A natural way out of the conflict between cosmic strings and inflation

Volume 212, number 3 PHYSICS LETTERS B 29 September 1988

A NATURAL WAY OUT OF T H E C O N F L I C T B E T W E E N C O S M I C STRINGS AND I N F L A T I O N

Jun' ichi YOKOYAMA Department of Physics, Faculty of Sciences, The University of Tokyo, Tokyo 113, Japan

Received 13 July 1988

The phase transition which produces cosmic strings is studied in curved spacetime. It is shown that cosmic string formation naturally takes place in the late inflationary stage if the string-forming scalar field is appropriately coupled with the spacetime curvature. As a result the cosmic string scenario of galaxy formation turns out to be compatible with inflation.

Recent developments in the cosmic string scenario has made it a major candidate for the formation mechanism of the large scale structures in the universe [ 1-3 ]. One of the remarkable features of this scenario is that it has only one parameter, namely, the line density of the string p. Much work has been done to estimate its value so that the cosmic string scenario can meet various observational data. A number of independent analyses have shown that the scenario is successful if we consider cosmic strings with dimensionless line density [2-5 ]

G~, =l,t/M~,~ ~ 1 0 - 6 , ( 1 )

where G and Mpl= 1.2× 1019 GeV are the gravita- tional constant and the Planck mass, respectively.

Cosmic strings are one-dimensional topological defects which appear as a result o f a certain class o f cosmological phase transitions predicted by unified theories [6]. The energy scale o f the phase transition which produces cosmic strings with energy density (1) is

v.~ x ~ = O ( 1016 GeV) , (2)

which is a typical grand unification scale. Although the description of string formation at this

energy scale has not yet been established well in the current unified theories of elementary interactions, the essential feature can be understood in terms of the following often-quoted toy model [ 7 ].

Consider a lagrangian with local U ( 1 ) symmetry

~ = - ½ (D~,z)*(D*'z) i p ~'~,; - 4 : , , ~ - - v [ z ] , ( 3 )

V[x] = go[x] ~ IR( I x l 2 - v 2) 2, (3 cont 'd )

where F~,=O~A,,-O~A~, and D u = 0 u - i e A ~ with e the gauge coupling constant whose magnitude at the grand unification scale is typically e = O ( 1 0 - ~ - 1 ). The complex scalar field X has the global min imum on the circle [XI = v and this system allows string so- lutions. The simplest configuration containing a string is the well-known Nielsen-Olesen solution [ 7 ].

It is described by the Kibble mechanism of how cosmic strings appear in the course of the cosmic evolution [6]. In the early universe when the cosmic temperature is high, the potential V[X] acquires a high temperature correction

v [ x ] = Vo[x] + vT[x], VT[Z] ~ (.~-ke2)T2x2 , (4)

and Z settles in the symmetric state Z= 0 provided the temperature is larger than the critical temperature Tc ~ ~ v. As the universe cools down below the critical temperature To, the symmetric state Z= 0 becomes unstable and the phase transition takes place. Since there is no correlation in Z on a scale larger than the horizon size, Z rolls down the potential hill towards random directions with a coherent size smaller than the horizon scale. As a result there typically appears one string within a horizon volume.

The strings thus produced evolve by intersecting with each other to form loops [8]. As a result the scaling solution is realized. That is, there is always about one infinite string in the horizon volume and about one loop of horizon size is formed per horizon

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volume per expansion time. Thus the distribution of string loops is uniquely determined. As the universe becomes matter-dominated, these loops begin to ac- crete dust to form galaxies and clusters. Comparing the present density of galaxies (clusters) with the predicted spectrum of string loop density, the size of loops which form galaxies (clusters) turns out to be

Lg,~5XlO-3(ffdh)Zteq (Ld~O.7(.Qh)Zteq), (5)

where s'2 is the density parameter, h is the Hubble parameter in units of 100 km/s Mpc, and /eq denotes the equipartition time [ 3,5 ].

The above is the standard cosmic string scenario based on the Kibble mechanism and seems to work well in the standard big bang cosmology associated with the simple lagrangian (3). Unfortunately, however, the evolution of the real universe is certainly not so simple. Since the energy scale of the cosmic string relevant to galaxy formation is as high as the grand unification scale, magnetic monopoles may also be produced at the same epoch of string formation. Then the energy density of monopoles much exceeds the critical density and such a universe would soon recollapse [ 9 ].

The usual way out of this difficulty is to dilute them by inflation [ 10 ]. Apart from the above monopole problem, inflation is a necessary ingredient of mod- ern cosmology as the only mechanism to explain the large-scale homogeneity, isotropy, and flatness of the universe [ 10 ].

Inflation, however, dilutes cosmic strings as well as monopoles and we can by no means utilize those strings produced before inflation for galaxy formation. One way to preserve the string density is to have a high enough reheat temperature to restore the symmetry of the string-forming field. This possibility, however, is rather difficult to realize, as seen below.

