Cosmological constraints on strongly coupled moduli from cosmic strings

Eray Sabancilar*

Institute of Cosmology, Department of Physics and Astronomy, Tufts University, Medford, Massachusetts 02155, USA(Received 28 April 2010; published 1 June 2010)

Cosmic (super)string loops emit moduli as they oscillate under the effect of their tension. Abundance of

such moduli is constrained by diffuse gamma ray background, dark matter, and primordial element

abundances if their lifetime is of the order of the relevant cosmic time. It is shown that the constraints on

string tension G� and modulus mass m are significantly relaxed for moduli coupling to matter stronger

than gravitational strength which appears to be quite generic in large volume and warped compactification

scenarios in string theory. It is also shown that thermal production of strongly coupled moduli is not

efficient, hence free from constraints. In particular, the strongly coupled moduli in warped and large

volume compactification scenarios and the radial modulus in the Randall-Sundrum model are found to be

free from the constraints when their coupling constant is sufficiently large.

DOI: 10.1103/PhysRevD.81.123502 PACS numbers: 98.80.Cq, 11.27.+d

I. INTRODUCTION

String theory requires the presence of scalar fields calledmoduli such as complex structure and Kahler moduliwhich parametrize the volume and the shape of a six-dimensional manifold representing the extra dimensionsin string theory. There is also the modulus called dilatonwhose expectation value determines the strength of thestring coupling constant. Moduli are originally masslessand their values are presumably fixed by the dynamics ofthe theory so that in the effective theory they becomemassive scalar fields. The idea of flux compactificationmade it possible to fix moduli by turning on some fluxesin the internal manifold [1]. The possibility of having alarge number of values for different fluxes leads to thepicture of string theory landscape where there exist 10500

different vacua [2]. In this large landscape of vacua, thereare attractive models where some of the long standingproblems are revisited such as the hierarchy [1,3,4], thepossibility of having a de Sitter vacuum in string theory[5], brane inflation as the origin of inflation [6,7], andcosmic superstrings [8–11].

Moduli can be produced by the oscillating loops ofcosmic strings.1 Such moduli can have effects on bigbang nucleosynthesis (BBN) and can also contribute todark matter and diffuse gamma ray background. In general,moduli are expected to have Planck mass suppressed cou-plings. These effects for the gravitationally coupled modulihave been studied in detail [12–14]. Recently, it was ar-gued that some moduli couple to matter more strongly thanthe Planck mass suppressed coupling in warped and largevolume flux compactification scenarios [15–17].References [15,16] argue that the dilaton is localized in

the IR region of a throat for a large warping. The dilatonmass is suppressed by the warp factor and coupling tomatter is stronger than the Planck mass suppressed cou-pling. Localization of wave functions and stronger cou-plings to matter are expected for other moduli as well[15,16]. They also show that there is a smooth interpolationbetween moderate and large warping cases which mimicsthe Randall-Sundrum (RS) model [18] as the effectivetheory in the large warping case.The Giddings-Kachru-Polchinski model [1] was the first

string theory realization of producing large hierarchiesfrom pure numbers, i.e., quanta of fluxes. It was arguedthat the RS model gives an effective description of thewarped compactification scenario with a large warp factorwhere the bulk space is replaced by the UV brane and allthe 4D physics except for gravity is localized on the IRbrane located at the bottom of the throat [19]. In theoriginal RS model, the radial modulus is not fixed andleft as a free parameter. A mechanism for stabilizing thismodulus was proposed by Goldberger and Wise [20], whoshowed that this modulus has a TeV suppressed couplingrather than Planck mass suppressed and has a TeV scalemass if the hierarchy problem is solved. Brummer et al.further showed that the RS model with the radial modulusstabilized by the Goldberger-Wise mechanism is the effec-tive description of the warped compactification scenario[19]. Therefore, moduli with strong coupling seemsgeneric in the warped type-IIB flux compactificationscenario.Another model where a strongly coupled modulus is

present is the so-called large volume compactificationwhere volume becomes exponentially large [3,4]. It wasshown in [17] that one of the Kahler moduli can have massm� 106 GeV and coupling to matter is suppressed by thestring mass scale ms � 1011 GeV for a particular value ofvolume which leads to TeV scale SUSY breaking. In thismodel, there is another Kahler modulus with m� 1 MeVand Planck mass suppressed coupling to matter which

