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STRINGS, TEXTURES, INFLATION

arXiv:hep-ph/9203214v1 16 Mar 1992SU-ITP-92-8August 14, 2018STRINGS, TEXTURES, INFLATIONAND SPECTRUM BENDINGAndrei Linde 1Department of Physics, Stanford University, Stanford, CA 94305ABSTRACTWe discuss relationship between inflation and various modelsof production of density inhomogeneities due to strings, globalmonopoles, textures and other topological and non-topologicaldefects. Neither of these models leads to a consistent cosmologi-cal theory without the help of inflation.

However, each of thesemodels can be incorporated into inflationary cosmology. We pro-pose a model of inflationary phase transitions, which, in additionto topological and non-topological defects, may provide adiabaticdensity perturbations with a sharp maximum between the galaxyscale lg and the horizon scale lH.1Onleavefrom:LebedevPhysicalInstitute,Moscow.E-mail:LINDE@PHYSICS.STANFORD.EDU

Modern cosmology has two apparently contradictory goals. First of all,one should explain why our universe is so flat, homogeneous, isotropic, whyit does not contain such defects as monopoles, domain walls, etc.

Then oneshould explain why our universe is not perfectly flat, homogeneous, isotropic,and why the deviation from perfection is so small ( δρρ ∼δTT ∼10−5).In our opinion, it is somewhat risky to suggest various solutions to thesecond problem without having at least an idea of how to solve the first one.Fortunately, a possible solution to the first problem is well known, it is in-flation. Now, ten years after the inflationary scenario was suggested, we stilldo not find any fundamental flaws in it.

Nor have we found any alternativesolution to the first problem. (Actually, we are speaking about ten differ-ent problems, which are solved simultaneously by one simple scenario; for areview see [1].) The only alternative solution of the homogeneity problemwhich I am aware of is based on quantum cosmology [2]: It is possible thatthe probability of quantum creation of the universe, like the probability oftunneling with bubble production, is particularly large for spherically sym-metric universes.

However, even if it is true, we will still need somethinglike inflation to make the newly born universe not only symmetric but alsoextremely large.It came as a great bonus to inflationary cosmologists when it was realizedthat inflation may solve not only the first problem, but the second one aswell: Quantum fluctuations produced during inflation lead to generation ofadiabatic density perturbations with a flat (almost exactly scale independent)spectrum [3]. Thus, inflation offers the most economical possibility to solveall cosmological problems by one simple mechanism.However, Nature is not very economical in the number of different cosmo-logical structures.

Even though it may be possible to explain the origin of allthese structures by one basic mechanism, one should keep in mind all otherpossibilities, such as adiabatic and isothermal perturbations with a non-flatspectrum, strings [4], global monopoles [1, 5], textures [6], late time phasetransitions [7], etc.Adiabatic perturbations with a flat spectrum is a natural consequenceof many inflationary models. Therefore, there is a tendency to identify per-2

turbations with a flat spectrum with inflation2. Some authors do it just forbrevity, to distinguish between inflation without strings and textures andinflation with string and textures [8].

However, some other authors, whenadvertising new types of density perturbations, represent them as a real alter-native to inflationary cosmology, see, e.g., [9]. Even though such an attitudeis understandable, we do not think that it is well motivated.

None of thenew mechanisms of generation of density perturbations offer any solution tothe first, basic cosmological problem, without the help of inflation. On theother hand, each of these mechanisms can be successfully implemented inthe context of inflationary cosmology [1].

Moreover, inflationary cosmologyoffers many other possibilities which do not exist in the standard hot BigBang theory. The list of new possibilities includes, in particular, productionof exponentially large domains with slightly different energy density insideeach of them, or with the same density but with different number density ofbaryons, or with the same density and composition but with different am-plitudes of density perturbations, etc.

[1, 10]. Therefore, even if later it willbe found that in addition to flat adiabatic perturbations one needs stringsor textures, or perturbations with a local maximum in the spectrum, or evensomething much more exotic, this by itself will not be a signal of an incon-sistency of inflationary cosmology.

