Quantum Inhomogeneities in String Cosmology
Abstract
Within two specific string cosmology scenarios –differing in the way the pre and postbig bang phases are joined– we compute the size and spectral slope of various types of cosmologically amplified quantum fluctuations that arise in generic compactifications of heterotic string theory. By further imposing that these perturbations become the dominant source of energy at the onset of the radiation era, we obtain physical bounds on the background’s moduli, and discuss the conditions under which both a (quasi) scaleinvariant spectrum of axionic perturbations and sufficiently large seeds for the galactic magnetic fields are generated. We also point out a potential problem with achieving the exit to the radiation era when the string coupling is near its present value.
pacs:
PACS number(s): 04.50.+h, 98.80.CqI Introduction
Perhaps the most appealing feature of standard inflationary cosmology [1] is its ability to stretch out generic/arbitrary initial classical inhomogeneities and to replace them by a calculable spectrum of cosmologically amplified quantum fluctuations. The latter behave, for all physical purposes, as a set of properly normalized stochastic classical perturbations. A much advertised outcome of slowroll inflation is a (quasi) scaleinvariant (HarrisonZeldovich (HZ)) spectrum of density fluctuations, a highly desirable feature for explaining both the CMB temperature fluctuations on large angular scales and the largescale structure of the visible part of our Universe.
The socalled prebig bang (PBB) scenario [2, 3] offers, within the context of string theory, an alternative to the usual inflationary paradigm. Provided a graceful exit can be achieved (see [4, 5] for recent progress on this issue), the PBB scenario exhibits several appealing advantages, e.g.

it naturally provides inflationary solutions through the duality symmetries [6] of string theory;

it assumes a natural, simple, initial state for the Universe, which is fully under control: the perturbative vacuum of superstring theory;

it needs no finetuning of couplings and/or potentials: the inflaton is identified with the dilaton, which is ubiquitous in string theory, is effectively massless at weak coupling, and provides inflation through its kinetic energy;

