DFTT-55/95 September 1995 hep-th/9509127

Status of String Cosmology:

Phenomenological Aspects
^{*}^{*}Lecture given at the 4th Course of the International
School of Astrophysics “D. Chalonge” on
String gravity and physics at the Planck scale
(Erice, 8-19 September 1995).
To appear in the Proceedings
(Kluwer Acad. Pub., Dordrecht), ed. by N. Sanchez.

M. Gasperini

Dipartimento di Fisica Teorica, Università di Torino,

Via P.Giuria 1, 10125 Turin, Italy,

and

INFN, Sezione di Torino, Turin, Italy

Abstract

I report recent studies on the evolution of perturbations in the context of the “pre-big-bang” scenario typical of string cosmology, with emphasis on the formation of a stochastic background of relic photons and gravitons, and its possible direct/indirect observable consequences. I also discuss the possible generation of a thermal microwave background by using, as example, a simple gravi-axio-dilaton model whose classical evolution connects smoothly inflationary expansion to decelerated contraction. By including the quantum back-reaction of the produced radiation the model eventually approaches the standard radiation-dominated (constant dilaton) regime.

STATUS OF STRING COSMOLOGY:

PHENOMENOLOGICAL ASPECTS

M. GASPERINI

Dipartimento di Fisica Teorica, Università di Torino,

Via P.Giuria 1, 10125 Turin, Italy

1. Introduction

Inspired by the basic ideas of string theory, we have recently started the investigation of a cosmological scenario in which the standard big-bang singularity is smoothed out and is replaced by a phase of maximal (finite) curvature. Such a phase is preceeded in time by a “pre-big-bang” epoch [1-3], which has (approximately) specular properties with respect to the present phase of standard decelerated evolution.

This scenario was originally motivated by the solutions of the string equations of motion in curved backgrounds [4], and by the duality symmetries of the string effective action [5-7] (see for instance [8] for a more detailed introduction). Its basic ingredients are an initial weak coupling, perturbative regime characterized by a very small background curvature (in string units), and a final transition to the radiation era controlled by the dilaton dynamics and by the two basic parameters (string length, string coupling) of string theory. The most revolutionary aspect of this scenario (with respect to the conventional, even inflationary, picture) is probably the fact that the early, pre-Planckian universe can be consistently described in terms of a semiclassical low energy effective action, with the vacuum as the most natural initial conditions for the quantum fluctuations of all the background fields (more details on this scenario can be found in Gabriele Veneziano’s contribution to these proceedings [9]).

According to the above scenario, the pre-big-bang epoch is characterized by an accelerated (i.e. inflationary) evolution. In all inflationary models, the most direct (and probably most spectacular) phenomenological predictions is the parametric amplification of perturbations [10], and the corresponding generation of primordial spectra, directly from the quantum fluctuations of the background fields. In a string cosmology context such an effect is even more spectacular, as the growth of the curvature scale during the pre-big-bang phase is associated with a perturbation spectrum which grows with frequency [11,12]; moreover, the growth of the curvature may also force the comoving perturbation amplitude to grow (instead of being frozen) outside the horizon, as first noted in [2]. This leads to a more efficient amplification of perturbations, but the amplitude could grow too much, during the pre-big-bang phase, so as to prevent us from applying the standard linearized formalism.

In view of this aspect, the aim of this lecture is twofold. On one hand I will discuss how, in some case, this anomalous growth can be gauged away, so that perturbations can consistently linearized in an appropriate frame. On the other hand I will show that, even if the growth is physical, and we have to restrict ourself to a reduced portion of parameter space in order to apply a linearized approach, such enhanced amplification is nevertheless rich of interesting phenomenological consequences.

I will discuss in particular the following points: i) the growing mode of scalar metric perturbations in a dilaton-driven background, which appears in the standard longitudinal gauge and which seems to complicate the computation of the spectrum [13]; ii) the production of a relic gravity wave background with a spectrum strongly enhanced in the high frequency sector, and its possible observation by large interferometric detectors [14] (such as LIGO and VIRGO); iii) the amplification of electromagnetic perturbations due to their direct coupling to the dilaton background, and the generation of primordial “seeds” for the galactic and extragalactic magnetic field [15]; iv) the generation of the large scale CMB anisotropy directly from the vacuum fluctuations of the electromagnetic field [16]. Finally, I will present some speculations about a possible geometric origin of the CMB radiation itself [17]. This lecture is an extended version of previous lectures given at the Observatory of Paris [18] and at the Gaeta Workshop on the “Very Early Universe” (Gaeta, August 1995, unpublished). The main results reported here are based on recent work done in collaboration with R. Brustein, M. Giovannini, V. Mukhanov and G. Veneziano.

