Cosmic strings on the verge of a gravitational breakdown

Volume 237, number 3,4 PHYSICS LETTERS B 22 March 1990

C O S M I C S T R I N G S ON T H E VERGE O F A GRAVITATIONAL B R E A K D O W N *

M. QUIR(3S lnstituto de Estructura de la Materia, Serrano 119, E-28006 Madrid, Spain

Received 3 January 1990

The theory of galaxy formation by cosmic string loops is endangered by experimental bounds on the background of gravitational radiation coming from recent measurements of the millisecond pulsar PSR 1937 + 21 timing stability and from nucleosynthesis. Galaxy formation and bounds on gravitational radiation can be reconciled if the phase transition is inflationary, and such that the largest scale crossing outside the horizon during the inflationary era lies in the range between Mo and 10 ~ ~Me. A simple model with local U ( 1 ) cosmic strings and chaotic inflation is analyzed, yielding the corresponding number of e-folds in the interval between 50 and 60.

Stable cosmic strings [1 ] arise at a phase transi- t ion when the manifold ~¢/of degenerate vacua has non-trivial first homotopy group, i.e. l-I, 6 t [ ) ¢ 1. The whole theory o f cosmic strings depends on a unique free dimensionless parameter, G/I, where ~t is the mass of string per uni t length, de te rmined by the vacuum expectat ion value o f the scalar fields along the degen- erate direct ion of the vacuum.

One o f the most appeal ing features of the local cosmic string theory is that it provides an explana- t ion of the origin o f galaxies, and clusters o f galaxies, by accret ion o f mat ter onto closed loops, provided that G / t~ 10 -6 [2] . However, this scenario has been recently challenged by an overproduct ion o f gravita- t ional radia t ion by loop decay into gravi tat ional waves. In part icular, the predicted background o f gravi ta t ional waves appears to be in conflict with re- cent measurements of mil l isecond pulsar t iming sta- bil i ty [3 -8 ] and with the measured abundance o f pr imordia l hel ium at the t ime o f nucleosynthesis [9,6] ~l

* Work partly supported by CICYT under contract AEN88-0040. ' Electronic mail: imtma27@emdcsicl.bitnet al For global cosmic strings gravitational radiation is not the

dominant decay mode because of the existence of the compet- ing Goldstone boson channel. In this way global strings can neither upset the millisecond pulsar stability and the nuclco- synthesis predictions nor make galaxies to condense around oscillating loops [ 10 ]. For those reasons we will not consider them here.

In this paper we will argue that both conflicts dis- appear if the phase t ransi t ion giving rise to cosmic strings is delayed in such a way that a certain amount of inflation is produced in it. I f this is so, cosmic strings are inflated away during the inflationary, ep- och and enter the horizon after the inflation, during the rad ia t ion-domina ted era. The condi t ion for the string loops entering the horizon not to dis turb the stabili ty of the mil l isecond pulsar t iming puts a lower bound on the number of nccessary e-folds, while the condi t ion for those loops to accrete mat ter and lead to the usual scenario of galaxy format ion puts an up- per bound on the number of c-folds o f inflation. For i l lustrative purposes we present the chaotic inflat ion generated in a phase transi t ion, where a U ( 1 ) gauge symmetry is spontaneously broken, producing local cosmic strings.

We will first review the bounds on G/t arising from the mil l isecond pulsar t iming stabil i ty and from the abundance of helium formed at the nucleosynthesis t ime.

The cosmological evolution of string loops is deter- mined [ 1 ] by three numerical coefficients o~, fl and y. The coefficient o~ is the ratio o f the initial size of the loop rclative to the horizon size. At the instant t~ a loop is created with size R given by

R ~ a t c . (1)

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Volume 237, number 3,4 PHYSICS LETTERS B 22 March 1990

The coefficient fl provides the time dependence of the number density of loops n(t) as

dn(t) =fit-4 (2) dt

The coefficient y gives the rate of power loss P of os- cillating loops by gravitational radiation as P=yGI~ 2, which results in a loop lifetime z of

• = (7Gp)-~R. (3)

The values of the coefficients a, fl and ), can be dc- termined only by numerical simulations. In this pa- per we will consider thrce cases: - Case (a) is the carliest simulation (presented here as a reference point) giving the generic values [ 1 ]

a ~ l , f l ~ l , y~ lO0 . (4)