On the contrary, it is much easier to findnew sources of density perturbations in the context of inflationary cosmologythan in the standard hot Big Bang theory.Moreover, it seems that for a consistent realization of the theory of per-turbations produced by strings or textures, or by any other mechanism, onestill needs inflation. Indeed, even if some as yet unspecified quantum gravityeffects at the Planck density will be able to solve the homogeneity, isotropy,horizon and flatness problems without any use of inflation (which does notseem likely), it is hardly possible that they will be able to solve the primordialmonopole problem, since the monopoles in GUTs are produced at a tempera-ture Tc ∼1015 GeV ∼10−4 Mp, when quantum gravity effects are negligible.Therefore, until we learn how to solve the primordial monopole problem with-out inflation, the theory of strings and textures produced in high temperaturephase transitions in non-inflationary cosmology will remain inconsistent.2Actually, even in simplest models the spectrum of perturbations is not absolutely flat.For example, in the theory λϕ4, density perturbations grow by about 1/3 in the intervalfrom the galaxy scale to the scale of horizon.3

The situation with strings and textures is not quite trivial even withinthe context of inflationary cosmology. Indeed, a typical critical temperatureof a phase transition in cosmologically interesting theories of strings andtextures is about 1016 GeV.

It is extremely difficult (though not impossible)to reheat the universe up to such temperature after inflation [1, 11, 12], and itwill require some additional fine tuning of parameters to make the reheatingtemperature smaller than the critical temperature of the phase transitionproducing monopoles.Of course, one may pretend that the problem does not exist, or suggestto postpone its discussion because of “our overwhelming ignorance” [13].One may even claim that the theory of textures by itself “seems to matchthe explanatory triumphs of inflation” [9]. A more constructive approachis to face the problem and to use specific possibilities offered by inflation torescue strings and textures.

Indeed, inflationary cosmology provides a simplemechanism which may lead to cosmological phase transitions during or afterinflation, without any need of reheating of the universe. This mechanismis particularly natural in the chaotic inflation scenario [10, 11, 14, 15], butit works in other versions of inflationary cosmology as well [16], and it caneasily explain why strings, textures and some other useful topological ornon-topological defects are produced, whereas monopoles are not.To make our arguments more clear and, simultaneously, to discuss somenontrivial examples of density perturbations in the standard Big Bang theoryand in inflationary cosmology, we will consider here a simple O(N)-symmetricmodel of an N-component field ⃗Φ = {Φ1, Φ2, ..., ΦN}, N > 1, with the effec-tive potentialV (Φ) = −12 M2ΦΦ2 + λΦ4 (Φ2)2 + Vo ,(1)where Φ2 = Φ21 + Φ22 + ... + Φ2N;Vo =M4Φ4 λΦ is added to keep the vacuumenergy zero in the absolute minimum of V (Φ).

At high temperature, theO(N) symmetry of this theory is restored, Φ = 0.As the temperaturedecreases, a phase transition with symmetry breaking takes place. Soon afterthe phase transition, the length of the isotopic vector ⃗Φ acquires the value v =MΦ/√λΦ.

However, its direction may differ in causally disconnected regions.Later on, all vectors tend to be aligned inside each causally connected domain(i.e. inside each particle horizon), but they cannot become aligned outside the4

particle horizon. Consequently, the field ⃗Φ always remains inhomogeneouson the horizon scaleRH = 2 H−1 =vuut3 M2p2 πρ .

(2)For N = 2 this model describes global strings [4], N = 3 global monopoles[1], [5], N = 4 textures [6]. For larger N, there are no topological defects.However, for all N > 1 this model produces additional density perturbationswith almost flat spectrum [17, 18, 19, 20] due to misalignment of the Gold-stone field on the horizon scale.

A somewhat better estimate of the scale ofinhomogeneity is just H−1, since it still takes some time of the order of H−1until the field becomes homogeneous inside the horizon. A typical variationof the scalar field Φ on this scale can be estimated by√2 v. This leads toan estimate of the energy density in the gradients of the fieldsδρ ∼8 v2πρ3 M2p.

