DF/UFJF-01/02

August, 2002

On the stability of the anomaly-induced inflation

A.M. Pelinson
^{1}^{1}1Electronic address: ,
I.L. Shapiro
^{2}^{2}2Electronic address:
F.I. Takakura
^{3}^{3}3Electronic address:

a. Departamento de Campos e Partículas, CBPF, Rio de Janeiro, Brazil

b. Departamento de Física – ICE, Universidade Federal de Juiz de Fora, MG, Brazil

c. Tomsk State Pedagogical University, Tomsk, Russia

Abstract. We analyze various phases of inflation based on the anomaly-induced effective action of gravity (modified Starobinsky model), taking the cosmological constant and topologies into account. The total number of the inflationary -folds may be enormous, but at the last 65 of them the inflation greatly slows down due to the contributions of the massive particles. For the supersymmetric particle content, the stability of inflation holds from the initial point at the sub-Planck scale until the supersymmetry breaks down. After that the universe enters into the unstable regime with the eventual transition into the stable FRW-like evolution with small positive cosmological constant. It is remarkable, that all this follows automatically, without fine-tuning of any sort, independent on the values of and . Finally, we consider the stability under the metric perturbations during the last 65 -folds of inflation and find that the amplitude of the ones with the wavenumber below a certain cutoff has an acceptable range.

## 1 Introduction.

In this paper, we are going to discuss the example of such a model - the new version [2, 3, 4] of the anomaly-induced inflation (Starobinsky model) [5, 6, 7, 8, 9, 10]. This model is based on the well known methods of quantum field theory in curved space-time (see, e.g. [11, 12]) and on the effective quantum field theory approach. The advantage of the anomaly-induced inflation is that it does not contain usual elements of the conventional cosmological phenomenology such as inflaton, the precise form of its potential and/or the necessity of a fine-tuning for the initial data. The anomaly-induced inflation provides the explanation for both the start of inflation and the graceful exit to the post-inflationary FRW-like phase [3]. Under a natural supposition about the supersymmetry breaking scale we arrive at the magnitude of the Hubble constant which does not generate unacceptable amplitudes of the gravitational waves [4].

The problems of stability play a great role in the anomaly-induced inflation. In particular, the transition between the stable and unstable regimes, which could take place due to the breaking of supersymmetry [3] and the quantum vacuum effects of massive fields [4], could be responsible for the graceful exit to the FRW evolution. In the next sections, we investigate the stability with respect to the small perturbations of the conformal factor for various regimes associated to the anomaly-induced inflation. One can distinguish three stages of the expansion:

For the almost exponential inflation that takes place at the very beginning (when the effect of particle masses is negligible) we consider the spaces with zero, positive and negative space curvatures, and with zero, positive or negative cosmological constant. The main part of the results can be obtained analytically such that some very important general features are established.

At the post-inflationary stage, after the supersymmetry breaking, the universe is approaching the stable fixed point with a small Hubble constant. In this case the FRW solution is a very good approximation for the theory with vacuum quantum corrections, such that these corrections do not play an essential role at the later time. We shall verify that the FRW-like solution is an attractor for the theory with quantum corrections if the cosmological constant has positive sign.

The most complicated is the intermediate transitional epoch between the inflationary and the post-inflationary evolutions. This period is characterized by the vacuum quantum effects of both massless and massive fields. Indeed, the exact analytic methods for the stability analysis [6] do not apply in this case, and one has to use numerical and approximate analytical methods.

The paper is organized as follows. In the next section, we present basic concepts of the anomaly-induced inflation. In part, we go beyond the results of the previous publications [2, 3, 4]. In particular, we take into account the effect of the renormalization of the cosmological constant (here, our approach is somehow similar to the one of [13]). In section 3 the stability under the small perturbations of the conformal factor is analyzed for the “massless” stage , mainly through the analytical methods. The main new aspect here is that we clarify the role of the cosmological constant. In section 4 we verify, using both approximate analytical and numerical methods, that the stability of the model with the supersymmetric particle content really holds until the scale of the supersymmetry breaking. In section 5 we discuss, using the approximate analytical methods tested in section 4 and also the numerical integrations, the stability under the metric perturbations. Finally, in section 6 we draw our conclusions and also present a general discussion about the possible theoretical significance of the model.