it can provide a hot big bang initial state as a latetime outcome of the prebig bang phase, through the amplification of vacuum quantum fluctuations generated in this latter phase.
In recent work [7, 8] we have discussed the conditions under which classical inhomogeneities get efficiently erased in string cosmology. In general, this does occur provided two moduli of the classical solutions at weak coupling and curvature (basically an initial coupling and an initial curvature scale) are bounded from above. Whether such conditions correspond to an acceptable degree of finetuning of the initial conditions or not is still the matter of some controversy [9, 10, 8].
An interesting outcome of these investigations has been a motivated conjecture [8] that, for negative spatial curvature, the prebig bang phase itself is generically preceded by a contracting “Milne” phase, corresponding to a particular parametrization of the past light cone of trivial Minkowski spacetime with a constant dilaton. Such a background, the trivial allorder classical vacuum of superstring theory, turns out to be an unstable earlytime fixed point of the evolution. Thanks to dilaton/metric fluctuations, it appears to lead, inevitably, to prebig bangtype inflation at later times.
In this paper we shall assume that the above classical picture effectively wipes out, during its long prebig bang phase, spatial curvature and classical inhomogeneities, and we move on to analyse the second alleged virtue of inflationary cosmology, the generation of an interesting spectrum of amplified quantum fluctuations. As several previous investigations have shown [11, 12, 13], achieving this is not at all automatic in string cosmology. It was soon realized that, in the simplest PBB scenario, tensor [11] and scalardilaton [12] perturbations tend to have steep spectra (typically a spectral index , as compared to HZ’s ). Perturbations of gauge fields coming from compactification of the extra 16 bosonic dimensions of heterotic string theory can have somewhat smaller spectral indices [13], but still in the range .
The situation can be improved by assuming [11] that a long string phase (during which the dilaton grew linearly in cosmic time while the Universe expanded exponentially) took place between the dilaton and the usual FRW phase. In such a case, it is possible to get either an interesting spectrum of gravitational waves [11] in the range of interest for detection, or enough EM perturbations to explain the magnetic fields [13], but not both, apparently. A flat spectrum of EM perturbations, which can possibly provide a new mechanism for generating large scale structure [14] is not excluded either.
Recently, however, Copeland et al. [15] made the interesting observation that axionic perturbations, even in the absence of a string phase, can have a flat spectrum, depending on how the internal dimensions evolve during the dilatonic phase. Unfortunately, Copeland et al. stopped short of computing the axionic spectrum after reentry. Nonetheless, their result hints at a possible dominance of axionic perturbations over all others and calls for a revision of the whole scenario and of the phenomenological constraints that must be imposed on it.
In this paper we analyse quantum fluctuations of various kinds in two distinct scenarios for the background, with or without an intermediate string phase. We may expect either possibility to occur, depending on the precise mechanism providing the transition (exit) from the PBB phase into the FRW phase.
An intermediate string phase is natural if we assume [4] that corrections provide a nonperturbative fixed point with a high constant curvature and a linearly growing dilaton. In this case we expect the transition to the FRW phase to occur during the string phase as soon as the energy stored in the quantum fluctuations reaches criticality (recall that the condition of criticality depends on the coupling). This is like saying that the final transition to the radiationdominated era will be induced by stringloop, backreaction effects (see, e.g. [5]).
We can imagine, instead, that corrections are sufficient to provide by themselves a sudden branchchange from the perturbative PBB phase to another dualityrelated vacuum phase, with the Hubble parameter making a bounce around its maximal value. In the language of [4] this would correspond to a squareroottype vanishing of a function. Again, the dual ( branch) phase will gently yield to a FRW Universe as soon as the energy stored in the quantum fluctuations becomes critical.
As already mentioned, an important ingredient of our approach is the (selfconsistency) requirement of criticality at the beginning of the radiation era. This provides a new relation between the moduli of the PBB background and the coupling and energy density (or temperature) at the beginning of the radiation era. As we will see, the dilaton at the beginning of this era is generically displaced from its eventual/present value; hence this primordial radiation era is not yet quite the one of standard cosmology. It may take a while before the nonperturbative dilaton potential makes its presence felt and forces the dilaton to its minimum. The detailed study of such postbig bang phase is left to further work.
One of the main conclusions of this paper is that, provided has a component in the KaluzaKlein gauge group produced in the compactification from to , sufficiently large seeds for galactic magnetic fields can be generated, even in the absence of a string phase. Furthermore, this happens in the same range of moduli for which a nearly scaleinvariant spectrum of axionic perturbations is generated. Such a range includes a particularly symmetric point in moduli space, the one corresponding to isotropy (up to Tduality transformations) in all nine spatial dimensions.
The outline of the paper is as follows: In Sec. II we give, for the sake of completeness, the fourdimensional lowenergy stringlevel heterotic effective action that we will work with. In Sec. III we fix our parametrization of the backgrounds for the two previously discussed scenarios. In Sec. IV we derive general formulae for the spectra of various perturbations, which get amplified by a generic background of the kind discussed in Sec. III. We will verify that our spectra satisfy a “duality” symmetry that can be shown to follow from general arguments [16]. In Sec. V we give the explicit form of the spectra for the two backgrounds discussed in Sec. III, and present them in various tables and plots. We will also impose the criticality condition and discuss its immediate consequences. Finally, Sec. VI contains a discussion of the results and some conclusions.
This paper is somewhat technical in nature and contains explicit general formulae that can be useful to the practitioner but do not carry easy messages: these can be better found in the tables and figures. At any rate, in order to help the reader, we have relegated the most complicated formulae to an appendix.
Ii String effective action from dimensional reduction
Following the notations of [17] we consider superstring theory in a spacetime , where , with Minkowskian signature, has four noncompact dimensions, and consists of six compact dimensions upon which all fields are assumed to be independent. Local coordinates of are labelled by , those of by . Moreover, all tendimensional fields and indices are distinguished by a hat.
We will limit ourselves to the case of a diagonal metric for the internal sixdimensional compact space, of a nonvanishing internal antisymmetrictensor and of one Abelian heterotic gauge field :
(1) 
(2) 
In the following we take .
The lowenergy fourdimensional effective string action is
(3) 
where is the stringlength parameter,
(4) 
(5)  
(6) 
and stands for the effective fourdimensional dilaton field:
The components of the antisymmetric tensor with can be rewritten in terms of the pseudoscalar axion as
(7) 
where is the covariant full antisymmetric LeviCivita tensor, which satisfies . Using Eq. (4), and imposing the Bianchi identity (), we get the equation of motion for the axion field
(8) 
The reduced action then becomes
(9) 
We are interested in fluctuations around a homogeneous background with ,
In the following we will also use the metric , where we have introduced the conformal time by .
Iii Two models for the background
If the initial value of the string coupling is sufficiently small, it is possible for the Universe to reach the high curvature regime, where higherderivative corrections are important, while the string coupling is still small enough to neglect loop corrections (). As discussed in the introduction, we will consider two extreme alternatives. In the first, corrections “lock” the Universe in a string phase with a constant and a linearly growing dilaton (with respect to cosmic time) [4]; in the second scenario, corrections induce a sudden transition from a perturbative branch solution to a perturbative branch phase. We will refer to the latter as the dualdilaton phase.
We will thus consider a PBB cosmological background in which the Universe starts in the perturbative string vacuum, reaches the string curvature scale while in a dilatonvacuum solution, goes either to the dualdilaton phase or to the string phase, and finally enters the radiation era as a result of the backreaction from the amplified quantum fluctuations. We now parametrize these two scenarios for the backgrounds, imposing the continuity of , , .
iii.1 Intermediate dual dilaton phase