Throughout this lecture the evolution of perturbations will be discussed in a type of background which I will call, for short, “string cosmology background”. At low energy such background represents a solution [3,5] of the gravi-dilaton effective action , at tree-level in the string coupling , and to zeroth order in the inverse string tension ,

(with the possible addition of string matter sources). The background describes the accelerated evolution from the string perturbative vacuum, with flat metric and vanishing dilaton coupling (), towards a phase driven by the kinetic energy of the dilaton field (), with negligible contribution from the dilaton potential (and other matter sources). In this initial phase the curvature scale and the dilaton coupling are both growing at a rate uniquely determined by the action (1.1), and the possible presence of matter, in the form of a perfect gas of non-mutually interacting classical strings, is eventually diluted [3,19].

The background evolution can be consistently described in terms of the action (1.1), however, only up to the time when the curvature reaches the string scale, namely when . At that time all higher orders in (i.e. all higher-derivative corrections to the string effective action) become important, and the background enters a truly “stringy” phase, whose kinematic details cannot be predicted on the ground of the previous simple action. The presence of this high-curvature phase cannot be avoided, as it is required [20] to stop the growth of the curvature, to freeze out the dilaton, and to arrange a smooth transition (at ) to the standard radiation-dominated evolution (where const).

In previous works (see for instance [3,8]) we assumed that the time scales and (marking respectively the beginning of the string and of the radiation era) were of the same order, and we computed the perturbation spectrum in the sudden approximation, by matching directly the radiation era to the dilaton-driven phase. Here I will consider a more general situation in which the duration of the string era () is left completely arbitrary, and I will discuss its effects on the perturbation spectrum.

During the pre-big-bang epoch, from the flat and cold initial state to the highly curved (and strongly coupled) final regime, the background evolution is accelerated, and can be invariantly characterized, from a kinematic point of view, as a phase of shrinking event horizons [1-3]. If we parameterize the pre-big-bang scale factor (in cosmic time) as

then the condition for the existence of shrinking event horizons

is simply . As a consequence, there are two physically distinct classes of backgrounds in which the event horizon is shrinking.

If we have a metric describing a phase of accelerated expansion and growing curvature,

of the type of pole-inflation [21], also called super-inflation (, and a dot denotes differentiation with respect to the cosmic time ). If we have instead a metric describing accelerated contraction and growing curvature scale,

The first type of metric provides a representation of the pre-big-bang scenario in the String (or Brans-Dicke) frame, in which test strings move along geodesic surfaces. The second in the Einstein frame, in which the gravi-dilaton action is diagonalized in the standard canonical form (see for instance [2,3]).

In both types of backgrounds the computation of the metric perturbation spectrum may become problematic, but the best frame to illustrate the difficulties is probably the Einstein frame, where the metric is contracting. It should be recalled, in this context, that the tensor perturbation spectrum for contracting backgrounds was first given in [22], but the possible occurrence of problems, due to a growing solution of the perturbation equations, was pointed out only much later [23], in the context of dynamical dimensional reduction. The problem, however, was left unsolved.

2. The “growing mode” problem

Consider the evolution of tensor metric perturbations, , in a -dimensional conformally flat background, parameterized in conformal time () by the scale factor

Define the correctly normalized variables ( is the Planck mass), whose Fourier components obey canonical commutation relations, . The modes satisfy, for each of the two physical (transverse traceless) polarizations, the well known perturbation equation [10]

(a prime denotes differentiation with respect to ). In a string cosmology background the horizon is shrinking, so that all comoving length scales are “pushed out”of the horizon. For a mode whose wavelength is larger than the horizon size (i.e. for ), we have then the general (to leading order) asymptotic solution

where and are integration constants.

The asymptotic behavior of the perturbation is thus determined by . If the perturbation tends to stay constant outside the horizon, and the typical amplitude at the scale , for modes normalized to an initial vacuum fluctuation spectrum,

can be given as usual [24] in terms of the Hubble factor at horizon crossing ()

In this case the amplitude is always smaller than one provided the curvature is smaller than Planckian. This case includes in particular , namely all backgrounds describing, according to eq. (2.1), accelerated inflationary expansion.

If, on the contrary, , then the second term is the dominant one in the solution (2.3), the perturbation amplitude tends to grow outside the horizon,

and may become larger than one, thus breaking the validity of the perturbative approach. Otherwise stated: the energy density (in critical units) stored in the mode , i.e. , may become larger than one, in contrast with the hypothesis of negligible back-reaction of the perturbations on the initial metric.

One might think that this problem - due to the dominance of the second term in eq. (2.3) - appears in the Einstein frame because of the contraction (which corresponds to ), but disappears in the String frame where the metric is expanding. Unfortunately this is not true because, in the String frame, the different metric background is compensated by a different perturbation equation, in such a way that the perturbation spectrum remains exactly the same [2,3].