(3)The relative amplitude of energy density of these fluctuations does not de-pend on the horizon scale,δρρ ∼8 πv23 M2p. (4)It gives the desirable value δρρ ∼10−5 for v ∼1016 GeV.

One should be alittle bit more accurate though, since if gradients of the scalar field are thesame everywhere, then the energy density is strictly homogeneous. A moredetailed study of this question performed in [19] shows that at large N theamplitude of density perturbations is suppressed by an additional factor of√N, which slightly increases the necessary value of v.We wish to note again, that we are discussing now essentially the samemechanism which is used in the theory of global strings, monopoles andtextures.

However, this mechanism is more general since it does not requireexistence of any topological defects. Moreover, one may expect that in manycases the contribution of the topological defects to density perturbations willbe subdominant, since the probability of their formation in this model issuppressed by a large combinatorial factor.5

Now let us study the phase transition in this model in more detail. Thephase transition occurs due to the high temperature corrections to the effec-tive potential (1) [21],∆V (Φ) = λΦT 212 (N + 2) .

(5)This gives the critical temperature of the phase transitionTc = vs12N + 2 . (6)This quantity is of the same order as v, it does not depend on λΦ andit only weakly depends on N.Thus, in order to haveδρρ ∼10−5 in thismodel, one should have the phase transition at an extremely large tempera-ture Tc ∼v ∼1016 GeV.

This temperature is close to the grand unificationscale, but the critical temperature in grand unified models typically is oneorder of magnitude smaller, Tc ∼1015 GeV [22]. This brings us back to thetwo problems mentioned in the beginning of the paper.

In non-inflationarycosmology all our achievements will be brought down by the basic inconsis-tency of the cosmological theory and by the primordial monopole problem.In inflationary cosmology it is very hard to reheat the universe up to thetemperature T > 1016 GeV [1, 11, 12], and if we are able to do it, we will getall our monopoles back.Still, if one is prepared to pay a high price for a new type of perturbations,then in inflationary cosmology this can at least be achieved in an internallyconsistent way. The most obvious possibility is to add to the model (1) someother fields (but not gauge fields!

), interacting with the field Φ with a couplingconstant much larger than λΦ. This will reduce the critical temperature inthis model.

Then one tunes the reheating temperature to make it smallerthan the critical temperature in grand unified models but larger than thecritical temperature in the extended model (1).There also exists another, less trivial possibility, which has certain advan-tages [10, 11, 14, 15]. Let us assume that the inflaton field ϕ, which drivesinflation, interacts with the field Φ with a small coupling constant g2:V (ϕ, Φ) = m2ϕ4 ϕ2 −12 M2ΦΦ2 + λΦ4 (Φ2)2 + g22 ϕ2Φ2 + Vo .

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The inflaton mass should be sufficiently small, mϕ <∼1013 GeV, to makestandard adiabatic perturbations produced during inflation smaller than 10−5[1]. The effective mass of the field Φ at Φ = 0 depends on ϕ:M2Φ(ϕ) = −M2Φ + g2ϕ2 .

(8)At large ϕ the effective mass squared M2Φ(ϕ) is positive and bigger than H2.This means that at the beginning of inflation, when the inflaton field ϕ isextremely large, the O(N) symmetry is restored, Φ = 0. However, at ϕ = ϕc,whereϕc = MΦg=√2λΦ vg,(9)the phase transition with the 0(N) symmetry breaking takes place.

Thus, theinflaton field ϕ plays here the same role as the temperature in the standardtheory of phase transitions. However, if it does not interact with the Higgsfields, which are responsible for the symmetry breaking in GUTs, its variationwill not lead to any phase transitions with monopole production.

Moreover,even if monopoles are produced, their density exponentially decreases afterthe phase transition. Strings and textures will lead to important cosmologicaleffects even if the universe inflates by a factor e50 after the phase transition,whereas even much smaller inflation makes monopoles entirely harmless.A similar mechanism may work in the new inflationary scenario as well[16].