*eray.sabancilar@tufts.edu1Moduli can also be produced thermally if the reheating

temperature is high enough. We shall comment on that possi-bility in Sec. V.

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suggests the presence of both strongly and weakly coupledmoduli together.

In this paper, we derive the constraints on strongly andweakly coupled moduli produced by oscillating loops ofcosmic strings and show that they are significantly relaxedfor large modulus coupling constants. Organization of thispaper is as follows: In Sec. II, modulus radiation fromcosmic string loops is summarized. In Sec. III, modulusand loop lifetimes are estimated and the density of loops inthe universe is given. In Sec. IV, we derive the abundanceof moduli produced by strings and obtain the upper limitson both strongly and weakly coupled moduli abundancesfrom diffuse gamma ray background [21], big bang nu-cleosynthesis [22–24], the dark matter density [25], anduse the lower limit on scalar field mass from Cavendish-type experiments [26]. The regions free from constraintsare shown in Figs. 1 and 2 for the parameter space in termsof string tension G� and modulus mass m for variousvalues of modulus coupling constant �. Finally, in

Sec. V, thermal production of moduli is discussed and itis shown that this mechanism is not efficient; therefore itdoes not lead to strong constraints on the free parametersmentioned above.

II. MODULUS RADIATION FROM STRINGS

Amodulus� couples to matter via the trace of its energymomentum tensor

L int � �

mp

�T��; (1)

where� is the modulus coupling constant,mp is the Planck

mass, and T�� is the trace of the matter energy momentum

tensor.We consider oscillating loops of cosmic strings coupled

to a modulus as a periodic source of moduli production.Modulus radiation from a loop of cosmic string occurs withthe power [12]

-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10

-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10

FIG. 1. logG� vs logmGeV for strongly coupled moduli. The region above the solid line is forbidden by the cosmological constraintsand the region below the dashed line is free from the constraints for the loops affected by plasma friction since such moduli are neverproduced because of friction domination. Note that if F- and D-strings do not interact with ordinary matter like solitonic cosmic stringsdo, the friction domination does not apply, and thus one should ignore the dashed line in that case.

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Pm � 30�2G�2; (2)

when the loop size L & 4�=m, where m is the modulusmass. This part of the spectrum corresponds to moduliproduced from small oscillating loops and so it is relevantto the early universe. We shall call this part of the spectrumas background moduli. The corresponding average particleemission rate is

_N � 13�2G�2

!; (3)

where ! is the energy of a modulus in the rest frame of theloop. Moduli are mainly produced in the fundamentaloscillation mode with ! ¼ 4�=L, where L is the size ofthe loop [12]. Thus, the particle emission rate can be ex-pressed as

_N � 13

4��2G�2L: (4)

When L � 4�=m, the main contribution to the radia-tion spectrum comes from cusps and has a different powerspectrum. Such moduli are produced in late epochs andhave larger lifetimes due to large boost factors of the cusps.Possible observable effects of such moduli are beingworked out [27]. Here, we shall only consider the back-ground moduli and their cosmological effects.

