Cosmic strings from superstrings

Volume 232, number 1 PHYSICS LETTERS B 23 November 1989

COSMIC STRINGS FROM SUPERSTRINGS ~r

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

Received 6 September 1989

In the low-energy theory, of four-dimensional superstrings the spontaneous breakdown of a pseudo-anomalous [J ( 1 ) can trigger (but not necessarily) the formation of global and /o r local cosmic strings. When cosmic strings are formed, supersymmetry is broken inside the string cores with a positive cosmological constant and an exponential growth of the cosmic scale factor. Strings are conformally stretched with the inflationary expansion, never enter the physics horizon and can have no observable (e.g. optical) effects. Off-string regions are in a (radiation-dominated) FRW universe. Most of our observable universe would still be inflating today unless a late (conventional) period of inflation transforms an off-string region into our present universe. The required amount of inflation coincides with the one needed to solve the usual flatness and horizon problems of the standard cosmological model. We consider, as an example, the chaotic inflation associated with the low-energy breaking ( ~ 1 TeV) of supersymmetry.

Four-dimensional strings [1] are (unique) ap- pealing candidates to unify all fundamental forces in nature, including gravity. However, the absence of unambiguous low-energy predictions that could be testable at present or future accelerators is one of the main troubles of string theories. A very interesting possibility is that string theories could have well defined astrophysical signatures (making it feasible to determine superstring parameters by astronomical observations) through the possible appearance of two kinds of macroscopic string-like objects: fundamental superstrings and vortex lines associated with sym- metry breaking (cosmic strings).

Fundamental superstrings (compactified from ten- dimensional heterotic strings) have a v e ~ large ten- sion [2], Glt=gZ/32n 2, and could endanger, when crossing the horizon, the isotropy of the microwave background radiation [3]. Fortunately such strings become [2], at the QCD scale, boundaries of axion domain walls and rapidly disappear. Thus we do not expect either cosmological troubles or astrophysical signatures from this kind of fundamental strings.

In compactifications of heterotic strings to four dimensions, and in four-dimensional fermionic strings,

¢r Work partly supported by CICYT under contract AEN88-0040. Electronic mail: imtma27 @ emdesicl

modular invariance prevents the appearance of anomalies [4] in semi-simple gauge group factors but says nothing about the presence of a pseudo-anomalous Ua( ! ) "~. Therefore, in the absence of a sym- metry protecting the U ( 1 )'s from gauge anomalies, the presence of a pseudo-anomalous Ua(1) can be considered as a generic feature in theories with U ( 1 ) gauge factors and, in particular, in those theories leading to the standard model with extra U( 1 )'s in four dimensions ~2

The presence of a pseudo-anomalous Ua( 1 ) triggers [7,8] a Fayet-Iliopoulos (FI) term in the four- dimensional effective (low-energy) theory. The F1 term breaks supersymmetry except if there are (as is usually the case [6] ) D- and/:-flat directions in the configuration space of scalar fields. In that case U~( 1 ) is spontaneously broken and global cosmic strings can

~1 By pseudo-anomalous U~( 1 ) it is understood a U( 1 ) gauge factor such that the trace of the corresponding charge over the fermionic fields is different from zero. This apparent anomaly is cancelled by the Grcen-Schwarz counterterms coming from string-loop corrections [ 5 ]. By redefinition of the U ( 1 ) charges one can readily show that there can be at most one pseudo- anomalous [J ( 1 ).

u2 This kind of theory contains the best candidates to realistic four-dimensional fennionic models and orbifold compactifications [6 ].

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form [5]. Such strings have been shown to act as gravitational lenses and produce a net angle between thc direction of polarization of the two images [5 ]. This optical activity could make it conceivable to determine superstring properties by astronomical observations.

