SU-ITP-92-8 |

February 1, 2021 |

STRINGS, TEXTURES, INFLATION

AND SPECTRUM BENDING

Andrei Linde ^{1}^{1}1On leave from: Lebedev
Physical Institute, Moscow.
E-mail: [email protected]

Department of Physics, Stanford University, Stanford, CA 94305

ABSTRACT

We discuss relationship between inflation and various models of production of density inhomogeneities due to strings, global monopoles, textures and other topological and non-topological defects. Neither of these models leads to a consistent cosmological theory without the help of inflation. However, each of these models can be incorporated into inflationary cosmology. We propose a model of inflationary phase transitions, which, in addition to topological and non-topological defects, may provide adiabatic density perturbations with a sharp maximum between the galaxy scale and the horizon scale .

Modern cosmology has two apparently contradictory goals. First of all, one should explain why our universe is so flat, homogeneous, isotropic, why it does not contain such defects as monopoles, domain walls, etc. Then one should explain why our universe is not perfectly flat, homogeneous, isotropic, and why the deviation from perfection is so small ( ).

In our opinion, it is somewhat risky to suggest various solutions to the second 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 inflation. Now, ten years after the inflationary scenario was suggested, we still do not find any fundamental flaws in it. Nor have we found any alternative solution to the first problem. (Actually, we are speaking about ten different problems, which are solved simultaneously by one simple scenario; for a review see [1].) The only alternative solution of the homogeneity problem which I am aware of is based on quantum cosmology [2]: It is possible that the probability of quantum creation of the universe, like the probability of tunneling with bubble production, is particularly large for spherically symmetric universes. However, even if it is true, we will still need something like inflation to make the newly born universe not only symmetric but also extremely large.

It came as a great bonus to inflationary cosmologists when it was realized that inflation may solve not only the first problem, but the second one as well: Quantum fluctuations produced during inflation lead to generation of adiabatic density perturbations with a flat (almost exactly scale independent) spectrum [3]. Thus, inflation offers the most economical possibility to solve all cosmological problems by one simple mechanism.

However, Nature is not very economical in the number of different cosmological structures. Even though it may be possible to explain the origin of all these structures by one basic mechanism, one should keep in mind all other possibilities, such as adiabatic and isothermal perturbations with a non-flat spectrum, strings [4], global monopoles [1, 5], textures [6], late time phase transitions [7], etc.

Adiabatic perturbations with a flat spectrum is a natural consequence
of many
inflationary models. Therefore, there is a tendency to identify
perturbations
with a flat spectrum with inflation^{2}^{2}2Actually, even in
simplest models
the spectrum of perturbations is not absolutely flat. For example,
in the
theory , density perturbations grow by about 1/3 in
the
interval from the galaxy scale to the scale of horizon.. Some
authors do it
just for brevity, to distinguish between inflation
without strings and textures and inflation with string and textures
[8].
However, some other authors, when advertising new types of density
perturbations, represent them as a real alternative to inflationary
cosmology,
see, e.g., [9]. Even though such an
attitude is understandable, we do not think that it is well
motivated. None
of the new mechanisms of generation of density perturbations offer
any
solution to the first, basic cosmological problem, without the help
of
inflation. On the other hand, each of these mechanisms can be
successfully
implemented in the context of inflationary cosmology [1].
Moreover,
inflationary cosmology offers many other possibilities which do not
exist in
the standard hot Big Bang theory. The list of new possibilities
includes, in
particular, production of exponentially large domains with
slightly
different energy density inside each of them, or with the same
density but
with different number density of baryons, or with the same density
and
composition but with different amplitudes of density perturbations,
etc.
[1, 10]. Therefore, even if later it will be found that in
addition
to flat adiabatic perturbations one needs strings or textures, or
perturbations with a local maximum in the spectrum, or even
something much
more exotic, this by itself will not be a signal of an
inconsistency of
inflationary cosmology. On the contrary, it is much easier to find
new
sources of density perturbations in the context of inflationary
cosmology
than in the standard hot Big Bang theory.