This important property of perturbations can be easily illustrated by taking, as a simple example, an isotropic solution of the -dimensional gravi-dilaton equations [2,3], obtained from the action (1.1) supplemented by a perfect gas of long, stretched strings as sources (with equation of state ).

In the Einstein frame the solution describes a metric background which is contracting for ,

and the tensor perturbation equation

has an asymptotic solution (for ) which grows, according to eq. (2.3), as

In the String frame the metric is expanding,

but the perturbation is also coupled to the time-variation of the dilaton background [12],

As a consequence, the explicit form of the perturbation equation is exactly the same as before,

so that the solution is still growing, asymptotically, with the same power as in eq. (2.9).

It may be noted that in the String frame the growth of perturbations outside the horizon is due to the joint contribution of the metric and of the dilaton background to the “pump” field responsible for the parametric amplification process [25]. Such an effect is thus to be expected in generic scalar-tensor backgrounds, as noted also in [26]. The particular example chosen above is not much relevant, however, for a realistic scenario in which the phase of pre-big-bang inflation is long enough to solve the standard cosmological problems. In that case, in fact, all scales which are inside our present horizon crossed the horizon (for the first time) during the dilaton-driven phase or during the final string phase, in any case when the contribution of matter sources was negligible [3,8].

We shall thus consider, as a more significant (from a phenomenological point of view) background, the vacuum, dilaton-driven solution of the action (1.1), which in the Einstein frame (and in ) can be explicitly written as

In such a background one finds that the growth of tensor perturbations is simply logarithmic [13],

so that it can be easily kept under control, provided the curvature scale at the end of the dilaton phase is bounded.

The problem, however, is with scalar perturbations, described in the longitudinal gauge by the variable such that [24]

The canonical variable associated to is defined (for each mode ) by [24]

and satisfies a perturbation equation

which is identical to eq. (2.2) for the tensor canonical variable, with asymptotic solution

Because of the different relation between canonical variable and metric perturbation, however, it turns out that the amplitude of longitudinal perturbations, normalized to an initial vacuum fluctuation spectrum,

grows, asymptotically, like . We have in fact, from (2.16),

This growth, as we have seen, cannot be eliminated by passing to the String frame. Neither can be eliminated in a background with a higher number of dimensions. In fact, in , the isotropic solution (2.13) is generalized as [3]

and the scalar perturbation equation in the longitudinal gauge [3]

has the generalized asymptotic solution

By inserting the new metric (2.21) one thus finds the same growing time-behavior, , exactly as before. The same growth of is also found in anisotropic, higher-dimensional, dilaton-dominated backgrounds [13].

Because of the growing mode there is always (at any given time ) a low frequency band for which . In , in particular, such band is defined [from eq.(2.20)] as . For such modes the linear approximation breaks down in the longitudinal gauge, and a full non-linear treatment would seem to be required in order to compute the spectrum. In spite of this conclusion, a linear description of scalar perturbations may remain possible provided we choose a different gauge, more appropriate to linearization than the longitudinal one.

A first signal that a perturbative expansion around a homogeneous background can be consistently truncated at the linear level, comes from an application of the “fluid flow” approach [27,28] to the perturbations of a scalar-tensor background. In this approach, the evolution of density and curvature inhomogeneities is described in terms of two covariant scalar variables, and , which are gauge invariant to all orders [29]. They are defined in terms of the momentum density of the scalar field, , of the spatial curvature, , and of their derivatives. By expanding around our homogeneous dilaton-driven background (2.13) one finds [13] for such variables, in the linear approximation, the asymptotic solution (),

This solution shows that such variables tend to stay constant outside the horizon, with at most a logarithmic variation (like in the tensor case), which is not dangerous.

As a consequence, the amplitude of density and curvature fluctuations can be consistently computed in the linear approximation (for all modes) in terms of and , and their spectral distribution (normalized to an initial vacuum spectrum) turns out to be exactly the same as the tensor distribution (2.14), which is bounded.

What is important, moreover, is the fact that such a spectral distribution could also be obtained directly from the asymptotic solution of the scalar perturbation equations in the longitudinal gauge [13],

simply by neglecting the growing mode contribution (i.e. setting ). This may suggests that such growing mode has no direct physical meaning, and that it should be possible to get rid of it through an appropriate coordinate choice.

A good candidate to do the job is what we have called [13] “off-diagonal” gauge,

which represents a complete choice of coordinates, with no residual degrees of freedom, just like the longitudinal gauge. In this gauge there are two variables for scalar perturbations, and , and their asymptotic solution is, in the linear approximation,

( and are integration constants). The growing mode is thus completely gauged away for homogeneous perturbations (for which ). It is still present for non-homogeneous perturbations in the off-diagonal part of the metric, but it is “gauged down” by the factor which is very small, asymptotically.