In this note wc will explore the cosmological consequences of such a pseudo-anomalous U , ( 1 ) trig- gering the presence of cosmic strings in superstring theories. The first ( tr ivial) obscrvation is to rcmind the reader that a spontancously broken pseudo- anomalous U~( 1 ) does not neccssarily lcad to the ex- istcnec of global cosmic strings ~3. The simplest example to illustrate this s tatement is the compactif i- cation of the Sp(32) / ff.~ heterotic string on a Calabi - Yau manifold with standard embedding of the spin connection [7]. In this compactif ication 0 ( 3 2 ) is broken down to O ( 2 6 ) ® U ( 1 ) with the following spectrum of light mat ter fields: b~[26~+l__2]+bt2126_~+12] . U ( 1 ) is pseudo- anomalous if and only i f b ~ #b~2. In that casc an FI tcrm is gcncratcd by onc-loop string cffects and U ( 1 ) is spontaneously broken at the supersymmetr ic vacuum. The potential has a supersymmetr ic m in imum at (26~ > = (26_ ~ ) =0 , which is energetically pre- ferred by the D-term corresponding to the 0 ( 2 6 ) gauge factor. At this m in imum only the singlets, # and ~ ( i=1 ..... b ~ ; j = l . . . . , b~2), do contribute to the U ( 1 ) D-term,

b 6 D,~: 8= 2 ' \ M c f l Z 1.0'12+ Y~ I~12

i = 1 j = l

(1)

(where M~ and .,'v/~ are the string and compactif ica- lion scales, respectively), and to the superpotential (allowcd by gauge invariance ~4)

f= ~ CJ),'..,J.r,~ i~ thi.,5 ,6 ,, ....... ~ ..'~" wj~...v./. , (2) r im2

with arbitrarv constants C~ ',--,j~ . il,...,in "

l f b ~ > b~2 the supersymmetr ic vacuum is given by

< ~ > = 0 ,

t,. 1 M 8 Z I # 1 2 - b t 2 ) ,=l = ~5n2 (bl, -7-6 - (3) M~

In fact the hypersurface defined by (3) satisfies the conditions of supersymmetiT: ( f> = < Of/O0 ~> =

( 0 J Y 0 ~ ) = ( D , ) = 0 . Here we can use the well known result that FI, ($2) =Y" and FI~ (S, ) = 1, n > 2, to infer that only if b, t = 1 (i.e. b~2 = 0) a global cosmic string is formed. I f b~z> b~ the supersymmetr ic degenerate vacuum can be constructcd in the same way, and only ifb~z= 1 (i.c. b,~ = 0 ) a global cosmic string will arise. However, this casc docs not appear in this kind of compactifications, wherc b~l >/1 #5. To con- clude, in the O ( 2 6 ) ® U ( 1 ) Calabi-Yau modcl only the case of one generation (b,, = 1, b~, = 0) gives rise to topologically stable vortex solutions. In the other cases the string loops are contractible through the manifold of degenerate supersymmetr ic vacua.

A second observation is that in (orbifold) compactifications with non-standard embcdd ing [ (0,2)- models] extra local U( 1 )'s, apart from the pseudo- anomalous one, can appear. In that case, as it was stressed in ref. [ 10], the breaking of the pseudo- anomalous U ( 1 ) can be accompanied by the breaking of local U ( 1 ) factors. Dcpending on the manilbld of dcgenerate vacua global and /o r local cosmic strings can appear. Examples of both cases were given in ref. [10].

In gcncral, in a theory with a pseudo-anomalous U~( 1 ) and a local gauge factor fq, thc contribution of the D-terms to the scalar potential ~6 can bc written as

V I ) ~ i j 2 i q,2 [thtT.oejt,5 ~2 ~g~ Qi l .Oi lZ-q 2 +26,~xv. - , w;: , (4)

where g~ and g , are the gauge coupling constants, q ~ M S / M ~ [7,8] as in eq. (1) , Q~ is the U~(1) charge of ,¢)g and T~J arc thc generators of f¢ in the rcprcscntation of the scalar fields. I f there are D- and

~3 Actually strings are formed only when the manifold ~t of degenerate vacua contains unshrinkable loops, and they arc then classified by their first homotopy group FI~ (.a¢) [9 ].

r.4 Terms with n = I in (2) are prevented by thc masslessness of the singlets at the string level.

