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Physics Letters B 614 (2005) 1–6

www.elsevier.com/locate/physlet

Constraining cosmic superstrings with dilaton emission

E. Babicheva,b, M. Kachelrießa

a Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 München, Germanyb INR RAS, 60th October Anniversary Prospect 7a, 117312 Moscow, Russia

Received 21 February 2005; received in revised form 16 March 2005; accepted 21 March 2005

Available online 7 April 2005

Editor: G.F. Giudice

Abstract

Brane inflation predicts the production of cosmic superstrings with tension 10−12 Gµ 10−7. Superstring theory predictalso the existence of a dilaton. We show that the emission of dilatons imposes severe constraints on the allowed evocosmic superstring network. In particular, the detection of gravitational wave burst from cosmic superstrings by LIGOpossible if the typical length of string loops is much smaller than usually assumed. 2005 Elsevier B.V. All rights reserved.

PACS: 04.30.Db; 11.27.+d; 98.80.Cq

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1. Introduction

The inflationary paradigm has become one ofcorner stones of modern cosmology. While this padigm solves many puzzles of the “old” big batheory, no convincing theoretical framework existspresent for the inflaton field and its potential. In tbrane inflation scenario[1], the inflaton is identifiedas a mode controlling the separation of two branesflation ends when the branes collide and heat-upuniverse. An intriguing prediction of most scenariof brane inflation is the copiously production of comic superstrings[2].

E-mail address: mika@mppmu.mpg.de(M. Kachelrieß).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.03.053

One of the few generic predictions of superstrtheory is the existence of the dilaton supermultipin its spectrum. Independent of the particular meanism of supersymmetry breaking, any realistic strtheory model should led to a low-energy theory wsoftly broken supersymmetry at the TeV scale in orto solve the hierarchy problem. This ensures togewith other phenomenological constraints that the dton mass is at most of the order of the gravitino mm3/2 [3]. In hidden sector models of supersymmebreakingm3/2 is around the electroweak scale, whin gauge-mediated supersymmetry breaking mom3/2 could be as low as 0.01 keV.

Cosmological consequences of dilatons in strtheories have been already widely discussed[3–5].The purpose of this work is the investigation of dilat

.

2 E. Babichev, M. Kachelrieß / Physics Letters B 614 (2005) 1–6

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emission from a network of cosmic superstrings.derive stringent limits on the tension of the stringsfunction of the dilation mass and discuss the maximgravitational wave signal from cosmic strings compible with these limits.

2. Dilaton radiation from a cosmic superstringnetwork

In this section, we reanalyze the emission of dtons by a cosmic string network. Our main aim is tgeneralization of the analysis of Damour and Vilenin Ref. [5] to the case of a more general network elution, where the energy losses of strings as weltheir reconnection probability can deviate from th“standard” values.

Let us first recall the main quantities describingstandard evolution of a string network. The netwois characterized by the typical length scaleLst(t) ∼t/

√ζ , whereζ ∼ 1/4 is a parameter that has to

determined in numerical simulations. Another charteristic quantity of the cosmic string network is tparameterβ characterizing the lengthl of loops whichare chopped-off long strings,

(1)l(t) ∼ βt.

In the standard scenario, this coefficientβ is deter-mined by the gravitational radiation losses from loo

(2)βst ∼ Γ Gµ.

Here,Γ is a numerical coefficient of the order 50. Aany given moment of timet , the typical size of loopsis given by inserting into Eq.(1) the appropriate valuof β. Then the density of such loopsnst can be writ-ten as

(3)nst ∼ ζ

Γ Gµt3.

The evolution of a string network can be modifiin two ways: first, the probability of intercommuting ointersecting strings may be different from one,p 1,as it happens for cosmic superstrings[6]. Second, thevalue of the coefficientβ can differ from the one usually assumed,β = Γ Gµ. Note that the first modifi-cation of the properties of cosmic strings (p = 1) isdue to the difference between superstrings and tological strings, while the possible deviation ofβ from

the standard value applies for both type of strings.troducing a non-standard value forβ is inspired byRef. [9] where it was argued thatβ could be muchsmaller than it was originally assumed. It is convenito introduce the ratio

(4)ε = β

βst .