Fortunately this is enough for the validity of the linear approximation, as the amplitude of the off-diagonal perturbation, in this gauge, outside the horizon,

stays smaller than one for all modes , and for the whole duration of the dilaton-driven phase, . We have explicitly checked that quadratic corrections are smaller than the linear terms in the perturbation equations, but a full second order computation requires a further coordinate transformation [13]. The higher order problem is very interesting in itself, but a complete discussion of the problem is outside the scope of this lecture. Having established that the vacuum fluctuations of the metric background, amplified by a phase of dilaton-driven evolution, can be consistently described (even in the scalar case) as small corrections of the homogeneous background solution, let me discuss instead some phenomenological consequence of such amplification. Scalar perturbations and dilaton production were discussed in [3,8,30]. Here I will concentrate, first of all, on graviton production.

3. The graviton spectrum from dilaton-driven inflation

Consider the amplification of tensor metric perturbation in a generic string cosmology background, of the type of that described in Sect. 1. Their present spectral energy distribution, , can be computed in terms of the Bogoliubov coefficient determining their amplification (see Sect. 5 below), or simply by following the evolution of the typical amplitude from the time of horizon crossing down to the present time . For modes crossing the horizon in the inflationary dilaton-driven phase (), and re-entering the horizon in the decelerated radiation era (), one easily finds, from eq. (2.14),

Here is the red-shifted proper frequency for the mode at time , is the critical energy density, = = is the radiation energy density in critical units, and is the maximum frequency amplified during the dilaton-driven phase. Finally, is a possible amplification factor due to the subsequent string phase (), in case that the perturbation amplitude grows outside the horizon (instead of being constant) during such phase. This additional amplification does not modify however the slope of the spectrum, as we are considering modes that crossed the horizon before the beginning of the string phase (see Sect. 5).

An important property of the spectrum (3.1) is the universality of the slope with respect to the total number of spatial dimensions, and with respect to their possible anisotropy. Actually, the spectrum is also duality-invariant [14], in the sense that it is the same for all backgrounds, including those with torsion, obtained via transformations [6] from the vacuum dilaton-driven background.

The spectrum (3.1) has also the same slope (modulo logarithmic corrections) as the low frequency part of a thermal black body spectrum, which can be written (in critical units) as

Here is a constant factor which depends on the time-gap between the beginning of the string phase and the beginning of the string era. We can thus parameterize the graviton spectrum (3.1) in terms of an effective temperature

which depends on the initial curvature scale , and on the subsequent kinematic of the high-curvature string phase.

For a negligible duration of the string phase, , we have in particular , and the spectrum (3.1) is peaked around a maximal amplified frequency Hz, while it is exponentially decreasing at higher frequencies (where the parametric amplification is not effective). Moreover, , so that this spectrum, produced by a geometry transition, is remarkably similar to that of the observed cosmic black body radiation [17] (see also Sect. 9 below).

The problem, however, is that we don’t know the duration and the kinematics of the high curvature string phase. As a consequence, we know the slope () of this “dilatonic” branch of the spectrum, but we don’t know the position, in the () plane, of the peak frequency . This uncertainty is, however, interesting, because the effects of the string phase could shift the spectrum (3.1) to a low enough frequency band, so as to overlap with the possible future sensitivity of large interferometric detectors such as LIGO [31] and VIRGO [32]. I will discuss this possibility in terms of a two-parameter model of background evolution, presented in the following Section.

4. Two-parameter model of background evolution

Consider the scenario described in Sect. 1 (see also [3,8,9]), in which the initial (flat and cold) vacuum state, possibly perturbed by the injection of an arbitrarily small (but finite) density of bulk string matter, starts an accelerated evolution towards a phase of growing curvature and dilaton coupling, where the matter contribution becomes eventually negligible with respect to the gravi-dilaton kinetic energy. Such a phase is initially described by the low energy dilaton-dominated solution,

up to the time , when the curvature reaches the string scale , at a value of the string coupling . Provided the value of is sufficiently negative (i.e. provided the coupling is sufficiently small to be still in the perturbative regime), such a value is also completely arbitrary, since there is no perturbative potential to break invariance under shifts of .

For the background enters a high curvature string phase of arbitrary (but unknown) duration, in which all higher-derivative (higher-order in ) contributions to the effective action become important. During such phase the dilaton keeps growing towards the strong coupling regime, up to the time (at a curvature scale ), when a non-trivial dilaton potential freezes the coupling to its present constant value . We shall assume, throughout this paper, that the time scale marks the end of the string era as well as the (nearly simultaneous) beginning of the standard, radiation-dominated evolution, where and const (see however Sect. 9 for a possible alternative).