III. LIFETIME AND LOOP DENSITY

The rate of decay of a modulus into the standard model(SM) gauge bosons can be estimated as

�� nSM

��

mp

�2m3; (5)

where nSM ¼ 12 is the total number of spin degrees offreedom for all SM gauge bosons and m is the modulusmass. The mean lifetime of such a modulus in its rest framecan be estimated as the inverse of the decay rate as

-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10

-35

-30

-25

-20

-15

-10

-5

0

-10 -5 0 5 10

FIG. 2. logG� vs logmGeV for weakly coupled moduli when mweak * mstrong. The region above the solid line is forbidden by thecosmological constraints and the region below the dashed line is free from the constraints for the loops affected by plasma frictionsince such moduli are never produced because of friction domination. Note that if F- and D-strings do not interact with ordinary matterlike solitonic cosmic strings do, then the friction domination does not apply, and thus one should ignore the dashed line in that case.

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�� 8:1� 1012��2m�3GeV s; (6)

where mGeV � m=ð1 GeVÞ.An oscillating loop of cosmic string also produces gravi-

tational radiation with the power [28]

Pg � 50G�2: (7)

The main energy loss mechanism for a loop of cosmicstring is the gravitational radiation provided that Pg * Pm

which occurs when � & 1 (the �� 1 case was worked outin [12–14]). However, when moduli are strongly coupled tomatter, i.e., � � 1, modulus radiation becomes the domi-nant energy loss mechanism for loops, hence this leads tosignificant modifications of the constraints obtained in[12–14].

When the modulus radiation dominates, the lifetime of aloop is given by

�L ��L

Pm

� L

30�2G�: (8)

The constraints we shall obtain in the next sectiondepend upon the length of the loops formed from thecosmic string network. There is still no consensus on theevolution of string network and analytical works [29,30]and different simulations [31–34] yield different answers.However, the biggest recent simulations [33,34] suggestthat a loop formed at cosmic time t has a typical length

L� �t; (9)

with �� 0:1.The loops of interest to us are the ones formed in the

radiation dominated era whose number density is given by[28]

nðL; tÞ � ��1=2ðtLÞ�3=2; (10)

where � � 16, 30�2G�t & L & �t.Loops cannot produce moduli efficiently at times t &

t� � tp=ðG�Þ2, where tp is the Planck time, since they lose

most of their energy via friction due to plasma of particlesscattering from strings [28]. Therefore, we consider latertimes where the loops reach a scaling solution and themodulus radiation becomes the main energy lossmechanism.

The particle emission rate (4) is valid for the loops ofsize L & 4�=m, which exist only at t & tm, and tm can beobtained from (8) as

tm � 4�

30��2ðG�Þ�1m�1: (11)

Thus, we shall be interested in moduli produced in the timeinterval

t� & t & tm: (12)

This implies tm * t�, which can be expressed as

G� *30

4��2 m

mp

� 2� 10�19�2mGeV: (13)

We represent this condition on the parameter space plotsgiven in the next section as a dashed line below which nomoduli are produced, and thus free from the constraints.2

IV. COSMOLOGICALCONSTRAINTSONMODULI

A. Abundance

Modulus abundance is given by YðtÞ ¼ nmðtÞ=sðtÞ,where nmðtÞ is the modulus number density and sðtÞ isthe entropy density given by

sðtÞ ¼ 0:0725N 1=4

�mp

t

�3=2

; (14)

where N � 100 is the total number of spin degrees offreedom in the radiation dominated era at time t.The total number of moduli produced by a single loop

until cosmic time t < �L can be obtained from (4) as

N � _Nt� 13

4��2G�2Lt: (15)

By using (10) and (15), the number density of moduli in theuniverse produced by the loops of size L can be found as

nmðtÞ � NnðL; tÞ � 13

4���1=2ðLtÞ�1=2�2G�2: (16)

Thus, the modulus abundance can be estimated as

Y � 4:5��1=2L�1=2tm�3=2p �2G�2: (17)

Note that the smallest loops of size Lmin � 30�2G�tdominate the abundance. After substituting L ¼ Lmin in(17), it can be seen from (12) that t ¼ tm gives the mostdominant contribution to the abundance. Using these facts,we obtain

Y � 2:7ðG�Þ�mp

m

�1=2 � 9:4� 109ðG�Þm�1=2

GeV : (18)

Note that the dependence on G� in Eq. (18) is differentfrom that found in [12], since there it was assumed that��G�. Although �� 0:1 seems to lead to more strin-gent constraints on string tension G�, we shall see that theconstraints are relaxed when the coupling constant � be-comes large enough.

