28] A. Avgoustidis and E. P. S. Shellard, Phys.Rev. D71,123513
(2005), arXiv:hep-ph/0410349 [hep-ph].
[29] E. J. Copeland and P. M. Saffin, JHEP 11, 023
(2005),arXiv:hep-th/0505110 [hep-th].
[30] S. H. H. Tye, I. Wasserman, and M. Wyman,Phys. Rev. D71,
103508 (2005), [Erratum: Phys.Rev.D71,129906(2005)],
arXiv:astro-ph/0503506 [astro-ph].
[31] P. A. R. Ade et al. (Planck), Astron. Astrophys. 571,A25
(2014), arXiv:1303.5085 [astro-ph.CO].
[32] A. Lazanu and E. P. S. Shellard, JCAP 2015, 024
(2015),arXiv:1410.5046v3 [astro-ph.CO].
[33] J. Lizarraga, J. Urrestilla, D. Daverio, M. Hind-marsh, and
M. Kunz, JCAP 1610, 042 (2016),arXiv:arXiv:1609.03386
[astro-ph.CO].
[34] T. Charnock, A. Avgoustidis, E. Copeland, and M.
A.,Phys.Rev. D93, 123503 (2016), arXiv:1603.01275
[astro-ph.CO].
[35] E. Witten, Nuclear Physics B 249, 557 (1985).
http://iopscience.iop.org/article/10.1088/0305-4470/9/8/029/meta;jsessionid=8C536D2333738F7208E64D30D08B0BC6.c3.iopscience.cld.iop.orghttp://dx.doi.org/10.1088/0034-4885/58/5/001http://dx.doi.org/10.1088/0034-4885/58/5/001http://arxiv.org/abs/hep-ph/9411342http://dx.doi.org/10.1088/1126-6708/2004/06/013http://dx.doi.org/10.1088/1126-6708/2004/06/013http://arxiv.org/abs/hep-th/0312067http://dx.doi.org/10.1016/S0370-2693(02)01824-5http://dx.doi.org/10.1016/S0370-2693(02)01824-5http://arxiv.org/abs/hep-th/0204074http://dx.doi.org/10.1088/1475-7516/2005/03/009http://arxiv.org/abs/hep-th/0501099http://dx.doi.org/10.1016/S0370-2693(03)00592-6http://dx.doi.org/10.1016/S0370-2693(03)00592-6http://arxiv.org/abs/hep-th/0303269http://dx.doi.org/10.1103/PhysRevD.68.103514http://arxiv.org/abs/hep-ph/0308134http://arxiv.org/abs/hep-ph/0308134http://dx.doi.org/10.1103/PhysRevD.77.043528http://arxiv.org/abs/arXiv:0709.0950v2http://arxiv.org/abs/arXiv:0709.0950v2http://dx.doi.org/10.1088/1126-6708/2004/12/043http://arxiv.org/abs/hep-ph/0411260http://dx.doi.org/
10.1088/1126-6708/2006/01/102http://arxiv.org/abs/hep-th/0511001http://arxiv.org/abs/hep-th/0511001http://dx.doi.org/10.1016/S0370-2693(99)00801-1http://arxiv.org/abs/hep-ph/9904392http://dx.doi.org/10.1088/1475-7516/2016/04/009http://arxiv.org/abs/arXiv:1505.07888http://dx.doi.org/10.1016/j.physletb.2016.02.067http://arxiv.org/abs/arXiv:1512.09138http://dx.doi.org/10.1088/1475-7516/2017/08/001http://arxiv.org/abs/1610.01639http://arxiv.org/abs/1610.01639http://dx.doi.org/10.1103/PhysRevD.78.043518http://arxiv.org/abs/arXiv:0804.3661http://arxiv.org/abs/arXiv:0804.3661http://dx.doi.org/10.1142/S0218271815300104http://dx.doi.org/10.1142/S0218271815300104http://arxiv.org/abs/arXiv:1412.0579http://dx.doi.org/10.1103/PhysRevD.83.083514http://arxiv.org/abs/arXiv:1101.5173http://arxiv.org/abs/arXiv:1101.5173http://dx.doi.org/10.1103/PhysRevD.96.023525http://arxiv.org/abs/1703.06696http://dx.doi.org/10.1088/1475-7516/2016/04/053http://arxiv.org/abs/arXiv:1602