During the string phase the curvature is expected to stay controlled by the string scale, so that

where is the Planck length. As a consequence, the curvature is increasing in the Einstein frame (where is constant), while it keeps constant in the string frame, where is constant and the Planck length grows like from zero (at the initial vacuum) to its present value . In both cases the final scale is fixed, and has to be of Planckian order to match the present value of the ratio . Using standard estimates [33]

In analogy with the dilaton-driven solution (4.1), let us now parameterize, in the Einstein frame, the background kinematic during the string phase with a monotonic metric and dilaton evolution,

representing a sort of “average” time-behavior. Note that the two parameters and cannot be independent since, according to eq. (4.2),

from which

(note also that the condition cannot be satisfied by the vacuum solutions of the lowest order string effective action [3], in agreement with the fact that all orders in are full operative in the high curvature string phase [20]).

The background evolution, for this class of models, is thus completely determined in terms of two parameters only, the duration (in conformal time) of the string phase, , and the shift of the dilaton coupling (or of the curvature scale in Planck units) during the string phase, . I will use, for convenience, the decimal logarithm of these parameters,

Here defines the total red-shift associated to the string phase in the String frame, where the curvature is approximately constant and the metric undergoes a phase of nearly de Sitter expansion. It should be noted, finally, that the parameters (4.7) are completely frame-independent, as conformal time and dilaton field are exactly the same both in the String and Einstein frame.

5. Parameterized graviton spectrum

Consider the background discussed in the previous Section, characterized by the dilaton-driven evolution (4.1) for , by the string evolution (4.4) for , and by the standard radiation-dominated evolution for . In these three regions, eq. (2.2) for the canonical variable has the general exact solution

where , and are the first and second kind Hankel functions. We have normalized the solution to an initial vacuum fluctuation spectrum, containing only positive frequency modes at

The asymptotic solution for is however a linear superposition of positive and negative frequency modes, determined by the so-called Bogoliubov coefficients which parameterize, in a second quantization approach, the unitary transformation connecting and states. So, even starting from an initial vacuum state, it is possible to find a non-vanishing expectation number of produced particles (in this case gravitons) in the final state, given (for each mode ) by .

We shall compute by matching the solutions (5.1) and their first derivatives at and . We observe, first of all, that the required growth of the curvature and of the coupling during the string phase (in the Einstein frame) can only be implemented for , i.e. [see eq.(4.5)]. This leads to an inflationary string phase, characterized in the Einstein frame by accelerated expansion (, , ) for , and accelerated contraction (, , ) for . As a consequence, modes which “hit” the effective potential barrier of eq.(2.2) (otherwise stated: which cross the horizon) during the dilaton-driven phase, i. e. modes with , stay under the barrier also during the string phase, since . In such case the maximal amplified proper frequency

is related to the highest frequency crossing the horizon in the dilaton phase, , by

For an approximate estimate of we may thus consider two cases.

If , i.e. if we consider modes crossing the horizon in the string phase, we can estimate by using the large argument limit of the Hankel functions when matching the solutions at , using however the small argument limit when matching at . In this case the parametric amplification is induced by the second background transition only, as and , and we get

(modulo numerical coefficients of order of unity). If, on the contrary, , i.e. we consider modes crossing the horizon in the dilaton phase, we can use the small argument limit of the Hankel functions at both the matching epochs and . This gives ( are numbers of order one), and

We can now compute, in terms of , the spectral energy distribution (in critical units) of the produced gravitons, defined in such a way that the total graviton energy density is obtained as . We have then

According to eqs. (4.6) and (4.7), moreover, and . The tensor perturbation spectrum (5.7) is thus completely fixed in terms of our two free parameters , of the (known) fraction of critical energy density stored in radiation at time , , and of the (in principle known) present value of the ratio , as

The same spectrum can also be obtained, with a different approach, working directly in the String frame (see [14]).

The first branch of this spectrum, with unknown slope, is due to modes crossing the horizon in the string phase, the second to modes crossing the horizon in the dilaton phase (I have omitted, for simplicity, the logarithmic term in eq. (5.9), because it is not much relevant for the order of magnitude estimate that I want to discuss here). Note that, as previously stressed, the cubic slope of the dilatonic branch of the spectrum is completely insensitive to the kinematic details of the string phase. Such details can only affect the overall intensity of the perturbation spectrum, and their effects can thus be absorbed by rescaling the total duration of the string phase. For the case , in which the amplitude of perturbations is constant outside the horizon, we recover in particular eq.(3.1) with .

