PRE-BIG BANG COSMOLOGY: AN INTRODUCTION

G. VENEZIANO Theoretical Physics Division, CERN, CH - 1211 Geneva 23

Switzerland

The aim of this talk is twofold: i) presenting the general picture/idea of pre-big bang cosmol­ogy; ii) providing a template/framework for the following four talks on the subject.

1 Preamble

Almost exactly 100 years ago, in 1897, J.J . Thompson discovered the electron. Even today the origin of the small electron inass is clouded in mystery. In the same year, French impressionist Paul Gauguin signed one of his most famous paintings, whose title consists of three questions:

D'ou venons-nous? Que sommes-nous? Ou allons-nous? 100 years later, Gauguin's questions are still very much with us. Modern versions of the second and third questions (What is the value of Q? Will the Universe expand forever?) have been discussed at length at this Conference. What about Gauguin's first question? In a recent article, E. Witten 1 , defending US participation in the LHC project, argued that discov

.ering

supersymmetry would "give a huge boost to . . . string theory." He then goes on to explain why non-specialists should care about this: it is because strings may hold "the key to answering some of the questions that people who are not scientists most often ask" . These modern versions of Gauguin's first question, prompted by experimental evidence that a Big Bang did occur in the early Universe, are (quoting again 1 ) :

"What happened before the Big Bang? What was the beginning of time?" i.e. precisely the subject of this talk.

2 Two attitudes towards two puzzles

It is commonly believed (see e.g. 2) that the Universe -and time itself- started some 15 billion years ago from some kind of primordial explosion, the famous Big Bang. Indeed, the experi-

mental observations of the red-shift and of the Cosmic Microwave Background (CMB) lead us quite unequivocally to the conclusion that, as we trace back our history, we encounter epochs of increasingly high temperature, energy density, and curvature. However, as we arrive close to the singularity, our classical equations are known to break down. The earliest time we can think about classically is certainly larger than the so-called Planck time, tp = v'GNfl, ,...., 10-43s (c = 1 throughout) . Hence, the honest answer to the question: Did the Universe -and time- have a beginning? is: We do not know, since the answer lies in the unexplored domain of quantum gravity.

Besides the initial singularity problem -and in spite of its successes- the hot big bang model also has considerable phenomenological problems. Amusingly, these too can be traced back to the nature of the very early state of the Universe. Let us briefly recall why.

The observable part of our Universe, our present horizon, is about 1028 cm large. At earlier times, the horizon was much smaller: a few Planck times after the Big Bang, the horizon was not much larger that a few Planck lengths, say about io-32 cm. Instead, the portion of space that corresponds to our present horizon was about 1 m m large, i.e. some 30 orders of magnitude larger than the horizon. In other words, at that time, the Universe consisted of (1030)3 = 1090 Planckian-size, causally disconnected regions.

The homogeneity problem of standard cosmology is the observation that there is no reason to expect that conditions in all those 1090 regions were initially the same, since there had never been any causal contact among them. Yet , today, all those regions make up our observable Universe and are homogeneous to one part in 105• A related puzzle is the so-called flatness problem: How come the primordial Universe had a (spatial) curvature radius 30 orders of magnitude larger that H-1?

In my opinion there are three possible attitudes towards solving the above puzzles, but only two of them are scientifically sound. These are:

• The Universe was not particularly homogeneous and flat at the big bang, but a period of accelerated expansion (inflation) after the big bang made it that way. This is the standard post-big bang inflation idea (see e.g. 3) .

• The Universe was already homogeneous and flat at the big bang, since a long period of inflation before the big bang cooked it up that way. This is basically the pre-big bang (PBB) idea.

• There is always, of course, a third answer: some unknown Planckian physics produced an incredibly homogeneous/flat Universe, which is essentially like saying that God decided things that way (see, in this connection, a nice picture in Penrose's book 4) .

How come string theory prefers the second answer in the above list?

3 String Theory hints

The following properties of superstrings strongly speak in favour of the PBB scenario over the one of conventional inflation:

• String theory does not automatically give Einstein's General Relativity at low energy /curvature. Rather, it leads to a scalar-tensor theory of the Jordan-Brans-Dicke (JBD) variety, more­over with a dangerously small WJBD parameter. The extra scalar particle/field, called the dilaton, is unavoidable in string theory.

• The dilaton, which we denote by </>, provides the overall strength of all interactions via relations like (here .\, = � is the fundamental string�length parameter) :

( 1 )

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from which we deduce that the weak coupling region is ¢i � - 1. At present, e<P � 1/25, implying .X, � 10/p � 10-32 cm.