.08014http://dx.doi.org/10.1103/PhysRevD.54.2535http://dx.doi.org/10.1103/PhysRevD.54.2535http://arxiv.org/abs/hep-ph/9602271http://dx.doi.org/10.1103/PhysRevD.65.043514http://dx.doi.org/10.1103/PhysRevD.65.043514http://arxiv.org/abs/hep-ph/0003298http://dx.doi.org/10.1023/A:1002040531185http://dx.doi.org/10.1103/PhysRevD.90.043518http://dx.doi.org/10.1103/PhysRevD.90.043518http://arxiv.org/abs/1405.7722http://dx.doi.org/10.1103/PhysRevD.57.7155http://arxiv.org/abs/hep-ph/9804378http://dx.doi.org/10.1103/PhysRevD.85.083515http://arxiv.org/abs/1201.5064http://arxiv.org/abs/1201.5064http://dx.doi.org/10.1103/PhysRevD.78.103510http://dx.doi.org/10.1103/PhysRevD.78.103510http://arxiv.org/abs/0705.3395http://dx.doi.org/10.1103/PhysRevD.71.123513http://dx.doi.org/10.1103/PhysRevD.71.123513http://arxiv.org/abs/hep-ph/0410349http://dx.doi.org/10.1088/1126-6708/2005/11/023http://arxiv.org/abs/hep-th/0505110http://dx.doi.org/10.1103/PhysRevD.71.103508,
10.1103/PhysRevD.71.129906http://arxiv.org/abs/astro-ph/0503506http://arxiv.org/abs/astro-ph/0503506http://dx.doi.org/10.1051/0004-6361/201321621http://dx.doi.org/10.1051/0004-6361/201321621http://arxiv.org/abs/1303.5085http://dx.doi.org/10.1088/1475-7516/2015/02/024http://arxiv.org/abs/1410.5046v3http://dx.doi.org/
10.1088/1475-7516/2016/10/042http://arxiv.org/abs/arXiv:1609.03386http://dx.doi.org/10.1103/PhysRevD.93.123503http://arxiv.org/abs/1603.01275http://arxiv.org/abs/1603.01275http://dx.doi.org/10.1016/0550-3213(85)90022-7
-
20
[36] S. C. Davis, A. C. Davis, and M. Trodden, Phys.Lett.B, 257
(1997), arXiv:hep-ph/9702360 [astro-ph.CO].
[37] S. C. Davis, A. C. Davis, and M. Trodden, Phys.Rev.D, 5184
(1998), arXiv:hep-ph/9711313 [astro-ph.CO].
[38] A. Everett, Phys.Rev.Lett. 61, 1807 (1988).[39] M.
Hindmarsh, K. Rummukainen, and D. J.
Weir, Phys.Rev.Lett. 117, 251601 (2016),arXiv:arXiv:1607.00764
[astro-ph.CO].
[40] B. Carter, Phys.Rev. D41, 3869 (1990).[41] B. Carter,
Phys.Rev.Lett. 74, 3098 (1995), arXiv:hep-
th/9411231v1 [hep-ph].[42] A. Vilenkin, Phys. Rev. D41, 3038
(1990).[43] L. Pogosian and T. Vachaspati, Phys.Rev. D60,
083504
(1999), arXiv:astro-ph/9903361 [astro-ph.CO].[44] B. Carter and
P. Peter, Phys.Rev. D, 1744 (1995),
arXiv:hep-ph/9411425 [hep-ph.CO].[45] B. Carter and P. Peter,
Phys.Lett. B, 41 (1999),
arXiv:hep-th/9905025 [hep-ph.CO].[46] N. Nielsen, Nucl.Phys.
B167, 249 (1980).[47] E. Babichev, P. Brax, C. Caprini, J. Martin,
and
D. Steer, JHEP 2009, 091 (2009),
arXiv:0809.2013[astro-ph.CO].
[48] C. Ringeval, Phys. Rev. D64, 123505 (2001),
arXiv:hep-ph/0106179 [hep-ph].
[49] M. Lilley, P. Peter, and X. Martin, Phys. Rev. D79,103514
(2009), arXiv:0903.4328 [hep-ph].
[50] C. Ringeval, Phys. Rev. D63, 063508 (2001),
arXiv:hep-ph/0007015 [hep-ph].
[51] M. Lilley, F. Di Marco, J. Martin, and P. Peter, Phys.Rev.
D82, 023510 (2010), arXiv:1003.4601 [hep-ph].