~5 There exists at lcast the ( l , l ) - form defining '.he K~ihlcr manifold.

N6 The F-term contribution to the scalar potential, V~., dcpcnds on the Kiihler potential G which is not well known in low-cn- ergy superstring theories.

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Ff la t directions [6 ], the minimum of the potential ~'. ; 2 = q 2 stays along the hypersurface ;Q~ 10;[ and

U,( 1 ) is spontaneously broken along some flat field direction 0 that takes a vacuum expectation value (VEV) i01 = q, while the other field directions 0t have vanishing VEVs. They will determine the manifold .//~' of degenerate supersymmetric vacua ~v. If Flt(.dd)# 1 global and/or local cosmic strings are formed.

Cosmic strings are endless tubes of false vacuum and so the VEV of the field 0 is, in cylindrical coordinates (r, 0, z),

O=qg(r) exp(in0) , (5)

where the string is along the z-axis and r is the distance from the string. The integer n is a topological number which gives the magnetic flux along the string. The function g(r) approaches linearly to zero as r ~ 0 and rises to its asymptotic value, equal to one, over a characteristic distance ~8 d equal to the inverse Higgs boson mass: c ~ 1/rlg~. So at the string core 101 < r/ and outside the string core 101 goes reD' quickly to its limiting value q. The existence of the string core is a topological feature that is due to the non-triviality ofthe background of degenerate vacua. Wc can therefore consider 0 as a background field, taking the value (5), and the other field directions 01 taking generic VEVs

10;12=C;r12[1 - - g a ( r ) ] , ( 6 )

where the value of the constants C~ (that can be equal to zero in particular cases) depends on the specific form of the scalar potential which depends, in turn, on the Kfihler potential G.

At the core of the string all standard model quantum numbers can be broken by the 0; fields. In particular the electric charge can be broken, producing an electric current up and down the string core carried by the Goldstone bosons and behaving like a super- conducting wire [ 12]. The behaviour of supercon- ducting strings can generate observable effects when

passing through astrophysical magnetic fields [13], and even alter the predictions of primordial nucleo- synthesis [ 14 ]. However, not only the electric charge can be broken at the corc of the string. Other quantum numbers, such as color, can bc broken and pro- ducc a color current also carried by the Goldstone bosons. This color current along the core of the string might also have some particle-physics and cosmological consequences, e.g. proton decay induced by baryon number violating processes with strong interac- tion sized strengths catalyzed by the string cores and the generation of the observed baryon to entropy ra- tio via cosmic strings induced fluctuations in ba~on number [15].

The breaking of the gauge quantum numbers at the core of the string is a modcl-depcndent feature. We will focus here on a modcl-indcpendent characteristic of superstring models: the breaking of supersym- mete,, by D- and F-terms, at the core of the string. In fact, at the "minimum" (5), (6) thc VEVs of Da, D,~ and OG/O0; arc all proportional to rfl( 1 _g2) and the energy at the minimum, Vmin, is proportional to r/n( 1 _g2)2. Supersymmetry is broken by the VEVs (5), (6) at the string core and the corresponding goldstino is proportional to the gauginos ).~, ).~ and to the chiral fermions Z~, with coefficients proportional to (D~) , ( D ~) and (OG/Oq),), ;espectively. Now, since the low-encrgy thcory of closed superstrings is N = 1 supergravity coupled to matter, the vacuum energy plays the role of a cosmological constant.