Moreover, it seems that for a consistent realization of the theory of perturbations produced by strings or textures, or by any other mechanism, one still needs inflation. Indeed, even if some as yet unspecified quantum gravity effects at the Planck density will be able to solve the homogeneity, isotropy, horizon and flatness problems without any use of inflation (which does not seem likely), it is hardly possible that they will be able to solve the primordial monopole problem, since the monopoles in GUTs are produced at a temperature GeV , when quantum gravity effects are negligible. Therefore, until we learn how to solve the primordial monopole problem without inflation, the theory of strings and textures produced in high temperature phase transitions in non-inflationary cosmology will remain inconsistent.

The situation with strings and textures is not quite trivial even within the context of inflationary cosmology. Indeed, a typical critical temperature of a phase transition in cosmologically interesting theories of strings and textures is about GeV. It is extremely difficult (though not impossible) to reheat the universe up to such temperature after inflation [1, 11, 12], and it will require some additional fine tuning of parameters to make the reheating temperature smaller than the critical temperature of the phase transition producing monopoles.

Of course, one may pretend that the problem does not exist, or suggest to postpone its discussion because of “our overwhelming ignorance” [13]. One may even claim that the theory of textures by itself “seems to match the explanatory triumphs of inflation” [9]. A more constructive approach is to face the problem and to use specific possibilities offered by inflation to rescue strings and textures. Indeed, inflationary cosmology provides a simple mechanism which may lead to cosmological phase transitions during or after inflation, without any need of reheating of the universe. This mechanism is particularly natural in the chaotic inflation scenario [10, 11, 14, 15], but it works in other versions of inflationary cosmology as well [16], and it can easily explain why strings, textures and some other useful topological or non-topological defects are produced, whereas monopoles are not.

To make our arguments more clear and, simultaneously, to discuss some nontrivial examples of density perturbations in the standard Big Bang theory and in inflationary cosmology, we will consider here a simple -symmetric model of an N-component field , , with the effective potential

(1) |

where ; is added to keep the vacuum energy zero in the absolute minimum of . At high temperature, the symmetry of this theory is restored, . As the temperature decreases, a phase transition with symmetry breaking takes place. Soon after the phase transition, the length of the isotopic vector acquires the value . 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 the particle horizon. Consequently, the field always remains inhomogeneous on the horizon scale

(2) |

For this model describes global strings [4], global monopoles [1], [5], textures [6]. For larger , there are no topological defects. However, for all this model produces additional density perturbations with almost flat spectrum [17, 18, 19, 20] due to misalignment of the Goldstone field on the horizon scale. A somewhat better estimate of the scale of inhomogeneity is just , since it still takes some time of the order of until the field becomes homogeneous inside the horizon. A typical variation of the scalar field on this scale can be estimated by . This leads to an estimate of the energy density in the gradients of the fields

(3) |

The relative amplitude of energy density of these fluctuations does not depend on the horizon scale,

(4) |

It gives the desirable value for GeV. One should be a little bit more accurate though, since if gradients of the scalar field are the same everywhere, then the energy density is strictly homogeneous. A more detailed study of this question performed in [19] shows that at large the amplitude of density perturbations is suppressed by an additional factor of , which slightly increases the necessary value of .

We wish to note again, that we are discussing now essentially the same mechanism which is used in the theory of global strings, monopoles and textures. However, this mechanism is more general since it does not require existence of any topological defects. Moreover, one may expect that in many cases the contribution of the topological defects to density perturbations will be subdominant, since the probability of their formation in this model is suppressed by a large combinatorial factor.

Now let us study the phase transition in this model in more detail. The phase transition occurs due to the high temperature corrections to the effective potential (1) [21],

(5) |

This gives the critical temperature of the phase transition

(6) |

This quantity is of the same order as , it does not depend on and it only weakly depends on . Thus, in order to have in this model, one should have the phase transition at an extremely large temperature GeV. This temperature is close to the grand unification scale, but the critical temperature in grand unified models typically is one order of magnitude smaller, GeV [22]. This brings us back to the two problems mentioned in the beginning of the paper. In non-inflationary cosmology all our achievements will be brought down by the basic inconsistency of the cosmological theory and by the primordial monopole problem. In inflationary cosmology it is very hard to reheat the universe up to the temperature GeV [1, 11, 12], and if we are able to do it, we will get all 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 internally consistent way. The most obvious possibility is to add to the model (1) some other fields (but not gauge fields!), interacting with the field with a coupling constant much larger than . This will reduce the critical temperature in this model. Then one tunes the reheating temperature to make it smaller than the critical temperature in grand unified models but larger than the critical temperature in the extended model (1).