Let us now impose on such spectrum the condition of falling within the possible future sensitivity range of large interferometric detectors (such as that of the “Advanced LIGO” project [34]), namely

It implies

These conditions define an allowed region in our parameter space which has to be further restricted, however, by the upper bound obtained from pulsar-timing measurements [35], namely

which implies

We have to take into account, in addition, the asymptotic behavior of tensor perturbations outside the horizon. During the dilaton phase the growth is only logarithmic, but during the string phase the growth is faster (power-like) for backgrounds with . Since the above spectrum has been obtained in the linear approximation, expanding around a homogeneous background, we must impose for consistency that the perturbation amplitude stays always smaller than one, so that perturbations have a negligible back-reaction on the metric. This implies at all and . This bound, together with the slightly more stringent bound required by standard nucleosynthesis [36], can be automatically satisfied - in view of the factor in eqs. (5.8), (5.9) - by requiring a growing perturbation spectrum, namely

The conditions (5.11), (5.13) and (5.14) determine the allowed region of our parameter space compatible with the production of cosmic gravitons in the interferometric sensitivity range (5.10) (denoted by LIGO, for short). Such a region is plotted in Fig.1, by taking as a reference value. It is bounded below by the condition of nearly homogeneous background (5.14), and above by the same condition plus the pulsar bound (5.13). The upper part of the allowed region corresponds to a class of backgrounds in which the tensor perturbation amplitude stays constant, outside the horizon, during the string phase (). The lower part corresponds instead to backgrounds in the the amplitude grows, outside the horizon, during the string phase (). The area within the full bold lines refers to modes crossing the horizon in the dilaton phase; the area within the thin lines refers to modes crossing the horizon in the string phase, where the reliability of our predictions is weaker, as we used field-theoretic methods in a string-theoretic regime. Even neglecting all spectra referring to the string phase, however, we obtain a final allowed region which is non-vanishing, though certainly not too large.

The main message of this figure and of the spectrum (5.7) (irrespective of the particular value of the spectral index) is that graviton production, in string cosmology, is in general strongly enhanced in the high frequency sector (kHz-GHz). Such a frequency band, in our context, could be in fact all but the “desert” of relic gravitational radiation that one may expect on the ground of the standard inflationary scenario. This conclusion is independent of the kinematical details of the string phase, which can only affect the shape of the high frequency tail of the spectrum, but not the peak intensity of the spectrum, determined by the fundamental string parameter . As a consequence, a sensitivity of in the KHz region (which does not seem out of reach in coincidence experiments between bars and interferometers [37]) could be already enough to detect a signal, so that a null result (in that band, at that level of sensitivity) would already provide a significant constraint on the parameters of the string cosmology background. This should encourage the study and the development of gravitational detectors (such as, for instance, microwave cavities [38]) with large sensitivity in the high frequency sector.

In the following Section I will compare the allowed region of Fig. 1, relative to graviton production (and their possible detection), to the allowed region relative to the amplification of electromagnetic perturbations (and to the production of primordial magnetic fields).

6. Parameterized electromagnetic spectrum

In string cosmology, the electromagnetic field is directly coupled to the dilaton background. To lowest order, such coupling is represented by the string effective action as

The electromagnetic field is also coupled to the metric background , of course, but in the metric coupling is conformally invariant, so that no parametric amplification of the electromagnetic fluctuations is possible in a conformally flat background, like that of a typical inflationary model. One can try to break conformal invariance at the classical or quantum level - there are indeed various attempts in this sense [39,40] - but it turns out that it is very difficult, in general, to obtain a significant electromagnetic amplification from the metric coupling in a natural way, and in a realistic inflationary scenario.

In our context, on the contrary, the vacuum fluctuations of the electromagnetic field can be directly amplified by the time evolution of the dilaton background [15,41]. Consider in fact the correct canonical variable representing electromagnetic perturbations [according to eq. (6.1)] in a , conformally flat background, i.e. , where . The Fourier modes satisfy, for each polarization component, the equation

obtained from the action (6.1) by imposing the standard radiation gauge for electromagnetic waves in vacuum, , . Such equation is very similar to the tensor perturbation equation (2.2), with the only difference that the Einstein scale factor, in the effective potential , is replaced by the inverse of the string coupling .

Consider now the string cosmology background of Sect. 4, in which the dilaton-driven phase (4.1) and the string phase (4.4) are followed by the radiation-driven expansion. For such background, the effective potential (6.2) is given explicitly by

The exact solution of eq. (6.2), normalized to an initial vacuum fluctuation spectrum ( for ), is thus

where , and .