• Dilaton exchange produces a "fifth force" , which threatens precision tests of the equivalence principle 5, hence string theory itself. The simplest way out of this problem (see 6 for an amusing alternative) is to assume that the dilaton has a mass of at least 10-4 eV, coming from supersymmetry-breaking non-perturbative effects. These are negligible at small coupling and therefore the dilaton, when large and negative, behaves like a massless particle.

• Interesting new symmetries (generically called "dualities") thrive on the existence of ¢i. For cosmology, their most relevant representative is the so-called Scale-Factor duality (SFD) , whose action is as follows.

An FRW cosmology at t > 0, i.e. a deceleroting expansion of the Universe (apparently) origi­nating from a singularity as t -t o+

gets mapped, via SFD, into an inflationary cosmology at t < 0, i.e. an acceleroted expansion of the Universe (apparently) going towards a singularity as t -t o- .

This second cosmology looks impossible at first since, in order to have inflation, one needs a peculiar equation of state (3p + p < 0), and a scalar field with no potential energy cannot satisfy that condition. The puzzle is solved, within string (or JBD) gravity, by observing that G'}.f1 � l� is controlled by ¢i through (1 ) . It is indeed perfectly possible for H to grow, even though the energy density decreases, provided ¢i is also growing. We thus come to an important conclusion:

Pre-Big Bang inflation needs a growing dilaton.

4 Pre-big bang's postulate/conjecture

If we wish to use a dual cosmology for the prehistory of the Universe, given the positive signs of H and ¢, we have to start from (very?) small initial values for H and e<P. Although not strictly necessary, we will also impose, for the sake of simplicity, an almost empty initial Universe. This leads us to the following basic postulate of PBB cosmology:

The Universe started its evolution from the simplest conceivable initial state in string theory, its perturbative vacuum. This corresponds to an (almost)

EMPTY, COLD, FLAT, FREE Universe as opposed to the standard

DENSE, HOT, HIGHLY-CURVED initial state of conventional cosmology.

For this assumption to make sense, the new initial conditions should be able to provide, at later times, a hot big bang with the desired characteristics. This looks like a very hard task, but, as the following speakers will explain, it is all but impossible.

The above postulate is good news for the theorist. Indeed, if both coupling and curvature are small, physics is adequately described by the tree-level low-energy effective action of string theory, which is something we know all about. Its form reads:

(2)

where we have included the contributions of the Kalb-Ramond antisymmetric tensor field Bµv through its field strength Hµvpi as well as some typical gauge fields.

The suggestion from string theory now becomes almost a compelling one 7: Can one combine a standard cosmology at t > 0 with a dual PBB cosmology at t < 0 to construct a smooth cosmology containing both? Since, in such a scenario, the Hubble parameter grows for t < 0 and decreases for t > 0, it should reach its maximum at t = 0, instant to be identified with the Big Bang of standard cosmology. The problem is that this maximum is actually infinite, if one works in the context of the low-energy effective theory (2). The pre- (post-) big bang phases have singularities in the future (past) . However, the low-energy approximation breaks down as soon as the Hubble parameter reaches values 0(>..;-1) , leading us to expect that the maximal value of H, reached at t = O, should actually be finite and of that order. All this leads to the main conjecture of PBB cosmology, the so-called graceful-exit assumption:

Pre-big bang and FRW phases smoothly join through a high-curvature/large-coupling (string) phase

Before discussing some recent work (and controversy) on the first (pre-big bang) phase, I wish to mention how the following talks on string cosmology can be situated within the general picture I have outlined. Alessandra Buonanno and Ramy Brustein will discuss cosmological particle production (mainly photons and gravitons, respectively) during the transition between the PBB and the FRW phase. Mairi Sakellariadou will discuss whether large-scale inhomogeneities in the energy density of those particles (around recombination) could explain CMB anisotropies. Finally, Burt Ovrut's talk may well contain the key to answering the graceful-exit problem through the latest developments of string theory.

5 Pre-big bang as classical gravitational collapse

The PBB postulate can be rephrased in more precise terms as a postulate of Asymptotic Triviality (AT) , where asymptotic means at (timelike and null) past infinity, and triviality refers both to space-time and to the dilaton. In analogy with QCD's asymptotic freedom AT is very different from just triviality. Actually, while the trivial vacuum of string theory is a very special point in the space of classical solutions, AT vacua are generic, in the technical sense that they depend on as many arbitrary functions as the general solution.

