Physics Letters B 267 ( 1991 ) 27-29 North-Holland PHYSICS LETTERS B

Nucleosynthesis constraints on cosmic strings

M. Qu i r6 s l Instituto de Estructura de la Materia, Serrano 123, E-28006 Madrid, Spain

Received 18 April 1991

We reanalyze the nucleosynthesis predictions in the presence of a background of gravitational radiation from string loop decay, The comparison with light elements abundance provides upper bounds on the dimensionless string mass per unit length Gp. These bounds are stronger than present bounds coming from the millisecond-pulsar timing stability. For Nv=3 the nucleosynthesis bounds are consistent with the values required for galaxy formation, provided that a < 10 -s (where a is the ratio of the initial size of the loop relative to the horizon size), and are stronger than present bounds coming from the angular distortions in the microwave background radiation. In the limit a~0 we find G/I< [2.6+ 12.9(3-N~) ] × 10 -6.

One of the most appealing features o f the theory of local cosmic strings is that it can provide [ 1-3 ] a nat- ural explanation (by accretion o f matter onto closed loops) of the large-scale structure o f the Universe, provided that Gp> 5 × 10 -7 [2 ] (p being the string mass per unit length). However this scenario can be endangered by an overproduction o f gravitational ra- diation by loop decay into gravitational waves. In particular the background of gravitational waves can be in conflict with the millisecond-pulsar t iming sta- bility [4 -8 ] and with the measured abundance o f primordial helium at the time of nucleosynthesis [9,10].

The millisecond-pulsar constraints on cosmic strings have been recently revised by Bennett and Bouchet [ 11 ] using the high-resolution numerical simulation of string evolution [ 12,13 ]. The latter shows evidence that most loops are formed with a ra- dius l at the time of formation much smaller than t, i.e. ot = l / t << I. Using seven years pulsar-timing data [ 14 ] they show that G# < 2 X 10- 7 for the smallest loops they can resolve (or~ 10-2), while for a ~ 0 (or ~ G/t is suggested by their simulation) G/I < 4 × 10-6, consistent with the string scenario for galaxy formation.

~" Work partially supported by CICYT under contract AE-88- 0040-01.

' E-mailaddress: quiros@cernvm.

In this letter we reexamine the constraints that pri- mordial nucleosynthesis impose on cosmic strings and compare them with those coming from the pulsar- timing stability. These bounds can be substantially improved with respect to previous estimates. First, the big-bang nuclear reaction network has recently been updated [ 15 ] and the neutron life-time deter- mined with much higher accuracy [ 16 ]. Second, there are new estimations of the primordial 4He mass frac- tion, Yp [ 17,7]. Third, we will use the last (high-res- olution) numerical simulation of string evolution [12].

The accumulated gravitational radiation from to~ 10 -30 s (the time when frictional effects become negligible in a standard cosmology with a G U T phase transition [ 9 ] ) to a later time t (during the radiation era) is given by

~gr (/) ~ pgr(t) 128~x/~ t f l r a d ( / ~ - - T X l 0 - 4 K f l n ~ ' (1)

where K = 5.26 + 1.49 [ 12 ] a n d f i s defined by

-- --1/2 f= (~5o ) G~6 (Ol"FTG'lz)3/2--Ol (•Gfl)3/2 (2)

The coefficient ), ( ~ 50) in eq. (2) gives the power loss o f oscillating loops by gravitational radiation ( P = y G # 2) and we use the definitions G/t6---- 106G/t and Yso-= ~,/50.

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Volume 267, number 1 PHYSICS LETTERS B 5 September 1991

At the time of nucleosynthesis, tNS~ 1 S, and in- cluding a suppression factor ~ 1.4 due to the extra red-shift of gravitational radiation when photons are reheated as the temperature falls through low-mass particle thresholds [ 10 ], we obtain

~r~gr= (1.640 + 0 .465) f . (3)

This amount of gravitational radiation will alter the expansion rate of the Universe at the time of nucleo- synthesis and increase the freeze-out temperature which, in turn, leads to an increase in the predicted value of Yo.