Dilaton phase
For , with we have
(10) (11) (12) (13) where is of order . We will consider the case and , i.e. a superinflationary solution with contracting internal dimensions. Because of the constraint between and , if , some of the must be nonvanishing. In what follows we will pick two extreme cases: i) the most isotropic case, with , or ii) the most anisotropic one, with . In figures we shall denote these two cases by a subscript and , respectively.

Dualdilaton phase
For , with we take
(14) (15) (16) where and we will fix and , i.e. a decelerated expansion for the external scale factor and a decelerated contraction for the internal ones. Again, we distinguish two cases, or .

Radiation phase
In the region , with the time of equivalence between radiation and matter density, we write
(17) (18)
iii.2 Intermediate string phase

Dilaton phase
We parametrize this phase exactly as before. Thus, for , Eqs. (10) to (3.4) hold.

String phase
For , with
(19) hence a constant Hubble parameter for the external scale factor.

Radiation phase
In the range we have
(20)
An important property of these backgrounds is that the derivative of the field is not continuous across the two transitions. This reflects the fact that we do not have as yet a satisfactory model for the transitions from one epoch to another. As discussed below, this discontinuity creates a technical problem, which has to be judiciously solved in order to correctly compute the spectrum of perturbations around this kind of backgrounds.
Iv Amplification of vacuum fluctuations
Let us consider a generic massless field, whose quadratic fluctuations are described by the action
(21) 
where a prime stands for derivative with respect to conformal time and , the socalled “pump” field, is a homogeneous background field that depends on the particular perturbation under study.
The safest way to analyse the amplification of the vacuum fluctuations of makes use of a canonical Hamiltonian approach and leads to the derivation [16] of certain duality symmetries of the spectra. We will use instead the simpler Lagrangian method and fix some ambiguity encountered in that approach by demanding agreement with the Hamiltonian treatment. We believe, of course, that our prescription can also be fully justified within the Lagrangian framework.
The equation of motion for the Fourier components of is
(22) 
Introducing the canonical variable
(23) 
Eq. (22) can be rewritten in the form
(24) 
In order to get general formulae for the spectrum we parametrize the pump field in the three epochs as follows^{4}^{4}4 In order to simplify the final expression of the Bogoliubov coefficients, we have slightly changed the constant parameters appearing in the pump field in the three phases. With the original parameters of Sect. 3, the Bogoliubov coefficients would just change by numerical factors , but the spectral slopes and the “duality” symmetry (see sect. IV.2) would still be the same.
(25)  
(26)  
(27) 
iv.1 Analytic form for the Bogoliubov coefficients
The solutions of the equation of motion (24) for the pump fields (25), (26) and (27) are respectively
(28)  
(29)  
(30) 
where
(31) 
and we have normalized (28) allowing only positive frequencies in the flat vacuum state at , so that
(32) 
and . For reasons explained below we impose the continuity of and of its first derivative at , , not the continuity of the canonical field . Using the relation
we obtain
(33)  
(34)  
and
(35)  
(36)  
where the prime stands for derivative with respect to the argument of the Hankel function. From the condition we get and , as needed for generic Bogoliubov coefficients.
iv.2 “Duality” of the Bogoliubov coefficients
In this section we analyse the behaviour of the Bogoliubov coefficients and under the “duality” transformation , and , under which the pump fields are reversed. We will thus check that, with a careful choice of the matching conditions, the symmetry that can be shown to be exact in the Hamiltonian approach [16] is preserved.
In our context we need the following relations among Hankel functions (see e.g. [18])
(37)  
(38)  
(39) 
Independently of the range of frequencies we get
(40)  
(41) 
There are two important comments to be made on the above formulae. The first is that differs from by just a phase. Hence the spectrum (being proportional to ) is identical for a given pump field or for its inverse. We stress that this duality property holds independently of the number and characteristics of the intermediate phases and thus, as argued in [16], is generally valid. The second observation is that duality depends crucially on having imposed the continuity of the field and of its derivative on and not on the canonical field . The difference in imposing continuity of or of arises from the discontinuous nature of the background itself (actually of ) and from the fact that and obey equations containing first and second timederivatives of the pump field, respectively. This gives rise to function contributions in the case of , making the requirement of continuity suspicious for that variable.
One welcome consequence of “duality” is the fact that the antisymmetric tensor field and the axion have identical spectra since their pump fields are the inverse of each other (see below). This must be so since they are just different descriptions of the same physical degree of freedom.
iv.3 General form of the spectral slopes
The parameters and in the formulae for define two characteristic comoving frequencies, , , which can be traded for two proper frequencies and by the standard relations
(42) 
Exponents  Bogoliubov coefficient  Leading contribution  Power of () 