[52] E. Allys, Phys.Rev. D93, 105021
(2016),arXiv:arXiv:1512.02029 [astro-ph.CO].
[53] N. Turok, U. Pen, and U. Seljak, Phys.Rev. D, 023506(1998),
arXiv:astro-ph/9706250 [astro-ph.CO].
[54] A. Albrecht, R. A. Battye, and J. Robinson,Phys.Rev. D,
023508 (1999), arXiv:astro-ph/9711121[astro-ph.CO].
[55] A. Avgoustidis, E. Copeland, A. Moss, and D. Skliros,
Phys.Rev. D, 123513 (2012), arXiv:1209.2461 [astro-ph.CO].
[56] U. Pen, D. Spergel, and N. Turok, Phys.Rev. D,
692(1994).
[57] J. Vieira, C. Martins, and E. Shellard, Phys.Rev.
D94,099907 (2016), arXiv:1611.06103 [hep-ph].
[58] J. N. Moore, E. P. S. Shellard, and C. J. A. P.
Martins,Phys. Rev. D65, 023503 (2002),
arXiv:hep-ph/0107171[hep-ph].
[59] J. N. Moore, E. P. S. Shellard, and C. J. A. P.
Martins,Phys. Rev. D65, 023503 (2002),
arXiv:hep-ph/0107171[hep-ph].
[60] C. T. Hill and L. M. Widrow, Phys. Lett. B189,
17(1987).
[61] M. Hindmarsh, Phys. Lett. B200, 429 (1988).[62] T. Damour
and A. Vilenkin, Phys. Rev. Lett. 85, 3761
(2000), arXiv:gr-qc/0004075 [gr-qc].[63] T. Damour and A.
Vilenkin, Phys. Rev. D64, 064008
(2001), arXiv:gr-qc/0104026 [gr-qc].[64] T. Damour and A.
Vilenkin, Phys. Rev. D71, 063510
(2005), arXiv:hep-th/0410222 [hep-th].[65] P. Binetruy, A. Bohe,
T. Hertog, and D. A. Steer, Phys.
Rev. D80, 123510 (2009), arXiv:0907.4522 [hep-th].[66] P.
Binetruy, A. Bohe, T. Hertog, and D. A. Steer, Phys.
Rev. D82, 126007 (2010), arXiv:1009.2484 [hep-th].[67] E.
O’Callaghan, S. Chadburn, G. Geshnizjani, R. Gre-
gory, and I. Zavala, Phys. Rev. Lett. 105, 081602
(2010),arXiv:1003.4395 [hep-th].
[68] E. O’Callaghan and R. Gregory, JCAP 1103, 004
(2011),arXiv:1010.3942 [hep-th].
[69] D. G. Figueroa, M. Hindmarsh, and J.
Urrestilla,Phys.Rev.Lett. 110, 101302 (2013),
arXiv:1212.5458[astro-ph.CO].
[70] L. Sousa and P. P. Avelino, Phys.Rev. D88, 023516(2013),
arXiv:1304.2445 [hep-ph].
[71] S. Kuroyanagi, K. Takahashi, N. Yonemaru, andH. Kumamoto,
Phys.Rev. D95, 043531 (2017),arXiv:1604.00332 [hep-ph].