## 2 Brief survey of the anomaly-induced inflation

Consider the theory which has real scalars, Dirac spinors, and vectors. One has to notice that the vacuum quantum effects originate from the virtual particles. Therefore, the numbers 9]. correspond to the particle content of the quantum theory, but they do not describe the real matter which might fill the Universe. As we shall see later on, the real matter has little importance for the anomaly-induced inflation. For simplicity, one can assume that the initial universe is empty and that all matter content of the Universe is created during the reheating period [

The renormalizable quantum theory of matter fields on curved background includes the action of vacuum, and its minimal necessary form is

(1) |

Here the first term contains higher derivatives of the metric

(2) |

and the second is the Einstein-Hilbert term
^{4}^{4}4Our notations are and

(3) |

where , and are the parameters of the vacuum action. The renormalization of these parameters and the quantum corrections to the action (1) are the basics of the inflationary model we are discussing here. The discussion in this section will perform in two stages: first we review the massless limit and then present a useful approximation [4] for the massive case.

### 2.1 Massless conformal case

The actions of the free real scalar, Dirac fermion and gauge vector fields are

(4) |

where ;

(5) |

and

(6) |

The massless matter fields possess local conformal invariance in the case where scalars couple to gravity in a nonminimal conformal way, with . In what follows, we consider only this value, because it provides the traceless stress tensor for the massless theory at the classical level. It is well known that at the one-loop level the condition holds under renormalization (see e.g. [12]), but at higher loop this is not the case [14]. However, even at higher loops the deviation from can be very small [15], such that can serve as a good approximation. Furthermore, massless fermions are conformal invariant, also the gauge fields can be treated as massless and conformal at very high energies.

The renormalization of the Einstein-Hilbert, cosmological and terms in (1) is absent in the case of conformal massless fields, and from this point of view their presence is not necessary. Indeed, we have to introduce the Einstein-Hilbert term because it is proved to fit with all experimental and observational data at the cosmic scale. For the same reason we introduce the cosmological term which was detected in the recent observations [16]. Regarding the term, we have to accept the presence of this term since it is important for the renormalizability at higher loops. But its coefficient may be safely taken very small [15] such that it is negligible compared to the quantum corrections of the same form (see Eq. (34)). For this reason, in what follows we shall simply take . Under this choice the action satisfies the Noether identity

(7) |

which is usually interpreted as zero trace for the stress tensor of vacuum . At quantum level, this identity is violated by the anomaly such that [17]

(8) |

Here is the quantum correction to the classical vacuum action (1), the coefficients , and are the -functions for the parameters correspondingly

(9) |

(10) |

(11) |

Some observation is in order. The direct application of the dimensional regularization [17] leads to the ambiguous result for the coefficient of the -term in the anomaly. We accept the expression for which emerges from other regularizations such as point-splitting [18] and -regularization [19].

The general solution for the anomaly-induced effective action is [20, 21]

(12) |

where

and

is the fourth derivative, conformal operator acting on scalars. is the “integration constant” for the equation (8). In general, there is no regular method for deriving . Fortunately, for some cosmological problems the form of this functional has no importance. Let us consider an isotropic and homogeneous metric

(13) |

where is conformal time and

(14) |

Then, the conformal functional is constant, for it does not depend on and does not contribute to the equations of motion. In this particular case, the solution (12) is an exact one-loop quantum correction. However, if one takes a more general background, for example

(15) |

where is a metric perturbation. Then the functional becomes relevant, because

and the solution (12) is only an approximation to the unknown general expression. In this and the next two sections we shall consider the metrics of the form (14) and in the section 5 the metric (15).

The total action with the quantum corrections

(16) |

leads to the following equation for the metric (13):

(17) |

where we used the standard notation for the Planck mass . The point over the variable designates the derivative with respect to the physical time , the last is related to the conformal time by the relation .

The equation (17) has remarkable particular solutions for all three cases :

(18) |

(19) |

(20) |

where the Hubble constant has the form (we do not consider solutions with negative )

(21) |

For a negative , there is only a particular solution with positive sign in (21).

The solutions for the case of have been found by Starobinsky [6] (see also [22]). According to (21) and (10), the two solutions with exist only for . Let us remark that in the high energy region the sign of is positive. Let us advocate this point. In the nowadays universe is also positive but very small, due to the cancelation of the induced and vacuum energy densities (see, e.g. [23, 13] and also [24] for the pedagogical introduction to the formal aspects of the cosmological constant problem).