However, in chaotic inflation this mechanism is much more natural andefficient, since the variation of the field ϕ in this theory is very significant. Atthe last stages of inflation in our model, when the structure of the observablepart of the universe was formed, and after the inflation, when the inflatonfield continued rolling down to the minimum of the effective potential, itdecreased by an extremely large value ∼3.5 × 1019 GeV, from 3.2 Mp to 0[1].

Correspondingly, the effective mass squared M2Φ(ϕ) changes by about10g2M2p.One cannot easily (without the help of supersymmetry) take the constantg2 in (7) arbitrarily large, since radiative corrections would induce an extraterm in the expression for the effective potential [1]:δV (ϕ)∼N M4Φ(ϕ)64π2ln M2Φ(ϕ)M2Φ∼N (g2ϕ2 −M2Φ)264π2ln g2ϕ2 −M2ΦM2Φ7

∼N g464π2ϕ2 −ϕ2c2 ln ϕ2 −ϕ2cϕ2c. (10)In order to have δρρ <∼10−5 for standard inflationary adiabatic perturbationsgenerated in a theory with such an effective potential, one should take g2 <∼10−6.

For definiteness, let us take g2 ∼10−7 and mϕ ∼1012 GeV. In thiscase we avoid large inflationary perturbations and make the additional termN g464π2ϕ2 −ϕ2c2 ln ϕ2−ϕ2cϕ2csmaller thanm2ϕϕ22at the last stages of inflation.During the rolling of the field ϕ from ϕ ∼3 Mp to ϕ = 0, the effective masssquared (8) of the field Φ changed from −M2Φ + (3 × 1016)2 GeV2 to −M2Φ.This leads to the desired phase transition for MΦ < 3 × 1016 GeV, which iscertainly the case in the theory under consideration.This scenario has a very interesting feature.

The wavelength of perturba-tions, which are produced when the inflaton field is equal to ϕ, later growsup to l(ϕ) ∼exp(2πϕ2/M2p) cm due to inflation and subsequent expansionof the universe [1]. These perturbations in our model have a flat spectrum,but only on a scale l >∼l(ϕc) ∼exp(2πϕ2c/M2p) cm.If the phase transi-tion occurs at ϕc > 3.2 Mp, all inhomogeneities produced by the gradients ofthe field Φ will be stretched away from the observable part of the universe.Perturbations produced at 2.8 Mp < ϕc < 3.2 Mp will form the superlargestructure of the observable part of the universe, but they will not contributeto perturbations on the galaxy scale.

Finally, if the phase transition occursat ϕc ≪2.8 Mp, all observational consequences will be the same as if it werethe ordinary finite temperature phase transition in the theory (1).For example, for v ∼1016 GeV, g2 ∼10−7, λΦ = 0.5, the phase transitionoccurs at ϕc ∼3.2 × 1019 GeV, and the corresponding density perturbationsonly appear on the horizon, at l >∼exp(2πϕ2c/M2p) ∼1028 cm. For λΦ = 0.1,perturbations with flat spectrum appear on all cosmologically interestingscales.

By increasing mϕ up to about 2×1012 GeV we obtain a mixture of thestandard inflationary perturbations with δρρ ∼10−5 and the new ones. Theamplitude of each of these components is controlled by mϕ and v respectively,and the cut-offof the spectrum of the new perturbations on small scalesis controlled by ϕc = √2λΦ v/g.

The model describes both inflation andthe phase transition in the theory (1), and it does not contain any couplingconstants smaller than 10−7−10−6. Since such coupling constants are known8

to exist even in the standard model of the electroweak interactions, this modelseems to be sufficiently natural.Note, that at the end of inflation in this model, the field ϕ still is extremelylarge, ϕ ∼Mp/5 ∼2 × 1018 GeV. Therefore, for λΦ <∼10−2 the phasetransition may occur even after the end of inflation.This indicates thatthe mechanism discussed in this paper is rather general, and that the fieldtriggering the phase transition may differ from the field which drives inflation.However, chaotic inflation provides a particularly natural framework for therealization of this mechanism for generation of density perturbations.Now let us try to see whether our model admits any interesting general-izations and/or simplifications.