The equation of motion for thc field 0z in the presence of the background field 0 givcn by (5) ~9 and a gravitational background gU,, with a Friedmann- Robertson-Walker (FRW) scale factor a (t) is given by ~o

0V 0"~-a-ZV20,+3H0,+ ,ss-~ = 0 , (7)

o~9/

where H=~t/a is the Hubble parameter, given by

w7 In realistic models the field ¢ is a standard model singlet while the fields ¢~; can (should) carD' standard model charges.

~s Of course g-=0 is an exact solution to the equation of motion of the field 0, while g=- 1 is a veD good approximation outside the string core [ 11 ]. In this way, the step function g = 0 (g= 1 ) for r ~< 6 ( r> 6) can be used as a valuable approximation to the exact solution for the field ¢~.

,9 We are using here for simplicity the step approximation for the function g(r).

,~o Using the step approximation, the scale factor becomes r-independent though it takes different values inside and Outside the siring core. Introducing an explicit r-dependence would make the equation of motion much more involved. The step approximation is enough for our purposes here.

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2 8zr II ='3,,~p. (P~ +Po) " (8)

In eq. (8) p~~ T 4 is the radiation energy density and

Po = ½ ~ ,0~+ ~ Y, (V,0l) 2+ V(O), (9) l l

the energy associated to the system of fields. The field 0¢ will evolve rapidly ,~1 towards its min imum ( 0 : ) , given by (6) with 0 V/30~= O, satisfying trivially the equation of mot ion (7). Inside the string core, pq,= ~ , i n ~ q 4 will dominate H 2, eq. (8) , at a given (critical) temperature T~. At the corresponding time, t~, the region of space with background field 0 given by (5) will enter a period of exponential expansion, with a cosmic scale factor a( t )=exp(Ht) , where H2=(8~z/3M~,I)Vmi,. Outside the string core V,.~,=0, the Hubble parameter is dominated by the radiation energy density p~ and a (t) ~c t 1/2, characteristic of a radiat ion-dominated universe.

We summarize now the main cosmological fea- tures accompanying cosmic strings based on a spontaneously broken pseudo-anomalous U( 1 ) in superstring theories. The phase transition associated with 0 happens at the critical temperature 7"~~gaq and with a correlation length ~~ 7"g -1~ 6. At the t ime of formation, strings have a core of false vacuum (5) with a typical radius ~ d and a typical distance between neighbouring string segments of order ~ 6. The off- string regions are in a radiat ion-dominated FRW universe (with zero cosmological constant ) while inside the core of the strings supersymmetry is broken and there is a non-vanishing cosmological constant. At the inner core of false vacuum, r<< J, the vacuum energy density dominates over the thermal radiation energy density at a temperature T ~ r / ~ 7"~ and the corresponding region of space enters an (inflationary) exponential expansion a( t )=exp(Ht) with a Hubble c o n s t a n t H ~ r l 2 / M w . Cosmic strings therefore seem to lead naturally to inflation .-~2

We will study now the evolution of the string in a De Sitter expanding background. Assuming that the

w~t Since the gauge coupling constants are O( 1 ) the phase transition associated with Ot is not inflationary and the amount of generated inflation negligible at the cosmic scale.

~ 2 This inflation will erase (except in off-string regions) any pre- existing fluctuation produced by previous periods of intla- tion, as for instance that driven by fundamental strings [ 16 ].

string movcs in the (x, y)-plane, its trajectory is de- scribed by the function y=y(x , r) , where r is the con- formal time, - o o < r < 0, given by d r = dt/a (t). In the case of a De Sitter universe, r = - ( 1 /H) e -m. The equation of motion for small perturbations on a straight string is [ 9 ]

y - 2 z - l ~ - y " = 0 , (10)

where 9 - 8y/3r and y' =- ~y/Sx. We can consider now the plane wave solution ansatz y = h ( r ) sin kx, where k is the co-moving wave number, and h ( r ) is given by the (real) function

h ( z ) = ( - " r ) 3 / 2 Z 3 / 2 ( - k r ) . ( 1 1 )