The induced cosmological constant appears due to the spontaneous symmetry breaking in the Standard Model, as a vacuum energy corresponding to the minima of the Higgs potential

(22) |

and has the classical value

(23) |

with the magnitude about 55 orders greater than the observable cosmological constant. Obviously, the vacuum counterpart must be positive in order the provide a cancelation at low energies. But, at high energies (temperatures) above the symmetry in the potential (22) gets restored such that the induced part (23) disappears. After that the value of the cosmological constant will be given by the vacuum term with the corresponding quantum corrections [13]. Similar increase of may be expected in any other phase transition which could occur at high energies. In particular, the contribution to the cosmological constant may be expected from the supersymmetry breaking in case this breaking is spontaneous. The last observation is that after inflation, when the matter content of the Universe gets closer to the equilibrium state, the high-temperature radiation dominates over the cosmological term [25].

In what follows, except if this is indicated explicitly, we assume that the cosmological constant in the high energy inflationary epoch satisfies the relation . Then the expansion of (21) in the small parameter gives the two values

(24) |

where the first solution is exactly that we meet without the quantum corrections. The error in both cases has the order of magnitude

As we shall see in the next section, the inflationary solutions (18), (19) and (20) are stable for and unstable for [6] (see also [2]). According to (11), is equivalent to the condition for a particle content of the theory

(25) |

The unique way to provide the non-stability of
inflation in the case of is to introduce
some additional term or terms into the classical action
of vacuum. For instance, the -term
with the sufficiently large positive coefficient does
this job. But, for the first sight, this term is not
necessary.
Let us notice, that the quantum correction (12)
is valid not only in the Early Universe but also at the
later stages of the evolution. However, the content
may be different, because for the small
energies the loops of the fields with large masses decouple.
Let us consider the present-day universe as an example.
The value of the Hubble constant today is extremely small,
about , that is about 30 orders
smaller than the mass of the lightest neutrino. Therefore,
the unique particle which makes a nontrivial contribution
to the anomaly is the photon
^{5}^{5}5If the neutrino would be massless, there
could be an equality in (25)..
Then, the relation (25) testifies a
non-stability and the present-day universe is safe without
the special -term. We shall continue
the discussion of this example in the section 3.

The condition (25) enables one to construct a very attractive inflationary scenario [3]. The universe could start in the stable phase, such that inflation starts independent on the initial data. The simplest way to provide stability in (25) is to assume supersymmetry in the high energy region and also suppose that the supersymmetry is absent at a lower energy because the sparticles are heavy [3]. If, in the course of inflation, the magnitude of the Hubble constant decreases, then at some point the loops of the sparticles decouple and the matter content gets modified. As a result, the sign of the inequality (25) changes to the opposite and the universe falls into the non-stable inflation with the eventual transition to the FRW evolution. The important problem is that why the Hubble parameter should decrease during the inflation? Indeed, the Starobinsky solutions (18), (19) and (20) are characterized by the constant . The answer to this question is that the value of really decreases if we go beyond the massless approximation [4]. We shall consider the effect of massive fields below.

Another interesting aspect is the possible role of matter. Here we have two different situations. The first is the possibility to have a set of heavy massive fields at the beginning of inflation. These fields can be remnants of the string phase transition. For our model it is important that this massive content does not destroy the stability which holds at the beginning of the stable inflation. Let us suppose that these fields create zero pressure. Then the equation (17) is modified by adding the term to the r.h.s.. The investigation of the equation with this term and with

(26) |

has been performed numerically. The result is such that even for the system does not loose stability and the stabilization performs in a few Planck times. The plots are almost identical to the matterless case (see Fig. 2). The reason of this output is that the new term decreases very fast during the inflation and becomes irrelevant.

Second aspect is the transition to the FRW evolution after some of the massive fields decouple and the inflation becomes unstable. In order to understand the situation better, let us just substitute the corresponding FRW solution into the equation (26). It is easy to see that the “classical” terms - both Einstein and matter, both behave like , while the quantum correction - with higher derivatives, behave like . Indeed, , as usual. Therefore, with the properly taken initial data for the unstable inflation, the FRW solution is a perfect approximation for the later epochs. In [4] we have checked that the “proper” initial data emerge naturally in the framework of the sharp cut-off approach to the decoupling.

