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

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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

∗ Ivan.Rybak@astro.up.pt† Anastasios.Avgoustidis@nottingham.ac.uk‡ Carlos.Martins@astro.up.pt

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

]).

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Semi-analytic calculation of cosmic microwave background anisotropies from wiggly and superconducting cosmic stringsAbstractIntroductionString model with currentsWiggly modelSuperconducting model (chiral case)ConclusionsAcknowledgmentsAppendixReferences