An obvious idea is to replace the field Φ ofthe model (1) by the fields Φ and H of the SU(5) model [23, 1, 14]. Thereexist some reasons to do it.

First of all, even though the models with brokenglobal O(N) symmetries may have interesting cosmological implications, sofar there is no independent reason to consider them as a part of a realistictheory of elementary particles. Moreover, recently it was argued that quan-tum gravity corrections may induce large additional terms in the effectivepotential (1), which will break the O(N) invariance [27].

If these terms leadto existence of one preferable direction in the isotopic space, they eliminatetextures. But if they lead to existence of several minima of equal depth,then domain walls will be produced after the phase transitions.

This is acosmological disaster, which can be avoided only if the universe inflates morethan e60 times after the phase transition.Meanwhile, the model (7), the fields Φ being interpreted as the SU(5)Higgs fields in a gauge theory with a spontaneous symmetry breaking, rep-resents the simplest semi-realistic model of chaotic inflation [1, 14]. In suchmodels we do not have textures, but we may have exponentially large strings.In addition to that, we may have density perturbations with a spectrumwhich grows on large scales, and then becomes flat on some scale l > lc.Indeed, one can easily show that standard inflationary density perturbationsgenerated in the model (7) on scale l <∼exp(2πϕ2c/M2p) cm are smaller thanperturbations on scale l >∼exp(2πϕ2c/M2p) cm.

The reason is the following.The amplitude of perturbations produced when the inflaton field was equal9

to ϕ is given by [1]δρρ = 485s2π3V 3/2(ϕ)M3p V ′(ϕ) ,(11)where V ′(ϕ) is the derivative of the effective potential, which is responsible forthe speed of rolling of the field ϕ. Before the phase transition V ′(ϕ) = m2ϕϕ.After the phase transition the field Φ rapidly falls down to the minimum ofits effective potential at v2(ϕ) = M2Φ(ϕ)/λΦ.

Effective potential along thistrajectory is given byV (ϕ) = m2ϕ2 ϕ2 +14λΦhM4Φ −(M2Φ −g2ϕ2)2i. (12)In the vicinity of the critical point ϕ ∼ϕc = MΦ/g the modification ofthe effective potential by the last term is very small, being quadratic in(M2Φ −g2ϕ2).

However, the derivative of the effective potential at ϕ < MΦ/gchanges more considerably,V ′(ϕ) = m2ϕϕ + g2 ϕλΦ(M2Φ −g2ϕ2) . (13)The last term in (13) increases the speed of rolling of the field ϕ and decreasesthe amplitude of density perturbations generated after the phase transition.The role of this term depends on the values of parameters; in some cases itjust decreases the amplitude of small scale density perturbations, in someother cases it may even lead to an abrupt end of inflation at the momentof the phase transition [24].

Other examples of the spectra bending due toinflationary phase transitions can be found in [10, 25].Even if there are no phase transitions and topological defects in our model(e.g., if the sign of the term M2ΦΦ2 in (1) is positive), inflation may stillproduce density perturbations with a non-flat spectrum [26, 12]. To give asimple example, let us consider an effective potential, which, for sufficientlylarge ϕ, looks as follows:V (ϕ) = m2ϕϕ22 1 + ϕ2ϕ2oln ϕϕo−34ϕ2ϕ2o!,(14)where ϕo is some normalization parameter.

Such potentials may appear inthe theory (7) and in GUTs at MΦ ≪gϕ due to radiative corrections to V (ϕ)10

[1]. This potential has a minimum at ϕ = 0, and it grows with an increaseof the field ϕ everywhere except the point ϕ = ϕo, where V ′ = 0.

Thispotential may lead to small density perturbations produced during inflationat ϕ ≪ϕo or at ϕ ≫ϕo, but, according to (11), it has a very high peakδρρ >∼1 corresponding to fluctuations produced near ϕ = ϕo. The height ofthe peak can be decreased by the decrease of the coefficient in front of thelogarithmic term in (14).