### 2.2 Massive case

The conformal invariance of the actions (4) and (5) is violated by the masses and we can not use the conformal anomaly to derive quantum corrections. But, this can be changed if we apply the conformal description of the massive theory in the framework of the cosmon model [27] (see also [28] for other applications of the cosmon model). Let us replace the massive parameters by the powers of a new auxiliary scalar field according to

(27) |

where is some dimensional parameter. In the original version of the cosmon model [27], the introduction of the new scalar fields is called to provide the dilatation invariance with the consequent breaking of the dilatation symmetry. In our case we need a more general, local conformal invariance. Therefore, we postulate that the auxiliary field transforms as

(28) |

independent of the space-time dimension
^{6}^{6}6Other fields transform
in a usual way:

The divergences of the theory in the conformal representation have the form

(29) |

where the coefficients and are given by Eqs. (9), (10) and (11) correspondingly, and we introduced useful notations for the dimensional quantities

(30) |

and

(31) |

Here the sums are taken over all massive fermion and scalar fields with the masses and correspondingly. and are multiplicities of fermions and scalars, for example and . In order to derive the conformal anomaly, one needs to consider the classical Noether identity for the vacuum part of the effective action

(32) |

It is easy to see that, in the theory under discussion, the conformal invariance does not mean that the stress tensor is traceless, but instead that the quantity is zero. Correspondingly, the conformal anomaly means instead of usual [29]. However, since both -dependent terms in (29) have the same conformal properties as the square of the Weyl tensor, they can be absorbed into the Weyl term such that the problem of deriving anomaly reduces to the standard one. Finally, we arrive at the expression

(33) |

In the standard way, using the parametrization and , one can derive the anomaly-induced effective action of the background fields and

(34) |

The last step is to fix the conformal unitary gauge , such that the classical Einstein-Hilbert and cosmological terms acquire their standard form. The one-loop effective action becomes

(35) |

It is worth noticing that the expression (35) does not depend on the magnitude of , but only on the fields and parameters of the theory. This effective action differs from the massless counterpart (16) in two respects. First, there are new terms which emerge due to the renormalization of the Einstein-Hilbert and cosmological term. Second, the integration constant is not irrelevant anymore, since we imposed the conformal unitary gauge. Therefore, the expression (35) can be regarded only as an approximation to the effective action of vacuum for the massive fields and we have to learn the limits of validity for this approximation.

The higher-derivative part of the Eq. (35) is identical to that for the massless fields, as it has to be (see, e.g. [11, 22]). Furthermore, there is a strong link between the anomaly-induced effective action Eq.(35) and the quantum corrections coming from the renormalization group. The expansion of the homogeneous, isotropic universe means a conformal transformation of the metric . On the other hand, the renormalization group in curved space-time corresponds to the scale transformation of the metric simultaneously with the inverse transformation of all dimensional quantities [12]. For any we have . One can compare the -dependence of the anomaly-induced effective action (35) and the -dependence of the renormalization-group improved classical action

(36) |

where denote vacuum parameters of the theory and . The expression (36) is a leading-log approximation for the solution of the renormalization group equation for the effective action [12]

(37) |

where are matter fields. It is easy to see that (35) becomes completely equivalent to (36) if we set . The coefficient is a factor of the -function for the inverse Newton constant and the coefficient is a factor of the -function for the cosmological term . Indeed, (37) can be considered as a generalization of the renormalization group improved classical action (36).

Therefore, we have strong reasons to regard Eq. (35) as a leading-log approximation for the effective action for the massive fields. This approximation picks up the logarithmic quantum corrections and is reliable in the high energy region where masses of the fields are much smaller than the Hubble constant . On the other hand, at low energies the massive fields decouple and their quantum effects become negligible. In this paper we shall interpolate between these two regimes using the MS scheme and the “sharp cut off” approximation at some specific scale . This scale is defined such that at a sufficient number of the sparticles decouple and the inequality (25) changes its sign. Within our approximation, the functional contains the sub-leading non-log corrections and can be disregarded. Let us remark that the scale can be different from the supersymmetry breaking scale . In particular, we can suppose that for a GUT model is about while the upper bound for is . This bound appears because the amplitude of the generated gravitational waves has an admissible range only for in the last stage (-folds) of inflation.