- Cases (b) [111 and (c) [61 are the most recent simulations, corresponding to a larger production of closed loops of sizes much smaller than the horizon. The corresponding valucs of a,/? and y are [ 7 ]

a ~ 0 . 0 2 , f l~600, y~50 (5)

for model (b), and

a ~ 0 . 0 2 8 , f l~850, y~50 (6)

for model (c). Using ( 1 ) - ( 3 ) we can write

dn(tc ~ a -IR) ~f la3R - 4 dR (7)

from where the number density of loops with size R, defined by

dn nR=-- dR ' (8)

becomes at an arbitrar3' time, during the radiation- dominated era,

]~01~ 312

n / ~ ( t ) = (R+TGIH)5/2t3/2, (9)

where the facts that the universe is expanding and the loops are not stable are already taken into account.

The presence of gravitational radiation will in- crease the rate of expansion of the universe at the time of nucleosynthesis [9,12 ], leading to an overabund- ance of primordial helium. In this way, to quanti~ the effects of gravitational radiation on nucleosyn-

thesis we need to know the accumulated gravita- tional radiation at that time. Using (8) and (9), the number density of loops with any size is

\ - - 3 / 2 --3/2

(10)

where to* is the time when the first loop is formed and 2 - a (7G/t)- ~. In a standard cosmology with a GUT phase transition (G#~ 10 -6) it has been estimated [9] that frictional effects become negligible at t* ~ l0 -3° S. in that case we can approximate (10) by the usual expression [91:

n(t) ~ ]floz3/2(TGl~ ) -3/2l-3

However, as we will see later, if the phase transition is delayed and generates some amount of inflation, t~ cannot be neglected when compared to tN/2 (where tN, 0.01 s<tN<300 s, is the time during which nu- cleosynthesis takes place). For that reason we will keep the complete expression (10).

The accumulated gravitational radiation from to* to a later time t is given by [ 9 ]

2

/ ) g r ( / ) = n(z)7Glt 2 dr, ( 11 ) t*

where the cosmological redshift in the radiation- dominated unverse is already taken into account. Us- ing (10) one gets

Pgr 64g - fla3/2"/-~/2(G#)'/2f(t) , (12) Pr 9

where pr = 3/32rcGt 2 is the energy density in radia- tion and

f ( t ) = 2 ( l +2)-1/2--2( l +) t*~cl ) - ' /2

/ [ ( .,,/2--, + [ l + ( l + ) . ) - ~ / 2 ] l n ~ + 2 1 n I+ 1 + 2 ~ ) J

- - 2 1 n [ l + ( l + ) . ) '/2] . (13)

The contribution to the energy density due to loops is

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Volume 237, number 3,4 PHYSICS LETTERS B 22 March 1990

pL ~ ~ l~Rnn(t)dR, (14) -/Grit

which translates, using (9), into

PL 160n o~.3/2, -l/z,.,~..~t2 p~ - 9~,/2 pet z t t ~

X ( 1 - 6x/~ ] + 2 "~ 5 ( 1=+-~3/2] " (15)

The stability of the nucleosynthesis predictions re- quires [9,12 ] that the total contribution to the cn- ergy density due to strings does not exceed 0.18p~ (as is the case with extra neutrino species), i.e. ~2

P~+PL <0.18 . (16) P~

In the case of the standard cosmology and in the re- gion 1 <<2<<ty/t'g we obtain from (12), (13), (15) and (16) the bound

G/l < 2 X 10 -s Od--3]~--2y . (17)

For the case (a) the bound ( 17 ) gives G# < 2 X 10- 6, which is marginally consistent with galaxy forma- tion, while for cases (b) and (c) it gives G,u<3.5×10 -7 and G/z<6X10 -s, respectively, which are clearly beyond the allowed edge. This re- sult agrees with the similar analysis performed in ref. [6]. However, as we will discuss later on, if the phase transition is somewhat inflationary t* can bc much greater than 10 -30 S and the bound (17) can be to a large extent weakened.