The length scale corresponding to the maximumin the spectrum is controlled by the parameter ϕo. One should emphasize,that there is nothing special in this potential; even much more complicatedpotentials of this type are often discussed in the standard electroweak theory[1, 28].In the presence of the phase transition (of any type, not necessarily leadingto textures), the effect discussed above is much more natural and pronounced.Let us consider for example the effective potential which may appear in themodel (7) due to one loop radiative corrections (10):V (ϕ) = m2ϕ2 ϕ2 + Ng464π2ϕ2 −ϕ2c2 ln ϕ2 −ϕ2cϕ2c+ Vo .

(15)The same potential may appear in GUTs, with N being replaced by someother combinatorial coefficient. Note, that the second term has a maximumat the critical point ϕ = ϕc.

This term may lead to a large modification ofV ′. In the vicinity of the critical point, at ∆ϕ = ϕ −ϕc ≪ϕc, the effectivepotential (15) can be represented in the following form:V (ϕ) = m2ϕ2 ϕ2 + N g4ϕ2c16π2 (∆ϕ)2 ln 2∆ϕϕc+ Vo .

(16)Let us take m2 ≪N g4ϕ2c8π2 . After some elementary algebra, one can show thatin this case the first derivative of the effective potential reaches its minimumat some point ϕm = ϕc + ∆ϕ, where ∆ϕϕc ≈12 e−3/2 ∼0.1.

This minimum willcorrespond to a maximum in the spectrum of δρρ at ϕ = ϕm. If one wishes,for example, to make this maximum C times higher than the value of δρρ atϕ = ϕc, one should takem2 ≈N g4ϕ2c8π2×C2(C −1) e−3/2 .

(17)11

This is quite consistent with our assumption that m2 ≪N g4ϕ2c8π2for C >∼2.For relatively small C, the width of the peak is comparable with ∆ϕ ∼0.1 ϕc.For large C, the effective width of the peak becomes smaller. If the phasetransition in our model occurs at ϕc ∼2.8 Mp, then the maximum of thespectrum will be displaced at some point ϕm in the interval 2.8 Mp < ϕ <3.2 Mp.

In other words, this spectrum grows C times and then decreasesagain when the length scale changes from the galaxy scale to the scale ofhorizon. But this is exactly the type of the spectrum which is necessaryin order to improve the theory of formation of large scale structure of theuniverse in the cold dark matter model, and, simultaneously, to avoid anexcessively large anisotropy of the microwave background radiation!A detailed theory of this effect strongly depends on relations betweenparticle masses and the Hubble parameter at the moment of the phase tran-sition.

In some cases, one may obtain a sharp maximum in the spectrumeven without any account of the one loop corrections to the effective poten-tial [25]. However, the fact that the one loop contribution δV typically has amaximum near the point of the phase transition (10), makes this effect moregeneral.To summarize our results, inflationary phase transitions in GUTs and/orin the theories with a global symmetry breaking may lead to production ofadiabatic perturbations with a spectrum which looks almost flat on very largescale, which has a relatively narrow maximum at l(ϕm) ∼exp(2πϕ2m/M2p)cm, and which decreases even further at l < l(ϕc) ∼exp(2πϕ2c/M2p) cm.

Inaddition to these perturbations, on a scale l > l(ϕc) one may have the samestrings, global monopoles, topological and nontopological textures whichwould be produced by the ordinary high temperature phase transitions. Theamplitudes of inhomogeneities of all types and the values of length scalesl(ϕm) and l(ϕc) are controlled by values of masses and coupling constants inthe underlying theory of elementary particles.One should remember also, that even the ordinary high temperaturephase transition in the SU(5) model occurs by a simultaneous productionof domains of four different phases: SU(3) × SU(2) × U(1), SU(4) × U(1),SU(3) × (U(1))2 and (SU(2))2 × (U(1))2 [29].

There is no reason to expectthat inflationary phase transitions are simpler. On the contrary, one may ex-12

pect that the inflaton field couples differently to different scalar fields, whichleads to a series of phase transitions at some critical values φi of the infla-ton field. This may create an exponential hierarchy of cosmological scalesl(ϕi) ∼exp(2πϕ2i /M2p) cm.These examples show that inflation is extremely flexible and can incor-porate all possible mechanisms of generation of density perturbations [1].This does not mean that one can do whatever one wishes; for example, it isvery hard to avoid the standard prediction that the density of the universeshould be equal to critical.