First, the density fluctuation constraint imposed by the microwave background isotropy sets a stringent constraint on the inflationary energy density U as [111

U l / 4 < 3 X 10-3Mpl . (6)

This means that the reheat temperature TR is bounded by

TR < 1X 1016 GeV. (7)

This constraint alone does not rule out the possibility of symmetry restoration of the string-forming field

after inflation. In realistic inflationary models, however, the inflaton is not coupled very strongly with other fields so that some of the inflaton's energy red- shifts away before it is thermalized. As a result the reheat temperature is generally not high enough for symmetry restoration [ 12 ].

A high enough reheat temperature may be realized if one considers a limited class of inflationary models such as superinflation models in higher dimensional theories [ 12,13 ]. In that case, however, we may not escape from the monopole problem. Furthermore, overproduction of gravitinos may also take place. In- deed, the reheat temperature should be smaller than 10 9 GeV to avoid an excess of gravitinos if their mass is in the weak scale [ 14 ]. Thus, to restore the symmetry in terms of a high reheat temperature seems not only difficult but undesirable.

In view of the above points, several people aban- doned the idea of cosmic string formation by means of a thermal phase transition and inquired the possibility of a non-thermal phase transition [ 15,16 ]. What is done in all of their work is to make a model in which the inflaton is directly coupled with the string-forming field and a phase transition is induced by the inflaton's variation. For example, Vishniac, Olive, and Seckel [ 16 ] introduced an inflaton field ~u and a string-forming field q~, which couples with a U( 1 ) gauge field, with the following potential:

Wo [ I//, ~P] = ~ 4 ( 1 - ~ 3 / a 3 ) 2

+ ¼2( ~2 + v2-a~u2) 2 (8)

where the first term is responsible for the new inflation. The effective mass squared of ~0 reads mZ=J.(vZ-a~2). So, at the beginning of inflation

2 is positive and the symmetric when ~ is small, m~ state ~a= 0 is the absolute minimum of the potential. I f we take aa2> v 2, then m 2 becomes negative as V evolves towards global minimum. Hence we can preserve a large enough string density for galaxy formation by adjusting the model parameters so that the phase transition occurs sufficiently late in the inflationary stage.

The above model, however, seems too unnatural to be acceptable, because what is done in this model is to couple the two fields artificially without being guided by any principle other than saving the string- inflation problem. In the present situation of high energy physics theories by which we should describe the

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early history of the universe, we cannot find what is the inflaton, nor what is the string-forming field. Hence it is not persuasive at all solving the problem by connecting the two unknown fields in such a spe- cific manner. A more natural mechanism is called for. This is what we present here.

The previous discussion of the phase transition given above has all been done in fiat Minkowski spacetime. However, in the early universe when such a phase transition takes place, the expansion of the universe is so fast that the finite curvature effect may play an important role. Thus it is necessary to re-ana- lyze the phase transition in curved spacetime.

In general, the scalar field Z may couple with the spacetime curvature scalar R. That is, the potential of Z may have an additional term Vc [Z] = ½~R IZI 2, where ~is the coupling strength between Z and R. Such a coupling term is, of course, meaningless in Minkowski spacetime where R = 0, but this term may affect or even control the phase transition in the universe [ 17 ]. What we intend to do here is to show that cosmic strings can be saved from inflation by its effect.

First let us consider what is the natural choice of~. There are two values of (which are particularly often considered in the literature, namely: ~= 0, the minimally coupled case, and ~= ~, the conformally coupled case. The latter value is aesthetically more appealing, because this is the only choice that pre- serves the conformal invariance of a massless scalar field. Even if the Z field is minimally coupled with curvature at the tree level, there naturally appears a non-zero value of ~ by radiative corrections. Since Z is coupled with the gauge field, a typical value of induced by quantum corrections is O (e 2 ) = O ( 10 -1 ) [ 17 ]. Thus taking the above two cases into account, the natural choice of ~ turns out to be O ( 10- ~ ).

Now let us investigate the effect of non-minimal coupling on the cosmic string-forming phase transition. To do this, we consider, for definiteness, the following potential:

V[z] = Vo[Z] + Vc[x]

= ~2( Izl 2--/32) 2+ ½~R Izl 2, (9)

assuming that ~ is O ( 10 - ~ ). The scalar curvature R generally decreases in the course of cosmic expansion. For example, it is given in terms of the scale factor a( t ) as

R = 6 ( ? i / a + d 2 / a 2) (10)

in the spatially flat Friedmann-Robertson-Walker spacetime. Thus in the radiation- or matter-dominated universe, it varies as t-2, while in the exact de Sitter phase it is constant: R = 12H 2, where H is the Hubble parameter. In most of the viable inflationary universe models the scalar curvature decreases grad- ually, even in the inflationary stage, keeping R ~ 12H(t) 2. In the early universe the scalar curvature is so large that the symmetry o f z is restored. To be more precise, Z=0 is the global minimum of the potential if ~.R > 2v 2. Hence as long as this inequality is satisfied, the symmetry remains unbroken, even if the cosmic temperature drops due to inflationary expansion. Thus a phase transition will not take place until the curvature becomes sufficiently small in the late or after the inflationary stage.