2Plasma friction may or may not affect cosmic F- and D-strings depending on whether they interact with ordinary matteror not. However, thermally produced bulk field backgrounds,such as moduli, might have a similar effect on cosmic F- and D-strings. If they are not affected by friction, the condition (13) isremoved for these types of cosmic strings.

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B. Cosmological constraints on strongly coupledmoduli

Short distance measurements of Newton’s law of gravityin Cavendish-type experiments give a lower bound on themodulus mass as m> 10�3 eV, i.e., mGeV > 10�12 [26].

If moduli are long lived, i.e., � * t0, they contribute tothe dark matter in the universe. Here, t0 � 4:3� 1017 s isthe age of the universe. Thus, we have the upper bound�mh

2 < 0:13 [25] or in terms of abundance Y < 9:6�10�10m�1

GeV.

If moduli are long lived, they also contribute to thediffuse gamma ray background [35]. When � * t0, theenergy density of moduli that decayed into photons untilthe present time can be estimated as

�m � Ysðt0Þm t0�� 2:2� 1017Ym4

GeV�2 eV cm�3; (19)

where t0=� is the fraction of the decayed moduli andsðt0Þ ¼ sðteqÞðteq=t0Þ2 � 2:9� 10�38 GeV3. According to

EGRET data, an approximate upper bound on the diffusegamma ray density for the photons of energy>MeV is� � 2:0� 10�6 eV cm�3 [21]. Using this upper bound,

we can estimate the limit on the abundance from theconstraint �m & � as Y & 9:1� 10�24��2m�4

GeV.

When tdec � 1013 s & � & t0, the most stringent con-straint comes from the diffuse gamma ray background[35]. Assuming all the moduli decay by the time �, theenergy density can be estimated as

�m � Ysð�Þm� 1:1� 1022Y�4m7GeV eV cm�3; (20)

where sð�Þ ¼ sðteqÞðteq=�Þ2 � 8:2� 10�29�4m6GeV GeV3.

Redshifting the photon energy density to time t ¼ �, wefind

�ð�Þ � �

�t0�

�8=3 � 1:0� 107�16=3m8

GeV eV cm�3;

(21)

which gives the upper bound on the modulus abundance as

Y & 9:1� 10�16�4=3mGeV.If the modulus lifetime is shorter than tdec, they can have

effects on primordial element abundances [22–24]. Whensuch moduli decay electromagnetically, they dissolve thelight elements created during nucleosynthesis. Besides,modulus-gluon coupling leads to hadron production whichcan also change the primordial light element abundances.To obtain upper limits on the modulus abundance, we madea piecewise power law approximation to the results of [22–24] and summarized them in Table I, where �s � �= sec .

Using the bounds obtained from Cavendish-type experi-ments, diffuse gamma ray background, BBN, and darkmatter constraints, we obtain the limits on string tensionG�, modulus mass m, and modulus coupling constant �.Using all these constraints, we obtained Fig. 1 for theparameter space of G� vs mGeV for various values of �.The analytic forms of the constraints in all parameter

ranges are given in Table II of the Appendix. As it canbe seen from Fig. 1, the constraints become weaker as �increases. The condition (13) shifts towards the upper endof the parameter space which leads to the region free fromthe constraints below the dashed line where no moduli areproduced from cosmic strings.