In eq. ( 11 ) Z3/2 is a linear combinat ion of first, .13/> and second, N3/2, order Besscl function

2 "]1/2(sin kr - c o s k r ) , .h/~(-k~)=\-(7-~) \ ~ N 3 / 2 ( - k r ) = ( - -~kr)'/2(sin kr+ c°s kr'] ~ 7 - z J " (12)

One can see from ( 1 1 ), (12) that when the physi- cal wave length 2 is much larger than the physics horizon (i.e. 2 H = - 2 7 r / k r >> 1 ), (11) reduces to a con- slant and the string does not move in co-moving coordinates. In that case the string is being conformally stretched with the exponential expansion of its core (its length growing as a(t) ) and will never enter the physics horizon. Since the volume of the string core scales as a ( l ) 3, the string density rapidly becomes negligible and any direct observable effects are excluded. Let us remark that the process of inflation of the string core is related to the very existence of the string and (because the string is topological in nature) looks like an endless phenomenon.

The off-string regions, of size ~ ~~ r/- ~, do not feel any cosmological constant and thus tbllow a FRW evolution. In that case the growth of the scale factor a (t) from the Planck time, 4,1 ~ Mb3' ~ 5.4 + 10-44 s, to the present time, to~8.8×106ntm, is given by a(to)/a(tpl)=(to/t~q)2/3(teq/tm) w2, where teq~ 1.6 X 1054tpl is the equilibrium time. In this way, the off- string regions, of size r/-l, are stretched to ~ 4 × 103~1-1. Taking for instance the natural vahie ~1~ 10-~Mm, an off-string region becomes today ~ 1 cm! This would mean that mosl of our observable

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universe today (of radius~ 1028 cm) would still be inflating ~13. The only way out would be a late (conventional) period of inflation providing the extra 28 orders of magnitude (65 e-folds) necessa~ to trans- form an off-string region into our observable universe today ~14. We should stress here that the required amount of inflation (roughly) coincides with the one needed to solve the other problems of the standard cosmological model: the flatness and horizon problems.

Where can such an ordinary period of inflation come from? One possibility is the phase transition associated with the spontaneous breaking of the pseudo- anomalous U ( 1 ). This possibility has been fully ex- plored in refs. [ 17,10]. There it was shown [ 10] that requiring enough inflation to solve all cosmological problems, the quantum fluctuations of the inflaton field would generate (for realistic values of the string parameters M~ and M~) too large density perturbations, 8p/p>> 10-4, inconsistent with galaxy formation. The remaining possibility is a (late) period of inflation after the pseudo-anomalous U ( I ) phase transition. There arc many ways of generating such a necessary period of inflation. We will consider here one of these possibilities: that associated to the breaking of supersymmetry.

Supersymmetry breaking is a very poorly known phenomenon at the string level. However, it is rela- tively well understood at the level of the low-energy effective theory. There, non-perturbative effects can break supersymmetD' at the desired scale. For instance, gluino condensation [ 18 ] can arise at a temperature Tcond < 7 c and break supersymmetry by giv- ing a Majorana mass term to the gluinos. This supersymmetry breaking is transferred to the scalar sector by one-loop gauge interactions and triggers a supersymmetry breaking mass term for all scalar fields (specially for those carrying color quantum number, i.e. the squarks). In this way, the D- and F-flat directions appearing in four-dimensional theories will take,

#13 The number of e-folds of inflation experienced by the inner core of the string (r<< ~) from the time of the phase transition until today is ~llto where to is the present age of the universe. Thus taking q~ 10-~Mp~ one obtains tlto~ 1059 e-folds of inflation!

~a This inflation would erase any pro--existing fluctuation [16] in off-string regions.

at scales below T~,,nd, a non-supersymmetric mass term. We will consider one of those flat directions, q/, and show it can provide, assuming chaotic inflationary initial conditions [19], the number of e-folds necessary for an off-string region to become our observable universe today.