The freeze-out temperature Tr is determined by the competition between the expansion rate H and the weak interaction rate (FweakOcG2TS). In turn, H is determined by the product GNeff and ~'2g r through

n o c [ GNefr( 1 "q- ~r'2g r ) ] 1 /2 . ( 4 )

Assuming a thermal bath with photons, electrons, positrons and Nv species of light neutrinos, the mod- ification of Tf produced by t'2gr and by variations with respect to three light neutrinos is given by

8Tr 7 Tf - ~ - ~ [ ( N v - 3 ) + ~ g 2 g r ] . (5)

Eq. (5) leads to a value of Yp given by ~' [ 17,18 ]

Yp =0.228+0.010 In r/lO +0.185 z, -889 .8 889.8

+ 0.072 [t2gr + 0.167 (N~- 3) ] , (6)

where z, is the measured neutron life-time in seconds and r/to the baryon-to-photon ratio in units of 10-1o.

The observational upper limit on D + 3He deter- mines the lower bound on r/lO. Using the numerical calculation of ref. [ 17 ] and the analysis of ref. [ 18 ] we obtain the bound

In r/10 >~ In 2.60 + 0.085 ( N v - 3) + 0.814 £2gr. (7)

Combining (6) and (7) with the neutron life-time [16]

rn =889.8+8.8 s ( 2 a ) , (8)

~t Notice, from the last term in eq. (6), that I2gr=0.17 is equiv- alent to one additional species of light neutrinos. This equiv- alence has been widely used in the literature to compute the nucleosynthesis constraints on cosmic strings [ 10].

and a conservative 2a estimation of the experimental value of Yp [ 17,7 ]

Yp=0.23±O.O1, (9)

we readily obtain the 2a upper bound for the gravi- tational radiation at the time of nucleosynthesis ,2

.Qgr < 0.033 +0.161 ( 3 - N ~ ) . (lO)

Replacing now (3) in (10) we obtain the allowed region in the Glz-a plane. The result is summarized in fig. 1. The upper bound on G# as a function of a is shown for Nv = 2 and 3. The bounds from the mil- lisecond-pulsar stability, using the most recent data analysis [ 19 ], and from the lack of angular distor- tions AT/T in the microwave background radiation [ 20 ] are also shown for the sake of comparison. They are weaker than the nucleosynthesis bounds. For Nv = 2 the upper bound on G# is consistent with the values required for galaxy formation for a < 10 -2 (the smallest loops numerical simulations can resolve).

~2 Notice that the bound (10) is saturated for Nv=3.2, which is the limit imposed by nucleosynthesis to the number of light neutrino species [ 17 ].

G/z

10 -4

10 -4

10 -~

10 -7

1 0 - *

1 0 " 10 -5

, i I I , , . u r ' ' ' ' ' ' " n , , . , , i . I J J I , H I

~ P S - B o u ndl,

kqBR-Bound ~ ~ "~ . . . . . . . . - -

10 4 10 4 10 -= 10 -'

CX

Fig. 1. Upper bounds on G# from nucleosynthesis for Nv= 2 and 3 (solid lines ), from the millisecond-pulsar timing stability (MPS- Bound) and from the lack of angular distortions in the micro- wave background radiation (MBR-Bound). The lower bound on Gp from galaxy formation and the upper limit on a from numer- ical simulations are also presented.

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Volume 267, number 1 PHYSICS LETTERS B 5 September 1991

For Nv= 3 and a = 10 -2 the nucleosynthesis bound gives G p < 9 X 10 -8, which is too small a value for galaxy formation. In fact, values of G/~ consistent with galaxy formation would require a < 10-3. In the limit a ~ 0 we obtain

GII6 <2.61 + 1 2 . 8 9 ( 3 - N v ) . ( 11 )

In conclusion, we have evaluated the nucleosyn- thesis constraints on G/t for local cosmic strings and found they are stronger than the bound from the mil- lisecond-pulsar stability. For Nv = 2 the bound on G/I is consistent with the values required for galaxy for- mation. For Nv=3 the nucleosynthesis bound is stronger than the bound from the angular distortions in the microwave background radiation and is con- sistent with galaxy formation for a < 10 -3.

I thank D. Schramm for urging me to investigate the problem analyzed in the present paper.

Note added. After submission of this paper I found a similar analysis by Bennett and Bouchet [20 ] where the upper limit on G/t from nucleosynthesis is com- puted for the case Nv = 3 and a = 0. The results of both papers agree within the uncertainties due to the ef- fects on big-bang nucleosynthesis of quark-hadron transition induced fluctuations [ 21 ].

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