The equation of motion for has the form

(38) |

where we introduced useful notations

(39) |

It proves useful to rewrite this equation, also, in terms
of ^{7}^{7}7We correct a small mistake in the
similar expression (without the cosmological constant
term) in [4]. This correction does
not modify the behavior of .

(40) |

The solution of the equation (38) can not be performed analytically in the general form, but it is obvious that the approximate solution for small can be obtained by the replacement

(41) |

in the expression for the Hubble parameter (21)
corresponding to the solutions (18),
(19) and (20).
Another possibility is the numerical integration of
the equation (38). We present the
plots produced by the numerical integration at
Figure 1 for the case . These plots were obtained
using Mathematica program [31] and also FORTRAN and
standard algorithms of [30].
It is easy to see that the initial parts of these numerical
solutions are identical to the ones which were obtained
in [4]. Let us make one more step in
the understanding of these plots and the quality
of the approximation (41). For the sake of
simplicity, we shall consider only the value
^{8}^{8}8The numerical analysis shows that this is
a good approximation for ..
In this case one can use (41) and
easily integrate the equation

(42) |

and find its solution in the form

(43) |

One might expect that this simple formula should serve as a reasonable approximation only at the initial stage when is small. However, if we compare the plots (a) and (b) at the Figure 1 and the parabola Fig. 1c (43), the numerical difference is remarkably small - about at the maximal point with (for the toy model with the MSSM field content and ). The two solutions are almost identical for all . Remarkably, this is not a specific feature of this particular value of . If we substitute (43) into the equation (38), the result is

that is indeed negligible for all situations of interest, when is very small. This unexpected feature of the solution is extremely useful, for we can sometimes use simple expression (43) instead of the complicated numerical solution of the equation (38).

In the last considerations we took into account that and are very small quantities. This leads to the condition for the magnitude of the quantum corrections to the cosmological constant

(44) |

Furthermore, one can request that the cosmological constant plays smaller role than the higher derivative induced contributions:

(45) |

Indeed, these conditions are satisfied at the beginning of inflation when is small and the higher derivative terms dominate over the cosmological constant which must be, at most, of the order . It is convenient to assume the soft supersymmetry breaking scheme such that the magnitude of the cosmological constant is much smaller than . When becomes large and the universe is close to the transition from stable to unstable inflation, we can not rely much on Eq.(41) and have to use the numerical methods to verify both stability of the tempered inflationary solution and the graceful exit to the FRW phase after the supersymmetry breaks down.

## 3 The conditions of stability in the massless case

(46) |

Without the matter fields the -equation is equivalent to the Eq. (17) which we used in the previous section [5, 6, 32]. The -equation can be solved analytically [6] and the phase diagram shows the existence of the single attractor corresponding to the inflationary solution in the stable case and various attractors, including in the non-stable case with . Unfortunately, our equations (38) or (40) are much more complicated than in the massless case and there it is not possible to solve them exactly. However, there is no real need to do that. At the beginning of inflation the stabilization of the exponential solution performs very fast [2]. Therefore, at the latest stages it is sufficient to check the asymptotic stability of the inflationary solution with respect to the small perturbations in the sense of Lyapunov (see, e.g. [33] regarding the stability issues in higher derivative models).

In this section we establish some important general properties of the stability under small perturbations and will apply this knowledge for the approximate analytical and numerical analysis of the massive case in the next section.

With respect to the criterion (25) of stability [6, 2], there are three relevant questions: i) which kind of perturbations one has to consider; ii) whether the choice of may affect (25); iii) whether the presence of the cosmological constant may affect this criterion.

In order to address the issue i), let us make small perturbations in the equation (17) and set, for the sake of simplicity, and . First we try the perturbations of the form

After simple calculations we arrive at the equation for the perturbations

(47) |

Indeed, the factors of can be easily removed by the simple rescaling of time, and then the standard Routh-Hurwitz conditions (see the Appendix) show that the stability can not be achieved for any values of . Of course, this contradicts our previous statement (25), and we can conclude that the perturbations of do not serve for our purposes.