There also exists another, less trivial possibility, which has certain advantages [10, 11, 14, 15]. Let us assume that the inflaton field , which drives inflation, interacts with the field with a small coupling constant :

(7) |

The inflaton mass should be sufficiently small, GeV, to make standard adiabatic perturbations produced during inflation smaller than [1]. The effective mass of the field at depends on :

(8) |

At large the effective mass squared is positive and bigger than . This means that at the beginning of inflation, when the inflaton field is extremely large, the symmetry is restored, . However, at , where

(9) |

the phase transition with the symmetry breaking takes place. Thus, the inflaton field plays here the same role as the temperature in the standard theory of phase transitions. However, if it does not interact with the Higgs fields, which are responsible for the symmetry breaking in GUTs, its variation will not lead to any phase transitions with monopole production. Moreover, even if monopoles are produced, their density exponentially decreases after the phase transition. Strings and textures will lead to important cosmological effects even if the universe inflates by a factor 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 and efficient, since the variation of the field in this theory is very significant. At the last stages of inflation in our model, when the structure of the observable part of the universe was formed, and after the inflation, when the inflaton field continued rolling down to the minimum of the effective potential, it decreased by an extremely large value GeV, from to [1]. Correspondingly, the effective mass squared changes by about .

One cannot easily (without the help of supersymmetry) take the constant in (7) arbitrarily large, since radiative corrections would induce an extra term in the expression for the effective potential [1]:

(10) | |||||

In order to have for standard inflationary adiabatic perturbations generated in a theory with such an effective potential, one should take . For definiteness, let us take and GeV. In this case we avoid large inflationary perturbations and make the additional term

This scenario has a very interesting feature. The wavelength of perturbations, which are produced when the inflaton field is equal to , later grows up to cm due to inflation and subsequent expansion of the universe [1]. These perturbations in our model have a flat spectrum, but only on a scale cm. If the phase transition occurs at , all inhomogeneities produced by the gradients of the field will be stretched away from the observable part of the universe. Perturbations produced at will form the superlarge structure of the observable part of the universe, but they will not contribute to perturbations on the galaxy scale. Finally, if the phase transition occurs at , all observational consequences will be the same as if it were the ordinary finite temperature phase transition in the theory (1).

For example, for GeV, , , the phase transition occurs at GeV, and the corresponding density perturbations only appear on the horizon, at cm. For , perturbations with flat spectrum appear on all cosmologically interesting scales. By increasing up to about GeV we obtain a mixture of the standard inflationary perturbations with and the new ones. The amplitude of each of these components is controlled by and respectively, and the cut-off of the spectrum of the new perturbations on small scales is controlled by . The model describes both inflation and the phase transition in the theory (1), and it does not contain any coupling constants smaller than . Since such coupling constants are known to exist even in the standard model of the electroweak interactions, this model seems to be sufficiently natural.

Note, that at the end of inflation in this model, the field still is extremely large, GeV. Therefore, for the phase transition may occur even after the end of inflation. This indicates that the mechanism discussed in this paper is rather general, and that the field triggering the phase transition may differ from the field which drives inflation. However, chaotic inflation provides a particularly natural framework for the realization of this mechanism for generation of density perturbations.

Now let us try to see whether our model admits any interesting generalizations and/or simplifications. An obvious idea is to replace the field of the model (1) by the fields and of the model [23, 1, 14]. There exist some reasons to do it. First of all, even though the models with broken global symmetries may have interesting cosmological implications, so far there is no independent reason to consider them as a part of a realistic theory of elementary particles. Moreover, recently it was argued that quantum gravity corrections may induce large additional terms in the effective potential (1), which will break the invariance [27]. If these terms lead to existence of one preferable direction in the isotopic space, they eliminate textures. But if they lead to existence of several minima of equal depth, then domain walls will be produced after the phase transitions. This is a cosmological disaster, which can be avoided only if the universe inflates more than times after the phase transition.