For this model of background the effective potential grows in the dilaton phase, keeps growing in the string phase where it reaches a maximum around the transition scale , and then goes rapidly to zero in the subsequent radiation phase, where const. The maximum amplified frequency is of the same order as before, , where is the last mode hitting the barrier (or crossing the horizon) in the dilaton phase. For modes with the amplification is thus due to the second background transition only: we can evaluate by using the large argument limit of the Hankel functions when matching the solutions at (which gives , ), using however the small argument limit when matching at , which gives

Modes with , which exit the horizon in the dilaton phase, stay outside the horizon also in the string phase, so that we can use the small argument limit at both the matching epochs: this gives ( are numbers of order of unity) and

We are interested, in particular, in the ratio

measuring the fraction of electromagnetic energy density stored in the mode , relative to the total radiation energy . By using the parameterization of Sect. 4 we have and , so that the electromagnetic perturbation spectrum is again determined by two parameters only, the duration of the string phase , and the initial value of the string coupling, . We find

for modes crossing the horizon in the string phase, and

for modes crossing the horizon in the dilaton phase. The same spectrum has been obtained, with a different approach, also in the String frame [15,16] (in this paper, the definition of has been rescaled with respect to [16], by absorbing into the factor). Finally, this spectrum can be easily generalized to include the effects of an arbitrary number of shrinking internal dimensions during the pre-big-bang phase [15]. The basic qualitative result presented in the following Sections hold however independently of such a generalization.

7. Seed magnetic fields

The above spectrum of amplified electromagnetic vacuum fluctuations has been obtained in the linear approximation, expanding around a homogeneous background. We have thus to impose on the spectrum the consistency condition of negligible back-reaction, at all . For this condition requires a growing perturbation spectrum, and imposes a rather stringent bound on parameter space,

[note that a growing spectrum also automatically satisfies the nucleosynthesis bound , in view of the factor which normalizes the strength of the spectrum, and of eq. (4.3)].

It becomes now an interesting question to ask whether, in spite of the above condition, the amplified fluctuations can be large enough to seed the dynamo mechanism which is widely believed to be responsible for the observed galactic (and extragalactic) magnetic fields [42]. Such a mechanism would require a primordial magnetic field coherent over the intergalactic Mpc scale, and with a minimal strength such that [39]

This means, in terms of our parameters,

Surprisingly enough the answer to the previous question is positive, and this marks an important point in favor of the string cosmology scenario considered here, as it is in general quite difficult - if not impossible - to satisfy the condition (7.2) in other, more conventional, inflationary scenarios.

The allowed region of parameter space, compatible with the production of seed fields [eq. (7.3)] in a nearly homogeneous background [eq. (7.1)], is shown in Fig. 2 (again for the reference value ). In the region within the full bold lines the seed fields are due to modes crossing the horizon in the dilaton phase, in the region within the thin lines to modes crossing the horizon in the string phase. In both cases the background satisfies , i.e. , so that the whole allowed region refers to perturbations which are always growing outside the horizon, even in the string phase.

We may see from Fig. 2 that the production of seed fields require a very small value of the dilaton coupling at the beginning of the string phase,

This initial condition is particularly interesting, as it could have an important impact on the problem of freezing out the classical oscillations of the dilaton background (work is in progress). It also requires a long enough duration of the string phase,

which is not unreasonable, however, when is translated in cosmic time and string units, , namely .

Also plotted in Fig. 2, for comparison, are the allowed regions for the production of gravitons falling within the interferometric sensitivity range, taken from the previous picture. Since there is no overlapping, a signal detected (for instance) by LIGO would seem to exclude the possibility of producing seed fields, and vice-versa. Such a conclusion should not be taken too seriously, however, because the allowed regions of Fig. 2 actually define a “minimal” allowed area, obtained within the restricted range of parameters compatible with a linearized description of perturbations. If we drop the linear approximation, then the allowed area extends to the “south-western” part of the plane (), and an overlap between electromagnetic and gravitational regions becomes possible. In that case, however, the perturbative approach around a homogeneous background could not be valid any longer, and we would not be able to provide a correct computation of the spectrum.

The above discussion is based on the low-energy string effective action (6.1). Of course there are corrections to this action coming from the loop expansion in the string coupling , and higher curvature corrections coming from the expansion. Loop corrections are not important, however, until we work in a region of parameter space in which the dilaton is deeply in its perturbative regime (), as is indeed the case for the production of seed fields (moreover, the non-perturbative dilaton potential is known to be extremely small, as in the weak coupling regime [9]). The corrections may be important, but only for modes which crossed the horizon during the string phase (for modes crossing the horizon before the slope of the spectrum is unaffected by the subsequent background kinematics, as already stressed in Sect. 5 for the case of tensor perturbations). It should be clearly stressed, therefore, that the main result of this Section - namely the existence of a wide region of parameter space in which seed fields are efficiently produced - is completely independent of the unknown details of the string phase, which can influence in a direct way only the high frequency tail of the spectrum, and only control the possible extension of the allowed region in the limit of very large .