C. Cosmological constraints on weakly coupled moduli

In the previous section, we analyzed the cosmologicalconstraints on strongly coupled moduli. In this section, weshall assume that there is at least one strongly coupledmodulus and one weakly coupled modulus (coupling sup-pressed by at least Planck mass) with coupling constants� � 1 and �W & 1, respectively. We shall estimate thecosmological constraints on weakly coupled moduli simi-lar to the previous section. Note that the dominant energyloss mechanism for the loops is still via strongly coupledmodulus radiation, thus the loop lifetime is given by (8)and the minimum size of the loops is Lmin � 30�2G�t. Onthe other hand, the modulus lifetime depends upon itscoupling constant to matter and is given by

�W � 8:1� 1012��2W m�3

GeV s: (22)

The abundance of weakly coupled moduli can be calcu-lated as

YW � 9:4� 109��2�2WðG�Þm�1=2

GeV ; (23)

which is valid for mweak * mstrong, where mweak and mstrong

are the masses of the strongly and the weakly coupledmoduli, respectively. However, ifmweak <mstrong, although

the strongly coupled moduli production terminates attmstrong

ð�Þ given by Eq. (11), weakly coupled moduli are

still produced and the process terminates at tmweakð� ¼ 1Þ.

At this point, gravitational radiation starts dominating and

the abundance becomes YW � 5:6� 109�2WðG�Þm�1=2

GeV .

Therefore, the constraints are the same as given inFig. 1 for the � ¼ 1 case when mweak <mstrong assuming

�W � 1.In the opposite regime, when mweak >mstrong, by using

(23) and all the constraints we discussed in the previoussection, we obtained the parameter space in Fig. 2 forvarious values of �W and�. Once again, the analytic forms

TABLE I. BBN constraints on the strongly coupled modulusabundance. This table shows the approximate upper bounds onthe strongly coupled modulus abundance as a function of modu-lus lifetime and modulus mass.

�s Y

104 & �s & 1013 10�14m�1GeV

102 & �s & 104 10�8��3=2s m�1

GeV

10 & �s & 102 10�11m�1GeV

10�2 & �s & 10 10�11��5=2s m�1

GeV

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of the constraints are given in Table III in the Appendix. Ascan be seen from Fig. 2, the constraints become lessimportant as � increases and �W decreases since theabundance is suppressed by ��2�2

W . Besides, the con-dition (13) becomes stronger for larger � and there is alarger region in the parameter space below the dashed linefree from the constraints.

V. THERMALLY PRODUCED MODULIBACKGROUND

So far we have discussed the production of moduli fromcosmic strings. Moduli can also be produced thermally ifthe reheating temperature is high enough. The photon-modulus interaction can be written as

L int � �

mp

�F�F�: (24)

Moduli should be in thermal equilibrium with photonsfor the thermal production to occur. The lowest orderprocess which contributes to the interaction ! �� issecond order and the cross section can be estimated as

����2

m2p

�2E2; (25)

where E� T is the energy of photons at temperature T.For the thermal production of moduli to occur, the rate

of thermal modulus production should be greater than theexpansion rate of the universe H, i.e.,

�th � �n * H; (26)

where n is the photon density at temperature T. Using

(25), n � T3 and H � T2=mp in (26), we obtain

T * ��4=3mp: (27)

For instance, when �� 109, (27) implies Trh *107 GeV. Since the reheating temperature is also modeldependent, strongly coupled moduli may or may not beproduced thermally. On the other hand, weakly coupledmoduli cannot be produced since Trh * mp is required for

� & 1.Assuming strongly coupled moduli are produced ther-

mally and dominate the universe, we can estimate thetemperature after their decay. The decay rate of moduli is

�dec ���

mp

�2m3; (28)

and when �dec �H� T2=mp, moduli will decay and re-

heat the universe to temperature T. Using that, we obtain

T � �

�m

mp

�1=2

m: (29)

The weakest constraint one can consider is that T shouldbe at least at the nucleosynthesis temperature �1 MeV.Using T * 1 MeV, we obtain the constraint

� * 106m�3=2GeV : (30)