We will now discuss the bounds on Glz from the stability of the millisecond pulsar timing. A loop of radius R created at time t~ decays at the time td=)tt¢ and produces gravitational waves of frequency ~ (cet~) -~. After its decay the frecuency is redshifted like I +z(t) . A convenient measure of the intensity of gravitational radiation is provided by the energy density per logarithmic frecuency per cosmic micro- wave energy

.0~(o9)- co dp~ (J8) Pr do) '

~ W e are neglecting lhe contribution to the energy due to infi- nite strings that goes like G,u.

where co is the angular frecucncy today. It was calcu- lated in ref. [ t3 ] yielding, for T<I .6×10SyG/ t X f2- ~h -~ yr, the result ,3

~ g (¢..O) = ~7"col3/2fly-l/2(Glz)l/2 (19)

On the other hand, assuming that all non-zero post- fit arrival time residuals R for millisecond pulsars are due to a stochastic background of gravitational waves, the stability of the pulsar timing over an observa- tional period provides an upper bound on ~g(Og). This bound depends on the measured time residuals R and on the total time T spanned by the observa- tions. The most recent results [ 3,14 ] from the milli- second pulsar PSR 1937 + 21 [ 15 ], corresponding to R=0.3 Its and T=6.25 yr, provide the bound ~g(CO) < 2 × 10- 3, which translates into

G / t < 2 × 10-90L-3fl-2~. (20)

The bound (20) gives G # < 2 X 10 -7 for case (a), G#<3.5X 10 -8 for case (b) and G/ t<6X 10 -9 for case (c). In all cases the bounds are again, as in the case of nucleosynthesis, past the edge permitted by galaxy formation. Similar results were obtained in refs. [6,7].

One can readily compute the creation time t c of a loop producing gravitational waves of period 7" to- day. Wc obtain

l c = 4 X 10 -17 (ayG/ . t "1-17"2 O2't~4 (21) teq

where tcq=4× 10*°f2-2h -4 s, h=H/lO0 km s- ' Mpc-* with H2=]nGp~, and T v~ is the period in years of the gravitational waves. Eq. (21 ) applies for Ty~< 1.6× 10SyG/z f2-~h-2 ~. Fixing G/z= 10 -6 we obtain the creation time ~ for loops producing gravitational waves with 7~.~=6.25 as

- 1.6× 10 -9 (c~y)-'(f2h 2) 2, (22) t~q

,3 A more detailed analysis of ~s(~o) has been performed re- cently [ 7 ]. For the purpose of the present paper, eq. ( 19 ) is a good estimation.

,4 Otherwise, for 1.6×lOSTGp~2-~h-2<T>~<l.6×tO s ×cd/3(TGp)2/3~-lh-2. eq. (21) becomes tdteq=2.5x 10 -25 a - ~ ( yG,t.t ) -2(12h2)3T~r.

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Volume 237, number 3,4 PHYSICS LETTERS B 22 March 1990

i . e .

- for case (a): ~ = 1.6× 10- l~g-22h4t~q=0.62 s, - for case (b): ~ = 1.6X 10-9-QZh4t~q= 62 s, - for case (c): ~ = 1.1 × 10-9922h4/cq=44 s. In all cases the creation of the corresponding loops occurs during the time of nucleosynthcsis. We will come back to this point later on in this paper.

Summarizing the results up to here, the bounds (17), from nucleosynthesis, and (20), from the sta- bility of the millisecond pulsar PSR 1937 + 21, seem to be inconsistent with the theory of galaxy forma- tion by cosmic strings, that require G/t~ 10 -6. We will show now there is a way out if the time t* of the first loop formation is larger than the time ~ of formation of loops disturbing the stability o f millisecond pul- sars: t * > ~ . In that case the bound (20) is not oper- ating. However, as we have noticed before, ~ is in- side the period o f nucleosynthesis and cosmic strings entering the horizon at times close to ~ could, in principle, accelerate the rate of expansion of the uni- verse and lead to an overproduction o f primordial helium. This does not happen because the accumu- lated gravitational radiation is not large enough to in- crease significantly the expansion rate of the uni- verse. Actually the bound ( 17 ) does not apply because it was calculated using the approximation t*/tN << 1, which is no longer true. Instead we have plotted ( p g r - ~ P L ) / D r a s a function oft*~/tN in fig. 1. There we can see that, for case (a), R is below the upper bound (16) already for G* = 10-35/N while, for cases (b) and (c), (Pgr+PL)/P~ is past the bound (16), in agree- ment with our bound ( 17 ) which was obtained within the approximation t* << t~. From fig. 1 we can also see that for the case (b) [ (c) ] (Pg,+PL)/P~ goes be- low (16) for G*>10-16tN Its*> 10-8tN]. However, since ~ > 10-StN, the proposed solution to the prob- lem of millisecond pulsar timing stability, i.e. t* > ~, automatically resolves the nucleosynthesis problem in all cases.