8. The anisotropy of the CMB radiation

In the inflationary models based on the low energy string effective action, the spectrum of scalar and tensor metric perturbations grows in general too fast with frequency [3,12,13] to be able to explain the large scale anisotropy detected by COBE [43,44]. If we insist, however, in looking for an explanation of the anisotropy in terms of the quantum fluctuations of some primordial field (amplified by the background evolution), a possible - even if unconventional - explanation in a string cosmology context is provided by the vacuum fluctuations of the electromagnetic field [16].

Consider in fact the electromagnetic perturbations re-entering the horizon () after amplification. At the time of reentry they provide a field coherent over the horizon scale, which can seed the cosmic magnetic fields, as discussed in the previous Section. If reentry occurs before the decoupling era, the perturbations may be expected to thermalize and to become homogeneous very rapidly soon after reentry, because of their interactions with matter sources in thermal equilibrium. Modes crossing the horizon after decoupling, on the contrary, contribute to the formation of a stochastic perturbation background with a spectrum which remains frozen until the present time , and which may significantly affect the isotropy and homogeneity of the CMB radiation. For a complete electromagnetic origin of the observed anisotropy, , at the present horizon scale, Hz, the perturbation amplitude should satisfy in particular the condition

namely

According to our spectrum [eqs.(6.8) and (6.9)] this condition can be satisfied consistently with the homogeneity bound (7.1), and without fine-tuning of parameters, provided the string phase is so long that all scales inside our present horizon crossed the horizon (for the first time) during the string phase, i. e. for (or ). If we accept this electromagnetic explanation of the anisotropy, we have then two important consequences.

The first follows from the fact that the peak value of the spectrum (6.8) is fixed, so that the spectral index , defined by

can be completely determined as a function of the amplitude at a given scale. For the horizon scale, in particular, we have from eqs. (8.1) and (5.3)

I have taken explicitly into account here the dependence of the spectrum on the the present value of the string coupling (which is illustrated in Fig. 3), to stress that such dependence is very weak, and that our estimate of from is quite stable, in spite of the rather large theoretical uncertainty about (nearly two order of magnitude, recall eq. (4.3)).

In order to match the observed anisotropy, , we obtain from eq. (8.4) (see also Fig. 3, where the relation (8.4) is plotted for three different values of )

This slightly growing (also called “blue” spectrum) is flat enough to be well compatible with the present analyses of the COBE data [43,44].

The second consequence follows from the fact that fixing a value of in eq. (8.3) amounts to fix a relation between the parameters and of our background, according to eq. (6.8). If we accept, in particular, a value of in the range of eq. (8.5), then we are in a region of parameter space which is also compatible with the production of seed fields, according to eq. (7.2). This means that we are allowed to formulate cosmological models in which cosmic magnetic fields and CMB anisotropy have the same common origin, thus explaining (for instance) why the energy density of the observed cosmic magnetic fields is of the same order as that of the CMB radiation:

A coincidence which is otherwise mysterious, to the best of my knowledge.

It is important to mention that the values of the parameters leading to eq.(8.5) are also automatically consistent with the bound following from the presence of strong magnetic fields at nucleosynthesis time [45], which imposes at the scale corresponding to the end of nucleosynthesis, Hz. By comparing photon and graviton production [eqs. (6.8) and (5.8)] we find, moreover, that for a background in which lies in the range (8.5) the graviton spectrum grows fast enough with frequency (, ) to be well compatible with the pulsar bound (5.12). Note that, with such a value of , the metric perturbation contribution to the COBE anisotropy is completely negligible.

It should be stressed, finally, that (in contrast with what discussed in the previous Section) the details of the string phase are of crucial importance for a possible electromagnetic explanation of the CMB anisotropy. However, once we accept a model in which the background curvature stays nearly constant in the String frame, the anisotropy discussed in this Section is again generated in a range of parameters for which the dilaton is deeply inside the perturbative regime (), so that the electromagnetic perturbation equations are certainly stable with respect to loop corrections. Moreover, since we are expanding around the vacuum background (), the perturbation equations are also stable in the linear approximation against corrections, provided such corrections appear in the form of powers of the Maxwell field strength with no higher derivative term. This behaviour, on the other hand, is typical of the Eulero forms (like the Gauss-Bonnet invariant) which are conjectured to contribute to the correct (ghost-free) higher curvature expansion of the gravity sector [46]. The possibility discussed in this Section can thus find consistent motivations in a string cosmology context.

9. Possible origin of the CMB radiation

In the standard inflationary picture the density perturbations of the homogeneous and isotropic cosmological model, and the