VI. CONCLUSIONS

We consider oscillating loops of cosmic strings as peri-odic sources of moduli production. When � & 1, gravita-tional radiation is the dominant energy loss mechanism forthe loops. The constraints for this case are identical to the� ¼ 1 case for the strongly coupled moduli as we haveshown on Fig. 1. Note that our results for �� 1 are morestringent than that of [12–14]. This is mainly because of thefact that they assume ��G� in their calculations,whereas we use �� 0:1 from the recent simulations[33,34].When a modulus is strongly coupled to matter, i.e., � *

1, the modulus radiation is the dominant energy lossmechanism for the loops. Hence, loop lifetimes dependupon �. Besides, modulus lifetime shortens as � is in-creased if the modulus mass is kept constant. These twoeffects make the cosmological constraints we obtained forthe strongly coupled moduli less severe. Basically, formoduli to have effects on BBN, and to contribute to darkmatter and diffuse gamma ray background, their lifetimeshould be long enough. As can be seen from Fig. 1, thisimplies that only strongly coupled moduli with smallmasses have effects on cosmology since they have longerlifetimes compared to the more massive ones.In addition, loops cannot radiate moduli effectively in

the friction dominated epoch since they lose their energymostly via friction. The condition for friction domination(13) becomes stronger when � is larger. Therefore, moreregion of the parameter space is allowed as � is increasedsince friction domination does not let moduli to be pro-duced by cosmic strings in that region. This may not be thecase for F- and D-strings since they may or may notinteract with ordinary matter depending on where theyare located in the bulk. However, if there is a thermallyproduced moduli background, a similar effect might occurto F- and D-strings which needs further investigation.We consider warped and large volume compactifications

as the two examples where at least one strongly coupledmodulus is present. In the warped compactification sce-nario, there is some evidence for moduli localization inlong throat regions which leads to stronger coupling tomatter and smaller moduli masses [15,16]. Besides, there isa smooth interpolation between large and moderate warp-ing. This suggests that warped compactification with a longthroat can be effectively described by the RS model with itsradion stabilized by the Goldberger-Wise mechanism [19].As it was argued some time ago by Goldberger and Wise,RS radion has TeV suppressed coupling to matter [20].Interpolating our results for this particular case, we see that

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the RS radions produced by cosmic strings are free fromthe cosmological constraints we considered in this work.

As a second example, we consider large volume com-pactification where a strongly coupled modulus is present.One of the Kahler moduli in this scenario has a mass m�106 GeV and a string mass scale suppressed couplingwhere ms � 1011 GeV [17]. In our notation, this meansthat �� 108, hence a strongly coupled modulus. As wecan see from Fig. 1, this modulus is free from the cosmo-logical constraints we considered.

In the large volume compactification scenario, there isanother Kahler modulus with Planck mass suppressedcoupling to matter, i.e.,�� 1. This suggests the possibilityof having at least one strongly coupled and a weaklycoupled modulus together. We also calculated the con-straints on weakly coupled moduli and show our resultsin Fig. 1 with � ¼ 1 for the mweak <mstrong case and in

Fig. 2 for the mweak * mstrong case. In particular, the con-

straints on the weakly coupled Kahler modulus of the largevolume scenario is given in the first plot of Fig. 1 (� ¼ 1)since mweak � 1 MeV<mstrong � 106 GeV. Note that in

this model, maximum string tension can only be G��10�16. If we take �W � 1 and mweak � 1 MeV, the con-straints on this weakly coupled modulus is not stringent.On the other hand, as can be seen from Fig. 2, constraintsare quite weak for the � ¼ 108, �W ¼ 1 case. If �W � 1,then weakly coupled moduli with mweak * mstrong are free

from constraints since their abundance is suppressed by��2�2

W .We also consider the possibility of producing moduli

thermally. We found that, if the universe has ever reached

the temperature of order T � ��4=3mp, then moduli can be

produced thermally. If the hierarchy problem is solved withwarped geometry, the RS radion couples to matter with�� 1015 and (30) implies m * 10�6 GeV for the RSradion mass. Since it is expected that m� 1 TeV [20],the RS radion is free from both the cosmological con-