Following [8], one can find the resulting differencin the properties of strings for the two extreme caε 1 andε 1. In the latter case, the typical lengof loops is given by

(5)l1 ∼ Γ Gµt,

and their density is

(6)n1 ∼ ε1/2ζ

pΓ Gµt3.

(Note that we are interested in the radiation domition epoch, while in[8] the matter domination epocwas considered. This leads to different expressionsthe loop density.) In the opposite case,ε 1, one ob-tains as typical length of loops

(7)l2 ∼ βt,

and as density of loops

(8)n2 ∼ ζ

pΓ Gµt3.

Let us now turn to the calculation of the numberdilatons radiated from a cosmic superstring netwoIn Ref. [5], it was shown that a single loop radiatdilatons of frequencyω with the rate

(9)Nφ = Γφα2Gµ2/ω.

The coefficientΓφ does not depend on the loop siand is typicallyΓφ ∼ 13, while α = ∂ ln

√µ(φ)/∂φ

measures the strength of the coupling of the dilato the string relative to the gravitational strength.simplify the analysis, we assume that all dilatonsemitted at the same fundamental frequencyω = 4π/l.Then the total number of radiated dilatons can be wten asNφ ∼ Nφτ , whereτ is the decay time of a loopFor ε 1, this time is simplyτ ∼ t . Since the typicalength of loops is forε 1 smaller by a factorε, thedecay time of a typical loop is by the same factor spressed,τ ∼ εt . As result we obtain from Eq.(9) as

E. Babichev, M. Kachelrieß / Physics Letters B 614 (2005) 1–6 3

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total number of emitted dilatons from a single loop

(10)N ∼

(4π)−1Γ Γφα2G2µ3t2, ε 1,

(4π)−1Γ Γφα2G2µ3t2ε2, ε 1.

It is convenient to introduce further the relative abudance of dilatons,Yφ = nφ(t)/s(t), wherenφ(t) is thedensity of dilatons ands(t) the entropy density. In thradiation epoch, the entropy density is given by

(11)s(t) = 0.0725[N (t)

]1/4(

MPl

t

)3/2

,

whereN (t) is the effective number of spin degreesfreedom andMPl the Planck mass. Loops which decat timet contribute toYφ the following abundance odilatons,

Yφ(t) ∼ nN

s

(12)

∼

p−1ε1/2ζΓφα2(Gµ)2(MPlt)1/2N−1/4,

ε 1,

p−1ε2ζΓφα2(Gµ)2(MPlt)1/2N−1/4,

ε 1.

The expression(12) differs from the one obtained iRef. [5] by a factorp−1 for ε 1 and by a fac-tor p−1ε3/2 for ε 1. Eq. (12) is valid as long asthe loop sizes are smaller than the critical sizelc =4π/mφ , or tc 4π/(Γ Gµmφ) for ε 1 and tc ∼4π/(εΓ Gµmφ) for ε 1. Substituting these timeinto Eq.(12)we find

(13)Yφ(t) ∼

20p−1ε1/2α2(Gµ)3/2(MPl/mφ)1/2,

ε > 1,

20p−1ε3/2α2(Gµ)3/2(MPl/mφ)1/2,

ε < 1.

In the last expression we replaced also the strongequalitiesε 1 andε 1 by ε > 1 andε < 1 toinclude the rangeε ∼ 1.

The derivation of the expression(13) has beenmade for the scaling regime of the network. Thissumes in particular that plasma friction is neglible. Since for timest∗ 1/MPl(Gµ)2 the motion ofcosmic strings is heavily dumped by the surroundplasma, Eq.(13) is only valid for

(14)Gµ >

Γ mφ/(4πMPl), ε > 1,

εΓ mφ/(4πMPl), ε < 1.