The scalar potential for the (real) q/direction is then

V(q/) = ½m2q/2 , (13)

where rn~<l TeV is the scale of supersymmetry breaking. We will assume [ 19] a chaotic initial distribution of the field q/in the universe, with the only constraint that (initially) V(q/o) <M41 ~5 and study the evolution of this initial distribution.

The equation of motion for the field q/(t) during the inflationary epoch is solved by qJ( t )=q/o-

( 1 / 3zr) 1/2m (t/tel), where q/o is the initial (chaotic) value of the field distribution, and the potential ( 13 ) is inflationary for values of the field q/>>q/r = ½ ( 1/370 l/2Mp~. The corresponding (quasi-exponential) scale factor of the universe is

a(t) = a ( 0 ) exp(Ilot- ~rn2t 2) , (14)

where llo-H(t=O)=2m(~n)l/2q/o/Mpl and the (total) number of e-folds, from q/o to q/r, is given by N.r= 27c( q/o/ Mpl ) 2 ,16

After the inflationary period the universe will be dominated by coherent field oscillations. Because the field q/has gauge interactions (it is a squark or a slep- ton field) all the vacuum energy at the end of inflation is converted into radiation. The universe reheats in less than an expansion time to a temperature "/RH LI 1/2 ~ 1/2 aArl/2 1011

.el f ~ frt 2v~ Pl ~ GeV. Density fluctuations at the time a scale q/<')-eq re -

enters the horizon after the inflationary period, when the universe is still radiation-dominated, are given by [21]

8p (16"~ 1'2 p - ~ k , ~ ] (y r / /Mp) ) JN~., ( 1 5 )

where Aq~41 + ln (2 /Mpc) is the number of c-folds of inflation from the time the scale 2 crosses out the

~ s I t is readily shown that the condition V<<M~,) leads to (~2(t) ) 1/2 evolving as a homogeneous field [20].

~6 We are assuming here that ~,2 >> ~,~. In fact N l > 68 implies q/o > 3.3Mpb which satisfies the former condition.

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horizon during the inflationary period until the end of inflation. For galactic scales ~7, ).~0.3 Mpc, ,~3.~ 40 and 5 p / p ~ 10 -15, for m ~ 1 TeV.

In short, if ~'o>~ (3-4)-V/e~ the potential (13) can produce enough inflation to make an off-string region to become our observable universe today, but it is not powerful enough to yield a spectrum of density perturbations of any relevance for galaxy formation. An additional source of density fluctuations would therefore be necessary. A recently analyzed possibility [20] is associated with the existence of another (genuine) chaotic inflation ~/, with a potential 17"= 12~/4 ( ) ~ I 0 - '3) or P=~rh2¢ 2 (rh~ 10-SMo,). In that case the combined cvolution of q/and ~ can generate density fluctuations of the correct magnitude (Sp/p ~ 10-4) under some special initial conditions of the fields ~Uo and @0. However, it is hard to figure out how the field @, with such a potential, could arise naturally in the low-energy limit of (compactified) four-dimensional superstring theories.

In conclusion, we have seen that cosmic strings do not necessarily accompany a spontaneously brokcn pseudo-anomalous U( 1 ) in the low-energy theory of four-dimensional superstrings. When cosmic strings are formed, the string core has broken supersymmetry and a positive cosmological constant, making it enter a De Sitter exponential expansion. The off-string regions evolve in a (radiation-dominated) FRW universe. Most of our universe would still be inflating today unless there is a period of conventional inflation capable of converting an off-string region into our presently observable universe. A mechanism of chaotic inflation associated with the breaking of supersymmetry has been considered. The observable (e.g. optical) effects of cosmic strings associated with the spontaneous breaking of the pseudo-anomalous U ( 1 ) in string theories disappcar.

~7 The scale dependence of (15) is logarithmic and so ve~, weak. Density perturbations are almost scale-independent.

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Cosmic strings from superstrings