straints from cosmic strings and the thermally producedradion background constraint. For the strongly coupledKahler modulus in the large volume scenario, the con-straint (30) implies m * 10�2 GeV. Since m� 106 GeVin this model [17], it is also free from the thermallyproduced moduli background constraint. Finally, we alsofound that weakly coupled moduli cannot be producedthermally since T * mp would be required for this to

happen.In this work, we assumed that the reconnection proba-

bility of strings is p ¼ 1 which is true for the ordinarycosmic strings. However, for cosmic F- and D-strings p <1which leads to an enhancement of the string density in theuniverse [9,11]. Therefore, the constraints are expected tobe a little bit stronger for the p < 1 case. However, thisturns out to be an insignificant effect [13].The main conclusion of this work is that, when there is at

least one type of strongly coupled modulus, both thecosmological and the thermally produced moduli back-ground constraints on strongly and weakly coupled modulibecome less severe and for sufficiently large values of �,they are free from the constraints considered in this work.

ACKNOWLEDGMENTS

I would like to thank Alexander Vilenkin for manyuseful comments and for carefully reading the manuscript.This work was supported in part by the National ScienceFoundation under Grant No. 0855447.

APPENDIX

1. Cosmological constraints on weakly and stronglycoupled moduli

The analytical forms of the cosmological constraints onstring tension G� for weakly and strongly coupled moduliare given in Tables II and III, respectively.

TABLE II. Constraints on the string tension G� for strongly coupled moduli. This table showsthe upper bounds we obtain from Cavendish-type experiments, diffuse gamma ray background,BBN, and dark matter density constraints on G� for strongly coupled moduli as a function ofmodulus mass m and modulus coupling constant �.

mGeV G�

10�12 & mGeV & 3� 10�2��2=3 1:0� 10�19m�1=2GeV

10�12 & mGeV & 3� 10�2��2=3 9:7� 10�34��2m�7=2GeV

3� 10�2��2=3 & mGeV & 9:3� 10�1��2=3 9:1� 10�26�4=3m3=2GeV

9:3� 10�1��2=3 & mGeV & 9:3� 102��2=3 1:1� 10�24m�1=2GeV

9:3� 102��2=3 & mGeV & 4:3� 103��2=3 4:6� 10�38�3m4GeV

4:3� 103��2=3 & mGeV & 9:3� 103��2=3 1:1� 10�21m�1=2GeV

9:3� 103��2=3 & mGeV & 9:3� 104��2=3 5:7� 10�54�5m7GeV

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TABLE III. Constraints on the string tension G� for weakly coupled moduli. This table showsthe upper bounds we obtain from Cavendish-type experiments, diffuse gamma ray background,BBN, and dark matter density constraints on G� for weakly coupled moduli when mweak *

mstrong as a function of modulus mass m and moduli coupling constants � and �W . When

mweak <mstrong, one should set � ¼ 1 in the above table.

mGeV G�

10�12 & mGeV & 3� 10�2��2=3W 1:0� 10�19�2��2

W m�1=2GeV

10�12 & mGeV & 3� 10�2��2=3W 9:7� 10�34�2��4

W m�7=2GeV

3� 10�2��2=3W & mGeV & 9:3� 10�1��2=3

W 9:1� 10�26�2��2=3W m3=2

GeV

9:3� 10�1��2=3W & mGeV & 9:3� 102��2=3

W 1:1� 10�24�2��2W m�1=2

GeV

9:3� 102��2=3W & mGeV & 4:3� 103��2=3

W 4:6� 10�38�2�Wm4GeV

4:3� 103��2=3W & mGeV & 9:3� 103��2=3

W 1:1� 10�21�2��2W m�1=2

GeV

9:3� 103��2=3W & mGeV & 9:3� 104��2=3

W 5:7� 10�54�2�3Wm

7GeV

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