Meanwhile, the model (7), the fields being interpreted as the Higgs fields in a gauge theory with a spontaneous symmetry breaking, represents the simplest semi-realistic model of chaotic inflation [1, 14]. In such models we do not have textures, but we may have exponentially large strings. In addition to that, we may have density perturbations with a spectrum which grows on large scales, and then becomes flat on some scale . Indeed, one can easily show that standard inflationary density perturbations generated in the model (7) on scale cm are smaller than perturbations on scale cm. The reason is the following. The amplitude of perturbations produced when the inflaton field was equal to is given by [1]

(11) |

where is the derivative of the effective potential, which is responsible for the speed of rolling of the field . Before the phase transition . After the phase transition the field rapidly falls down to the minimum of its effective potential at . Effective potential along this trajectory is given by

(12) |

In the vicinity of the critical point the modification of the effective potential by the last term is very small, being quadratic in . However, the derivative of the effective potential at changes more considerably,

(13) |

The last term in (13) increases the speed of rolling of the field and decreases the amplitude of density perturbations generated after the phase transition. The role of this term depends on the values of parameters; in some cases it just decreases the amplitude of small scale density perturbations, in some other cases it may even lead to an abrupt end of inflation at the moment of the phase transition [24]. Other examples of the spectra bending due to inflationary 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 in (1) is positive), inflation may still produce density perturbations with a non-flat spectrum [26, 12]. To give a simple example, let us consider an effective potential, which, for sufficiently large , looks as follows:

(14) |

where is some normalization parameter. Such potentials may appear in the theory (7) and in GUTs at due to radiative corrections to [1]. This potential has a minimum at , and it grows with an increase of the field everywhere except the point , where . This potential may lead to small density perturbations produced during inflation at or at , but, according to (11), it has a very high peak corresponding to fluctuations produced near . The height of the peak can be decreased by the decrease of the coefficient in front of the logarithmic term in (14). The length scale corresponding to the maximum in the spectrum is controlled by the parameter . One should emphasize, that there is nothing special in this potential; even much more complicated potentials 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 leading to textures), the effect discussed above is much more natural and pronounced. Let us consider for example the effective potential which may appear in the model (7) due to one loop radiative corrections (10):

(15) |

The same potential may appear in GUTs, with being replaced by some other combinatorial coefficient. Note, that the second term has a maximum at the critical point . This term may lead to a large modification of . In the vicinity of the critical point, at , the effective potential (15) can be represented in the following form:

(16) |

Let us take . After some elementary algebra, one can show that in this case the first derivative of the effective potential reaches its minimum at some point , where

(17) |

This is quite consistent with our assumption that for . For relatively small , the width of the peak is comparable with . For large , the effective width of the peak becomes smaller. If the phase transition in our model occurs at , then the maximum of the spectrum will be displaced at some point in the interval . In other words, this spectrum grows times and then decreases again when the length scale changes from the galaxy scale to the scale of horizon. But this is exactly the type of the spectrum which is necessary in order to improve the theory of formation of large scale structure of the universe in the cold dark matter model, and, simultaneously, to avoid an excessively large anisotropy of the microwave background radiation!

A detailed theory of this effect strongly depends on relations between particle masses and the Hubble parameter at the moment of the phase transition. In some cases, one may obtain a sharp maximum in the spectrum even without any account of the one loop corrections to the effective potential [25]. However, the fact that the one loop contribution typically has a maximum near the point of the phase transition (10), makes this effect more general.

To summarize our results, inflationary phase transitions in GUTs and/or in the theories with a global symmetry breaking may lead to production of adiabatic perturbations with a spectrum which looks almost flat on very large scale, which has a relatively narrow maximum at cm, and which decreases even further at cm. In addition to these perturbations, on a scale one may have the same strings, global monopoles, topological and nontopological textures which would be produced by the ordinary high temperature phase transitions. The amplitudes of inhomogeneities of all types and the values of length scales and are controlled by values of masses and coupling constants in the underlying theory of elementary particles.