How can one achieve such a delayed (t ~* > ~) phasc transition? The natural answer is inflation. If thc phase transition is inflationary., loops with size greater than the horizon at that timc ( ~ HF ~ ) are confor- mally stretched with the De Sittter expansion of the universe [ ~exp(Hl / ) ] [16] and follow, after the end of inflation, the expansion of a radiation-dominated universe with cosmic scale factor a (t) ~c t'/2. When the loops are smaller than the horizon the effect of

1 . 0 , 1- -1 . . . . . .

c 0 . 8 ~ 3 \

\

5 ~ ,. 0 . 6 \

~ 0.4 --...c~.~ (b)

r, '" "--. Bm x~nd~ ~-):e --- ~; 0 2 - N u c l e o s y n t h e s i s - ~

e~ t-, :-~-~ . . . . . . ~ -- . . . . . . .

i 00 c,,o,,l . . . . . . . . . . . . . . . .

- 3 5 - 3 0 2 5 - 2 0 - 1 5 - 1 0 - 5 0

[ .og l o ( F i r s t C r o s s i n g T i m e / N u c l e o s y n t h e s i s T i m e )

Fig. I. Plot of (pg +PL )/P~ as a function of In (t*/tN) for models (a), (b) and (c) in the text (solid lines). The dashed line shows the nucleosynthesis limit ofeq. (16).

expansion becomes unimportant and they collapse to a point. I f the number of e-folds of inflation is able to solve the horizon problem, the inflated away loops will never enter our physical horizon and could not produce any observable effects. However, if the phase transition giving rise to cosmic strings is unable, by itself, to solve the horizon problem, the inflated loops with initial sizes closest to the horizon will enter the physical horizon after inflation and can produce ob- servable effects. We will call, in agreement with our previous notation, t~* the time the first loop enters the horizon after the inflationary era. No loop will enter the horizon at times earlier than G*. Ift~*> ~ the loops entering the horizon would escape the millisecond pulsar detection and would not produce any trouble to nucleosynthesis. Of course we will require those loops to be able to accrete galaxies onto them, and this will provide an upper bound on t*.

Galaxies and clusters condense around oscillating closed loops, while the loops gradually decay by grav- itational radiation with lifctime given by (3). Loops entering the horizon at time tc~ t~,~ (with size R ~ c~tc) will produce density fluctuations on a comoving scale l~ f l - '/3 ( tjeQ) 1/2, i.e. a comoving mass Mt~ [3- i ( t~/ t~q)3/'-Meq (wherc Mcq is the mass within a sphere o f diameter tcQ), given by

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Volume 237, number 3,4 PHYSICS LETTERS B 22 March 1990

to ~-~/2, (~)eq ~ ~'O1~' I 1/6G. Ix t~q j (23)

where t~ is in the interval J- l<~tc / teq<~l , corre- sponding to a comoving mass Mt in the interval fl-t).-3/2<~Mt/Meq<~fl -l. Loops entering the hori- zon at times tJt~q<2 -~ have decayed before the equilibrium time and their density fluctuations, which are scale-independent, do not grow (or are damped) between the loop decay and t~q. Finally, loops enter- ing the horizon at times tc> t~q produce fluctuations behaving like afl(tdtcq)-~GIu, for all kinds of dark matter, and redshifting with the expansion of the uni- verse like ( 1 + z ) -~.

The evolution of fluctuations produced by loops entering the horizon at t~ < t~q is different for differ- ent types of dark matter due to different kinds of damping produced by free-streaming of the dark matter. In a baryon-dominated universe, loops pro- duced at t~<2-~t~q are erased by Silk damping. In the case of a neutrino-dominated universe (hot dark- matter), the neutrinos erase their density fluctua- tions on scales smaller than their Jeans mass, corre- sponding to tc<t~ (fl)-i)2/Stcq. For cases (a), (b) and (c) the inequality 2-tteq << t~ holds. Therefore, if the time t~* of the first loop crossing is t* ~<;t- ~tCq, the theory of galaxy formation by accretion onto closed loops still should hold in the cases of baryon- dominated and neutrino-dominated universes. In the case of cold dark-matter, where there is no efficient damping mechanism by axion free-streaming, the spectrum of density fluctuations would be cut-off at a scale -fl-l(t*/leq)3/2Meq. If t* is close to its lower bound ~, given by eq. (22), small scales are the first to go non-linear and the spectrum would correspond to a gravitational clustering picture, as in usual cold dark-matter scenarios. However if t* is close to 2 - l leq , then the spectrum of fluctuations is cut off at scales close to the scale of galaxies, which would then be the first ones to be formed ~5. In that case the cold dark- matter scenario would resemble that of a baryon- dominated universe. In summary we will adopt 2- l/eq as a safe upper bound on t* to keep the benefits of