, ,  ,  
, ,  ,  
, , ,  ,  
, , ,  ,  
, ,  , 
It is easy to see that the two scenarios for the background, intermediate dual dilaton phase and intermediate string phase, lead to and , respectively. In the case (fluctuations that exit in the dilaton phase and reenter in the radiation phase), which is common to both scenarios, we can approximate the exact result (36) for the Bogoliubov coefficient in the following way^{5}^{5}5We have used the following relations for not integer:
where
(43)  
(44)  
(45) 
(46) 
Since, by their definition (31), , gives the leading contribution unless the coefficients appearing in front of it vanish. Table 1 shows which one of the is dominant for different choices of the background parameters.
In the case (fluctuations that exit in the dilaton phase and reenter in the dualdilaton phase) we get instead:
(47) 
Table 2 shows the leading contribution to in this case. The explicit form of the coefficients , and for both cases is given in the appendix.
Exponents  Bogoliubov coefficient  Leading contribution  Power of () 

,  
,  ,  
,  ,  
, ,  ,  
, ,  ,  
,  , 
Tables (1 and 2) also show the leading power of appearing in the Bogoliubov coefficient , in the two abovementioned cases, i.e. reentry in the dualdilaton or in the radiation phase. We can summarize this behaviour as follows
(48)  
(49) 
In the case of an intermediate string phase the Bogoliubov coefficient for (fluctuations that exit in the string phase and reenter in the radiation phase) is given by Eq. (47) after the substitution (hence ).
Particles ()  Pump field ()  Spectral slope  

reentry dual phase  reentry radiation phase  
Gravitons  4  3  
Axions  
Heterotic photons  
The spectrum of fluctuations for a generic field is
(50) 
where is the number of polarization states. We have found it convenient to use a “spectral slope” parameter defined by the relation
(51) 
where is the exponent appearing in the dependence of (see Tables 1 and 2). The spectral slope, which is simply related to the usual spectral index by , is more convenient to describe the main property of the spectrum, since its sign tells us whether the spectrum is increasing or decreasing with . We will now apply the above general results to various possible backgrounds and perturbations occurring in string theory.
Particles ()  Pump field ()  Spectral slope  

Exit in dilaton phase  Exit in the string phase  
Gravitons  3  
Axions  
Heterotic photons  