One should always keep in mind the possibilitythat some new observational data will contradict all versions of inflationarytheory, including the versions with strings, textures and non-flat spectra ofperturbations. However, this did not happen yet.

On the contrary, one maybe afraid that those who wish to have simple and definite predictions to becompared with observations will feel embarrassed by the freedom of choicegiven to us by inflation. But do we ever have too much freedom?

In order tounderstand the present situation, let us try to draw some parallels with thedevelopment of the standard model of electroweak interactions.The four-fermion theory of weak interactions had a very simple structure,but it was unrenormalizable. In the late 60’s many scientists hoped that onemay make sense out of this theory by performing a cut-offat the Planckenergy.

However, this theory had problems even on a much smaller energyscale (violation of the unitarity bound on the electroweak scale). just as allnon-inflationary models had problems with monopoles on scales much smallerthan Mp.

The model suggested by Weinberg and Salam [30] is not particu-larly simple; just remember how long it takes to write a complete Lagrangian.It has anomalies, which are to be cancelled. It has about twenty adjustableparameters, some of which look extremely unnatural.

For example, most ofthe coupling constants are of order 10−1, whereas the coupling of the electronto the Higgs boson is Ge ∼2×10−6. Therefore this model remained relativelyunpopular for 4 years after it was proposed, until it was realized that gaugetheories with spontaneous symmetry breaking are renormalizable [31].

Soonafter that, Georgi and Glashow proposed an O(3)-symmetric theory of elec-troweak interactions, which was much simpler and which did not have anyanomalies [32]. Then, neutral currents were discovered, which could not bedescribed by this model.

However, nobody considered the problems of the13

simplest models of electroweak interactions as a signal of a general failureof gauge theories with spontaneous symmetry breaking. The possibility todescribe neutral currents and the existence of many adjustable parametersmade the Weinberg-Salam model flexible enough to survive and to win.

Andnow, instead of speaking about fine-tuning of parameters of this theory, weare speaking about measuring their values.This teaches us several interesting lessons. First of all, as stressed byAbdus Salam many years ago, Nature is not economical in particles, it iseconomical in principles.

After we learned the principle of constructing con-sistent theories of elementary particles by using spontaneously broken gaugetheories, there was no way back to the old models. Similarly, after we learnedthe principle of constructing internally consistent cosmological models, it isvery hard to forget it and return to old cosmological theories.

In order toaccount for the abnormal smallness of density perturbations, δρρ ∼10−5, oneshould consider theories with very small coupling constants. But the samehappened in the electroweak theory, when, in order to account for the small-ness of the electron mass, it was necessary to adjust the coupling constantGe to be 2 × 10−6.

Nobody says now that this is a fine tuning. We expectthat in the next decade observational cosmology will provide us with a largeamount of reliable data.

It will be absolutely wonderful if the simplest versionof inflationary cosmology with adiabatic perturbations with a flat spectrumis capable of describing all these data. One should continue investigationof this attractive possibility.

However, there is no special reason to expectthat the future cosmological theory will be much simpler than the theory ofelectroweak interactions, with its twenty adjustable parameters. On the con-trary, it may be extremely difficult to suggest any simple theory which woulddescribe new observational data, see e.g.

[33]. One should be prepared tomost radical changes in the cosmological theory, but one should try to makethese changes without breaking the internal consistency of the theory.

Webelieve that the large flexibility of inflationary cosmology in providing manydifferent sources of density perturbations is a distinctive advantage of thistheory.It is particularly interesting that most of the sources of nontriv-ial density perturbations are related to inflationary phase transitions. Thisshould make it possible to use cosmology as a powerful tool of investigationof the phase structure of the elementary particle theory.14

I am grateful to Rick Davis, Kris Gorski, Lev Kofman and David Schrammfor many useful discussions.This research was supported in part by theNational Science Foundation grant PHY-8612280.15

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