~s This scenario could be destabilized if the adiabatic density perturbations on small scales, produced by quantum fluctua- tions during the inflationary, era, have large amplitudes, as we will comment later on.

galaxy formation by matter accretion onto closed cosmic string loops.

Let us therefore assume that the phase transition giving rise to cosmic strings is inflationary and that the first loops to enter the horizon after being inflated away do it at the time t~, bounded to be in the inter- val [~, 2-~tcq]. The largest scale 2 ( 0 , crossing out- side the horizon Hi- ~ during the inflationary era and re-entering the physics horizon after inflation at the time t*~2( t*) , is the scale 2c(t) that entered the ho- rizon at the time tc ~2c( tc) . The comoving mass corresponding to a perturbation 2 (t), M;. u), evolves as t -~/z in a radiation-dominated universe and be- comes time-independent in the matter-dominated era. This constant comoving mass Ma_= M;.tt~,~) character- izes the corresponding perturbation. In this way the comoving mass Mo corresponding to the perturba- tion 2c entering the horizon at time t* is given by

i,v/~ = 5.5 × 1013 (g2h z) -2 ( t*~3/2~o. (24) \ / c q , ]

Thc bounds on to* from nuclcosynthesis and pulsar stability [ t* > ~ givcn by eq. (22) ] and from accrc- tion of matter onto closcd loops (to* <). -~t~q) trans- lates into the following interval on the largest co- moving mass crossing outside the horizon Hi- ~ dur- ing the inflationary era:

0.88(aY) -3/2< .~r (y )3 /2 <8.8X 105 , (25)

where we have fixed G/z= 10 -6, and 12= 1 and h = ½ for definiteness. The interval (25) corresponds to 9× 10-41~V/'o < :~V[c< 9× 108Mo for case (a), 0.9Mo < M c < l . l X l 0 1 t M o for case (b), and 0.5M o < M ~ < 6 . 6 × 101°Mo for case (c).

The total number of e-folds of inflation is given by

M~ ltl Trm : \~=50 .9+J I n ~ - ° +½ In M 3 ~ , (26)

where 7"~n is the reheat temperature after inflation. Therefore the interval (25) corresponds to an inter- val AA% given by

AA~=4.6+In 7, (27)

i.e. AN¢ = 9.2 for case (a) and A N = 8.5 for cases (b) and (c). This shows that there is room for this mech- anism to work without any fine-tuning of the model

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Volume 237, n u m b e r 3,4 PHYSICS LETTERS B 22 March 1990

parameters. We will now present a particular exam- ple of how this can be achieved.

Suppose cosmic strings are produced in a phase transition breaking a U( I ) gauge symmetry with a scalar potential

V(0) =~:(1012-a2) ' (28)

with x a coupling constant and the string tension Glt~ (a/Mm) 2. We will assume a chaotic initial dis- tribution of the field 0 in the universe, with the only constraint that initially V(00) <M~,~, and study thc evolution of this initial distribution. Fixing arbitrar- ily the beginning of inflation at t = 0, the field evolu- tion during the inflationary epoch is 0( t) =00 exp[ - (~t¢) I/3t]. The end of inflation cor- responds to a field value #6

0~ ~ 4q- O'2-k--~( 1 " 3 - 2 ' 1 / 2 , - - io ) ~8 (29)

for a z << 1. The total number of e-folds of inflation is (in the same approximation)

hVrT 1 2 ~0o-~, (30) which corresponds to the comoving mass Me given by NT=N¢ in (26) with HI'I"'RI.I~O(M31)K23/4. The interval (25) translates then into a range in the num- ber of e-folds of inflation given by

50.8--½1n(ot?)<NT--llnx<55.5+½1n 7 , (31) O~

which means a range of initial conditions ¢o given by

4 0 9 . 1 - 4 ln(ot7) < 0 ~ - 2 In x<446.1 + 4 In --7 . O~

(32)

IDc=O( 1 ), which is the natural strength of a strongly (gauge) interacting field, the range (31) and (32) gives 48.5<NT<57.8 and 19.7<0o<21.6 for case (a), 50.8<N, < 59.4 and 20.8<0o<21.9 for case (b) and 50.6 <NT< 59.2 and 20.1 <0o<21.9 for case (c). If x<< 1, corresponding to the case of a very weakly interacting field, the corresponding values of N-r and 0o z are shifted by ¼ In x and 2 In x, respectively.