http://dx.doi.org/10.1016/S0370-2693(97)00642-4http://dx.doi.org/10.1016/S0370-2693(97)00642-4http://arxiv.org/abs/hep-ph/9702360http://dx.doi.org/10.1103/PhysRevD.57.5184http://dx.doi.org/10.1103/PhysRevD.57.5184http://arxiv.org/abs/hep-ph/9711313http://dx.doi.org/10.1103/PhysRevLett.61.1807http://dx.doi.org/10.1103/PhysRevLett.117.251601http://arxiv.org/abs/arXiv:1607.00764http://dx.doi.org/10.1103/PhysRevD.41.3869http://dx.doi.org/10.1103/PhysRevLett.74.3098http://arxiv.org/abs/hep-th/9411231v1http://arxiv.org/abs/hep-th/9411231v1http://dx.doi.org/10.1103/PhysRevD.41.3038http://dx.doi.org/10.1103/PhysRevD.60.083504http://dx.doi.org/10.1103/PhysRevD.60.083504http://arxiv.org/abs/astro-ph/9903361http://dx.doi.org/10.1103/PhysRevD.52.R1744http://arxiv.org/abs/hep-ph/9411425http://dx.doi.org/10.1016/S0370-2693(99)01070-9http://arxiv.org/abs/hep-th/9905025http://dx.doi.org/10.1016/0550-3213(80)90130-3http://dx.doi.org/
10.1088/1126-6708/2009/03/091http://arxiv.org/abs/0809.2013http://arxiv.org/abs/0809.2013http://dx.doi.org/10.1103/PhysRevD.64.123505http://arxiv.org/abs/hep-ph/0106179http://arxiv.org/abs/hep-ph/0106179http://dx.doi.org/10.1103/PhysRevD.79.103514http://dx.doi.org/10.1103/PhysRevD.79.103514http://arxiv.org/abs/0903.4328http://dx.doi.org/10.1103/PhysRevD.63.063508http://arxiv.org/abs/hep-ph/0007015http://arxiv.org/abs/hep-ph/0007015http://dx.doi.org/
10.1103/PhysRevD.82.023510http://dx.doi.org/
10.1103/PhysRevD.82.023510http://arxiv.org/abs/1003.4601http://dx.doi.org/10.1103/PhysRevD.93.105021http://arxiv.org/abs/arXiv:1512.02029http://dx.doi.org/10.1103/PhysRevD.58.023506http://dx.doi.org/10.1103/PhysRevD.58.023506http://arxiv.org/abs/astro-ph/9706250http://dx.doi.org/10.1103/PhysRevD.59.023508http://arxiv.org/abs/astro-ph/9711121http://arxiv.org/abs/astro-ph/9711121http://dx.doi.org/10.1103/PhysRevD.86.123513http://arxiv.org/abs/1209.2461http://arxiv.org/abs/1209.2461http://dx.doi.org/10.1103/PhysRevD.49.692http://dx.doi.org/10.1103/PhysRevD.49.692http://dx.doi.org/10.1103/PhysRevD.94.096005http://dx.doi.org/10.1103/PhysRevD.94.096005http://arxiv.org/abs/1611.06103http://dx.doi.org/10.1103/PhysRevD.65.023503http://arxiv.org/abs/hep-ph/0107171http://arxiv.org/abs/hep-ph/0107171http://dx.doi.org/10.1103/PhysRevD.65.023503http://arxiv.org/abs/hep-ph/0107171http://arxiv.org/abs/hep-ph/0107171http://dx.doi.org/10.1016/0370-2693(87)91262-7http://dx.doi.org/10.1016/0370-2693(87)91262-7http://dx.doi.org/10.1016/0370-2693(88)90147-5http://dx.doi.org/10.1103/PhysRevLett.85.3761http://dx.doi.org/10.1103/PhysRevLett.85.3761http://arxiv.org/abs/gr-qc/0004075http://dx.doi.org/10.1103/PhysRevD.64.064008http://dx.doi.org/10.1103/PhysRevD.64.064008http://arxiv.org/abs/gr-qc/0104026http://dx.doi.org/10.1103/PhysRevD.71.063510http://dx.doi.org/10.1103/PhysRevD.71.063510http://arxiv.org/abs/hep-th/0410222http://dx.doi.org/
10.1103/PhysRevD.80.123510http://dx.doi.org/
10.1103/PhysRevD.80.123510http://arxiv.org/abs/0907.4522http://dx.doi.org/
10.1103/PhysRevD.82.126007http://dx.doi.org/
10.1103/PhysRevD.82.126007http://arxiv.org/abs/1009.2484http://dx.doi.org/10.1103/PhysRevLett.105.081602http://arxiv.org/abs/1003.4395http://dx.doi.org/10.1088/1475-7516/2011/03/004http://arxiv.org/abs/1010.3942http://dx.doi.org/10.1103/PhysRevLett.110.101302http://arxiv.org/abs/1212.5458http://arxiv.org/abs/1212.5458http://dx.doi.org/10.1103/PhysRevD.88.023516http://dx.doi.org/10.1103/PhysRevD.88.023516http://arxiv.org/abs/1304.2445http://dx.doi.org/10.1103/PhysRevD.95.043531http://arxiv.org/abs/1604.00332
Semi-analytic calculation of cosmic microwave background
anisotropies from wiggly and superconducting cosmic
stringsAbstractIntroductionString model with currentsWiggly
modelSuperconducting model (chiral
case)ConclusionsAcknowledgmentsAppendixReferences