One should remember also, that even the ordinary high temperature phase transition in the model occurs by a simultaneous production of domains of four different phases: , , and [29]. There is no reason to expect that inflationary phase transitions are simpler. On the contrary, one may expect that the inflaton field couples differently to different scalar fields, which leads to a series of phase transitions at some critical values of the inflaton field. This may create an exponential hierarchy of cosmological scales cm.

These examples show that inflation is extremely flexible and can incorporate all possible mechanisms of generation of density perturbations [1]. This does not mean that one can do whatever one wishes; for example, it is very hard to avoid the standard prediction that the density of the universe should be equal to critical. One should always keep in mind the possibility that some new observational data will contradict all versions of inflationary theory, including the versions with strings, textures and non-flat spectra of perturbations. However, this did not happen yet. On the contrary, one may be afraid that those who wish to have simple and definite predictions to be compared with observations will feel embarrassed by the freedom of choice given to us by inflation. But do we ever have too much freedom? In order to understand the present situation, let us try to draw some parallels with the development 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 one may make sense out of this theory by performing a cut-off at the Planck energy. However, this theory had problems even on a much smaller energy scale (violation of the unitarity bound on the electroweak scale). just as all non-inflationary models had problems with monopoles on scales much smaller than . The model suggested by Weinberg and Salam [30] is not particularly simple; just remember how long it takes to write a complete Lagrangian. It has anomalies, which are to be cancelled. It has about twenty adjustable parameters, some of which look extremely unnatural. For example, most of the coupling constants are of order , whereas the coupling of the electron to the Higgs boson is . Therefore this model remained relatively unpopular for 4 years after it was proposed, until it was realized that gauge theories with spontaneous symmetry breaking are renormalizable [31]. Soon after that, Georgi and Glashow proposed an -symmetric theory of electroweak interactions, which was much simpler and which did not have any anomalies [32]. Then, neutral currents were discovered, which could not be described by this model. However, nobody considered the problems of the simplest models of electroweak interactions as a signal of a general failure of gauge theories with spontaneous symmetry breaking. The possibility to describe neutral currents and the existence of many adjustable parameters made the Weinberg-Salam model flexible enough to survive and to win. And now, instead of speaking about fine-tuning of parameters of this theory, we are speaking about measuring their values.

This teaches us several interesting lessons. First of all, as stressed by Abdus Salam many years ago, Nature is not economical in particles, it is economical in principles. After we learned the principle of constructing consistent theories of elementary particles by using spontaneously broken gauge theories, there was no way back to the old models. Similarly, after we learned the principle of constructing internally consistent cosmological models, it is very hard to forget it and return to old cosmological theories. In order to account for the abnormal smallness of density perturbations, , one should consider theories with very small coupling constants. But the same happened in the electroweak theory, when, in order to account for the smallness of the electron mass, it was necessary to adjust the coupling constant to be . Nobody says now that this is a fine tuning. We expect that in the next decade observational cosmology will provide us with a large amount of reliable data. It will be absolutely wonderful if the simplest version of inflationary cosmology with adiabatic perturbations with a flat spectrum is capable of describing all these data. One should continue investigation of this attractive possibility. However, there is no special reason to expect that the future cosmological theory will be much simpler than the theory of electroweak interactions, with its twenty adjustable parameters. On the contrary, it may be extremely difficult to suggest any simple theory which would describe new observational data, see e.g. [33]. One should be prepared to most radical changes in the cosmological theory, but one should try to make these changes without breaking the internal consistency of the theory. We believe that the large flexibility of inflationary cosmology in providing many different sources of density perturbations is a distinctive advantage of this theory. It is particularly interesting that most of the sources of nontrivial density perturbations are related to inflationary phase transitions. This should make it possible to use cosmology as a powerful tool of investigation of the phase structure of the elementary particle theory.

I am grateful to Rick Davis, Kris Gorski, Lev Kofman and David Schramm for many useful discussions. This research was supported in part by the National Science Foundation grant PHY-8612280.

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