Fortunately, from the theory of ordinary differential equations (see, e.g. [34]) we know that the stability may depend on the choice of the variables. Therefore, let us consider the same problem in terms of . The corresponding equation can be obtained from (38) by setting and making small perturbation of the form

The equation for has the form

(48) |

or, taking the solution for into account,

(49) |

It is easy to see that in this case the stability is
achieved when is positive that immediately gives
(25)
^{9}^{9}9Of course, the conditions of
stability for are exactly those for ..
The illusory contradiction between
the two results can be clarified if we identify

Since grows up very fast, the growing modes in may be negligible because they are suppressed by the factor of . Of course, this is the case when , while for there is a mode in which is growing up faster than . Finally, we learned a very important lesson for the consequent study of the complicated massive case: one has to perturb , but not . The calculations for the and cases are a bit more complicated, and we will not bother the reader with the details. The condition of stability is always the same as in the simplest situation described above. But, it is important that this condition must be the same for and must be independent on . Let us formulate two simple theorems:

Theorem 1. The value of does not influence the asymptotic stability.

Proof. The equation for the perturbation or does not change if the original equation for or is multiplied by the factor of or . Therefore, we can always isolate the cosmological term such that its contribution to the perturbation equation is zero. The dependence on can be also seen in , but the magnitude of is irrelevant for stability, because we can always renormalize time such that in the new equation .

Observation. This theorem is valid only for .

Theorem 2. The choice of does not influence the asymptotic stability and the condition (25) is valid for all three cases.

Proof. It is sufficient to apply the corresponding theorem about the asymptotic stability in the limit (see Appendix). In the equation for the perturbations one can always replace the coefficient functions by their asymptotic values. Since both and behave like an exponential at the late times, the conditions of stability for all three cases are the same.

Illustration. The equation for the perturbation in the general massless case is

If we neglect those terms in the coefficients of the perturbations which tend to zero at , the - dependence disappears.

Thus, we have learned two lessons from the analysis of the massless case. It is sufficient to consider the case and , for this does not modify the stability of the solution. These statements can be used in the more realistic theory of inflation based on the effective action for the massive fields, but with a proper caution. In fact, the negligible role of depends on that it is a constant. As we shall see in the next section, this is not true for the effective action of massive fields.

Until now, we have considered only the stable inflationary solution with . In other words, we were interested in the high energy region where the supersymmetry is unbroken. Obviously, another end of the energy scale also represents a great interest, for it enables one to perform a simple and efficient test of the model. Let us consider, again, a nowaday universe with . According to the recent data [16], this magnitude of the Hubble constant is mainly due to the contribution of the cosmological constant. As we have already mentioned in the last section, the value of in the present-day universe is negative due to the contribution of the single massless particle - photon. Therefore, we have to check whether the higher derivative terms do not destroy the stability of the first solution in Eq. (24). Using (48), we arrive at the following characteristic equation for the perturbation of .

(50) |

Since the explicit solution of this equation is tedious, let us start from the case. Then and the roots of the equation (50) are

(51) |

At this level, we do not have definite answer to our question about stability. Now, let us look for the solution of Eq. (50) making perturbations in (51). Due to the huge difference between and , this approximation is perfect. The result is

(52) |

It is very nice to see that the cosmological constant really stabilizes the solution in the low-energy region, exactly as we should optimistically expect. The stability can be verified also for the earlier post-inflationary epoch, when the energy density of vacuum played smaller role than the density of radiation [25, 23]. For the later epoch, as we have already noticed, when we substitute the FRW solution into Eq. (17), the “quantum” terms decrease as compared to the Einstein-Hilbert and matter terms. Therefore, for the later epoch of the expanding post-inflationary universe the FRW is a very good approximation.

## 4 The stability in the interpolation regime

Before starting the numerical analysis, let us perform some analytic investigation of the stability in a useful approximation framework. It is sufficient to investigate the behavior of the perturbations in the physically important region corresponding to the last 65 -folds before achieves the value . Looking at the plot at Fig. 1, we see that the dependence is rather smooth. Hence, we can divide this region into small intervals and consider inside each of the intervals. Indeed, will change from one interval to another, and this may be the source of new perturbations. Of course, all this consideration is done for the stable case , while for the non-stable case this approach has no sense. For the purpose of analytic study, we accept the approximation (41). Then, the direct calculations give the following equation for the perturbations :