Now let us examine whether it is possible to preserve a large enough string density under a reasona- ble choice of parameters. Clearly, the above mechanism of curvature-induced phase transitions is applicable to all classes of viable inflationary universe models. Here, as a typical inflationary model, we consider chaotic inflation [ 18 ] with an inflaton having the potential

U[O]=½m2~ 2. (11)

The inflaton's mass m is usually taken to be rn~ 10 TM GeV in order to produce an appropriate density fluctuation 8p /p~ 10 -4 after inflation [ 19]. In our scenario, however, the relic density fluctuation is not responsible for galaxy formation so that the value m ~ 1014 GeV should be regarded as an upper bound.

According to the chaotic inflationary scenario, the natural initial value of 0 at the Planck epoch is (~o~M~l/m with the potential energy U[0o] ~ M ~ . At this epoch Z may also be randomly distributed but the non-minimal coupling term prevents this field from serving as an inflaton [ 20 ]. Hence ~'s potential energy U[0] will soon dominate the energy density of the universe and inflationary expansion sets in. In this stage ¢( t ) , a( t ) , and R ( t ) are approximately given by

~(t) = 0 o - ( m M p , / Z x / ~ ) t , (12)

a( t )=aoexp{ (Zn /M~, , ) [q )g -O( t )2]} , (13)

R ( t ) = ( 16~rmZ / M~,~)O( t ) Z - 2m 2

~- (16~rm2/MZe,)q)(t) 2 , (14)

respectively. Thus the scalar curvature decreases as

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evolves towards 0 = 0 . Therefore Z = 0 becomes unstable when ~R <2/) 2, o r

02 <2M2m/)2 /167r~rn2=_O2 . (15)

In exponentially expanding spacetime, the effect of quan tum fluctuation may play an impor tant role in determining the evolution of a scalar field. Hence even if the inequality ( 15 ) is satisfied, the phase o f z will not be fixed at once. The effect of quan tum fluc- tuations on the evolution ofx becomes negligible [ 21 ] when I V" [Z= 0][ > H 2, or

02< 32M2plV2/4rc( 12~+ 1 )m2=-02 . (16)

Thus the phase of Z is practically fixed when 0 satis- fies 0 2 < 0 < 0~. For ~ = O ( 10 -~) and / )= 10 ~6 GeV, we yield from these inequalities

0 ~- 30,,~Mp, ( 10 '4 G e V / m ) ---0s,, (17)

for the phase-fixing epoch ofz. After the phase transition there will appear about one string in a volume corresponding to the horizon volume at this epoch.

In order for the galaxy formation by string loops to be successful, there should exist at least one long string in the horizon volume when the horizon scale is ~ L 0 so that loops of this size can be properly formed. This is equivalent to demanding that the phase transi t ion

should take place at or after the length scale corresponding to the mean separation of present galaxies leaves the de Sitter horizon. More properly, the number of e-folding n from 0 = 0s~ till the end of inflation, when 0 -~ Mpl/x~ 8g, should satisfy

2 2 2 n = ( 2 ~ / M p , ) ( O s t - M p , / 8 ~ ) < ~ O ( 5 0 ) , (18)

o r

0sl ~< 3Mpl. (19)

This turns out to be a constraint on 2, which is

2 < l O - 2 ( m / l O 14 GeV) 2 . (20)

As long as the above inequality is satisfied, the inflaton's energy dominates the energy density of the universe and hence it determines the scalar curvatm e throughout the inflationary phase, as we have implic- itly assumed. The inequality (20) is by no means a stringent constraint. In fact, if we consider non-supersymmetric theories, the natural value of Z given by the radiative corrections is 2 ~ (e2/47r) 2 ~ 10 -4. In

supersymmetric theories 2 can be taken even smaller.

Thus we may conclude that the constraint (20) is naturally satisfied.

In conclusion, we have considered the cosmic string-forming phase transi t ion in curved spacetime and studied the effect of non-min imal coupling between the spacetime curvature and the scalar field Z. As a result the cosmic string scenario of galaxy for- mat ion has been shown to be compatible with infla- t ion under a natural choice of model parameters.

The author is grateful to Professor K. Sato for his cont inuous encouragement and to Dr. K. Maeda, Dr. M. Izawa and Dr. M. Kawasaki for valuable discussions.

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A natural way out of the conflict between cosmic strings and inflation