3. Observational bounds and the detection ofgravitational waves

Having derived the abundance of dilatons(13) wecan now apply different astrophysical constraintsthe parameters of cosmic strings. First, ultra-light dtons are excluded by tests of the gravitational invesquare law[10],

(15)mφ > 10−3 eV.

Another limit comes from the bounds on the densitydark matter. Dilatons interact with the gauge bosthrough their mass terms in the Lagrangian,L =(1/2)αF κφF 2

µν , whereκ = √8π/MPl and αF para-

meterizes deviations of the coupling at low-energfrom the tree-level coupling at the string scale. Texact value ofαF is model-dependent, but generaof order one or larger[11]. Therefore dilatons decayinto gauge bosons with the lifetime[5]

(16)τφ ∼ 3.3× 1013 s

(12

NF

)(GeV

mφ

)3

,

whereNF is the number of gauge bosons with masbelowmφ/2 and we assumedαF = 1.

If the lifetime of dilatons is larger than the agof the universe,τ > t0 ∼ 4 × 1017 s, the total den-sity of dilatons is bounded by the observed densitydark matter. According to the WMAP observationthe relative fraction of dark matter isΩmh2 = 0.13[12], whereh = 0.7 parameterizes the Hubble costant. Thus

(17)Ωφh2 = nφmφh2

ρcr< 0.13,

whereρcr is the critical density of the universe. Usin(17)one can easily obtain

(18)Yφ <0.13ρcr

h2s(t0)mφ

4.5× 10−10(

GeV

mφ

).

This constrain is valid forτ > t0 or dilaton massemφ < 0.1 GeV. For lifetimesτφ < t0 one can useobservations of the diffuseγ -ray background[13].Decaying dilatons produce photons with number dsity nγ ∼ nφ(t0)t0/τφ . Using this relation and the approximate value for the upper bound of total enedensity of photons with energy> 1 MeV, ρ ∼ 2 ×

γ

4 E. Babichev, M. Kachelrieß / Physics Letters B 614 (2005) 1–6

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ofan-s

the

ece.he

10−6 eV/cm−3 [13], we find

(19)Yφ <τφργ

t0s(t0)mφ

∼ 7× 10−22(

GeV

mφ

)4

.

For lifetimes tdec < τ < t0 of the dilaton (whichcorresponds to dilaton masses 0.1 GeV < mφ <

1.6 GeV), wheretdec∼ 1013 s is the decoupling timewe can also use the constraints from the densityγ -ray background. The energy density of photons cated due to decay of dilatons at a timeτ is ργ (τ ) =mφnφ(τ) = mφYφs(τ ). Noting that the energy densitof radiation scales with the time asργ ∝ t8/3, we ob-tain

(20)Yφ < 3× 10−16(

mφ

GeV

).

For the lifetime of dilatonτφ < tdec we obtain con-straints onYφ from the abundances of4He,3He, D and6Li. Using the results of Ref.[14] we can smoothlyinterpolate the results for decaying of relic particand approximate the maximal allowed abundancethe following way:

(21)Yφ <

10−14m−1φ ,

1.6 GeV< mφ < 690 GeV,

5.5× 10−19m1/2φ ,

690 GeV< mφ < 15 TeV,

9× 10−38m5φ,

15 TeV< mφ < 150 TeV.

For very heavy dilaton masses,mφ > 150 TeV, thereare no limits on the abundance of dilatons.

Substituting the resulting expression for the diton abundance(13) into the obtained constraints foYφ (18)–(21), one can find the constraints on the stritensionµ as function of dilaton massmφ , couplingconstantα, the probabilityp of reconnection of longstrings, and the parameterε, determining the size othe closed loops produced by the network. For“standard” valuesp = 1 andε = 1 the resultingFig. 1is similar to the one in Ref.[5]. The constraints onGµ are slightly more stringent, because of more pcise observational data. However, the bounds onGµ

may be strongly modified in comparison with Ref.[5]by the multiplierp−1ε1/2 for ε > 1 andp−1ε3/2 forε < 1.