Quantum fluctuations during the inflationary pe- riod will produce adiabatic density perturbations, for scales corresponding to comoving masses smaller than Me, with amplitude

~6 From now on we will work in uni ts in which M ~ = 8n.

5p ( 3 2 " ] 1/1 p ~ \~5~3] N3xl/2, (33)

which is O(10 -5 ) only for x=O(10-~5) , while ~p/p~lO 2 for to=O(1). Howcver, because M~< 10 ~ ~-'v/o in all cases, and since adiabatic pertur- bations on scales smaller than the Jeans mass of the dominant dark-matter component are exponentially damped by free-streaming [ 17], density perturba- tions are harmless in baryon-dominated and neu- trino-dominated universes. In the case of cold dark matter there is no exponential suppression of density perturbations, and the condition for not disturbing the scenario of matter accretion onto closed loops at small scales would require ~c~< O(10 - t s ) .

In summary we have shown in this paper that the theory of galaxy formation by matter accretion onto closed loops is consistent with the nucleosynthesis and millisecond pulsar timing stability bounds if the phase transition giving rise to cosmic strings produces a certain amount of cosmological inflation, but not enough to solve the horizon problem. We have illus- trated these general results with the simplest example of cosmic strings produced in a phase transition breaking a U ( 1 ) gauge symmetry and with a chaotic initial distribution of the Higgs field. In that case, and for the most recent numerical simulations [ 11,6] of cosmic string evolution, eqs. (5) and (6), the re- quired number of e-folds of inflation lies in the inter- val 50 < NT< 60, corresponding to the largest scale crossing outside the horizon during the inflationary era being in the range Mo < Mc < 10 t t Mo, and to the initial value of the field in the range 20 < 0o < 22. Of course, in our example only the (weak) anthropic principle [18 ] could help us to understand why we arc living in a universe which had such an initial dis- tribution of fields.

We will conclude with a few final comments: (i) In other, possible more realistic, examples cosmic strings can be associated with grand unified phase transi- tions, the scalar potential can be very different from (28) and the results from (29) and (33) can change reasonably. For instance, the scalar potential can be generated b y radiative corrections and look like a Coleman-Weinberg [ 19 ] or Shafi-Vilenkin-Pi [ 20 ] potential. (ii) As we have observed, the phase tran- sition giving rise to cosmic strings cannot solve the horizon problem. In particular, the interval (25)

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providcs a defect in the necessary, n u m b e r o f e-folds to solve the hor izon p rob lem be tween 17 and 9. This a m o u n t of in f la t ion should be p rov ided by one (or several) previous pcr iod o f inf la t ion. This first epi- sodc o f inf la t ion ( tha t might be d r iven by f u n d a m c n -

tal strings [21 ] ) should, by q u a n t u m fluctuations, fix the ini t ia l values o f the fields in the second episode o f inf la t ion ( tha t p roduc ing cosmic s t r ings) and gcn-

crate ad iaba t ic f luc tua t ions on scales larger t han Me. In that case, these f luc tua t ions on large scales should be compe t ing with those generated by cosmic strings. The final spec t rum of f luc tua t ions should resemble that p roduced by cosmic strings a lone since the den- sity f luctuat ions generated dur ing the inf la t ionary era

are expected to be a lmost scale- independent . ( i i i ) The scenar io d rawn a long this paper could bc, in turn , chal lenged by o ther observable effects o f cosmic strings. O n e is the direct obse rva t ion of gravi ta t ional

lensing by str ings [6 ,22] . An o t h e r is fur ther obser- va t ion of mi l l i second pulsar s tabil i ty or the future obse rva t ion of the b ina ry pulsar PSR 1913 + 16 [ 23 ] t im ing stabil i ty, co r re spond ing to gravi t ional waves of per iods in the range be tween l04 an d l06 years. In par t icu lar the lat ter obse rva t ion could close the al- lowed window given by eq. (25) .

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