Fig. 1. A log–log plot of upper bounds onGµ as function ofmφ forseveral values ofε; the other parameters were chosen to have t“standard” values:α = 1 andp = 1. Below the dashed lines, plasmfriction invalidates our approximation.

Next we apply the obtained constraints on the vues ofGµ to examine the prospects to detect gravtional wave bursts (GWB) from the cosmic superstrnetwork. For each value of the dilaton massmφ , wefind the maximal allowed valueGµ and then, using theresults of[8], the maximal possible signal for GWBWe choose as frequency of the wave signal 150i.e., the frequency range preferable for LIGO andsume one GWB event per year. The results formaximum possible amplitude of the GWB signalsshown inFig. 2 for different values of parameterpand inFig. 3 for different values ofε. The sensitivityof the gravitational wave interferometer LIGO (andadvanced configuration) is also shown in these figuWe can see that the constraints from dilaton radiasignificantly restrict the prospects for the discoverycosmic strings by the detection of GWBs. In particufor the range of parametersε = 1 and 10−3 < p 1,GWBs from a string network would hardly be dtected by LIGO for dilaton masses 10−6 GeV< mφ <

105 GeV. Only in the case that the typical sizeclosed string loops is much smaller than in the stdard scenario (ε 1), the prospects to detect GWBfrom cosmic superstrings improve: for instance, incase ofp = 1 andε = 10−6 the detection of GRBsis possible for dilaton masses 10−12 GeV mφ 10−2 GeV and formφ < 104 GeV. In the case of verysmall values ofε, ε < 10−10, there are no limits on thGWBs amplitude coming from the dilaton abundanWe also checked the limits on GWBs coming from tstochastic gravitational wave background[15]. How-

E. Babichev, M. Kachelrieß / Physics Letters B 614 (2005) 1–6 5

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Fig. 2. Maximal allowed values for the amplitudeh of thegravitational wave signal emitted in the LIGO frequency baf ∼ 150 Hz, versus the dilaton massmφ . The parameters of cosmsuperstring network are chosen to beε = 1 andp = 1 (blue curve),p = 10−1 (green curve),p = 10−3 (red curve). The sensitivity oLIGO is shown in the upper dashed line (initial configuration) alower dashed line (advanced configuration). (For interpretatiothe references to color in this figure legend, the reader is referrethe web version of this Letter.)

Fig. 3. Maximal allowed values for the amplitudeh of thegravitational wave signal emitted in the LIGO frequency baf ∼ 150 Hz, versus the dilaton massmφ . The parameters of thcosmic superstring network are chosen to bep = 1 andε = 1 (bluecurve),ε = 10−6 (green curve),ε = 104 (red curve). The sensitivity of LIGO is shown in the upper dashed line (initial configuratioand lower dashed line (advanced configuration). (For interpretaof the references to color in this figure legend, the reader is refeto the web version of this Letter.)

ever, this bounds is modified only by a factor< 3 inthe range of dilaton massesmφ < 150 TeV and cosmicstring parameters, 10−3 < p 1 and 104 < ε < 10−6,therefore theFigs. 2 and 3change only slightly.

4. Conclusions

We have examined the emission of dilatons bnetwork of topological string or cosmic superstringDepending on the particular mechanism of supersmetry breaking, one expects a dilaton mass betw10 eV and 10 TeV if supersymmetry solves the hiarchy problem. We have derived stringent limitsthe string tension as function of the dilation macf. Fig. 1, for the case of a non-standard evolutiof a string network. We have found that for dilatmass in the favored range between 10 eV and 10and values of the string tension 10−12 Gµ 10−7

predicted in Ref.[7], the detection of a gravitationawave signal from cosmic (super)strings is only posble when the evolution of the string network deviastrongly from the standard case.

Acknowledgements

We are grateful to Alex Vilenkin for useful comments. This work was supported by the Deutsche Fschungsgemeinschaft within the Emmy Noether pgram and the Russian Foundation for Basic Reseagrant 04-02-16757-a.

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