Determining the properties of a generic solution to the field equations implied by (2) is a formidable task, even if one neglects the Kalb-Ramond field. Fortunately, however, we can map our problem into one that has been much investigated, both analytically and numerically, in the literature: the problem of gravitational collapse of a scalar field. This is simply done by "going to the Einstein frame" , i.e. by the field redefinition

(3)

in terms of which (2) becomes:

(4)

where </>o (Ip = et<Po >.,) is the present value of the dilaton (of Planck's length) . This problem has been considered precisely in the regime of interest to us, i.e. starting from

very "weak" initial data with the aim of finding under which conditions gravitational collapse must occur later. Gravitational collapse basically means that the (Einstein) metric shrinks to zero at a spacelike singularity. However, typically, the dilaton blows up at that same singularity. Given the relation (3) between the Einstein and the (physical) string metric, we can easily imagine that the latter blows up near the singularity, thus giving inflation and a big bang. What can we say at the moment about details on all this? Not much, since this part of the game is just starting, but here it is.

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Assume that, deeply into the PBB phase, the Universe was not particularly homogeneous, spatial gradients and time derivatives being comparable and, in accordance with the PBB pos­tulate, both tiny in string units. It can be argued 8 that such initial conditions can lead to a chaotic version of PBB inflation since, if a certain patch develops where time derivatives slightly dominate over spatial gradients, then inflation turns on in that patch. Thereafter, the evolution of the system can be studied by analytic methods 9, since the approximation of neglecting spatial gradients becomes increasingly accurate within the inflating patch.

In the general case, no analytic methods are known for analysing the system at even earlier times (i.e. before the inflationary patch forms), while numerical codes are still largely inade­quate. Both analytic and numerical methods are instead available for solutions exhibiting some symmetry, e.g. spherical symmetry. In that case, recent numerical work 10 has fully confirmed the occurrence of inflation and the validity of the gradient expansion 9• However, in the spher­ically symmetric case, the most promising avenue 11 appears to consist in making use of the powerful analytic results by Christodoulou 12, which provide sufficient criteria for collapse (or for its absence). Those criteria are "scale-free" : in no way does the collapse theorem of12 fix the absolute curvature scale at the onset of collapse (i.e. when a closed trapped surface first forms). This curvature scale becomes 1 1 the scale of inflation (the Hubble parameter) at the onset of inflation in the original PBB problem. But why is this scale so important?

The point is that, in order to solve the homogeneity/flatness problems, dilaton-driven infla­tion has to last sufficiently long. Its duration is not infinite, since it is limited in the past by the conditions of collapse and, in the future, by the time at which curvatures become so large that we can no longer trust the low-energy approximation. Thus, as was actually noticed from the very beginning 7 , a successful PBB scenario does require very perturbative initial conditions, so that it takes a long time to reach the BB singularity. Fortunately, both the initial coupling and the initial curvature are free classical parameters, possibly with a random distribution throughout space.

A particular case of this "fine-tuning" was discussed recently by Turner and Weinberg and by Kaloper et al. 13 . These authors consider a homogeneous -but not spatially flat- Universe and notice that the duration of PBB inflation is limited in the past by the initial value of the spatial curvature. This has to be taken very small in string units if sufficient inflation is to be achieved. The issue is whether or not this is fine-tuning. String theory has a single length parameter, >-,, but, fortunately for us, it also has massless states and low-energy vacua (such as Minkowski space-time) , whose characteristic scale is arbitrarily larger than A8• Thus I see no fine-tuning in starting the evolution of the Universe in a state of low-energy, small curvatures, and small coupling. On the contrary, I find it very amusing/exciting to realize that a well-known classical instability pushes the Universe from low energy (curvature) and small coupling towards high energy (curvature) and large coupling.

But this is certainly not the last word on the subject . . .

6 Conclusions

Pre-big bang cosmology appears to have survived its first 6 or 7 years of life. Interest in (and criticism of) it is growing. It is perhaps time to make a balance sheet.

Conceptual (technical?) and phenomenological problems include:

• Graceful exit from dilaton-driven inflation to FRW cosmology is not fully understood, in spite of recent progress 14 • Possibly, new ideas borrowed from M-theory and D-branes could help in this respect (see B. Ovrut's talk) .

• A scale-invariant spectrum of large-scale perturbations is not automatic, although, thanks to possibly flat axion spectra, it does not look inconceivable either (see M. Sakellariadou's

talk).

Attractive features include:

• No need to "invent" an inflaton, or to fine-tune potentials.

• Inflation is "natural" , thanks to the duality symmetries of string cosmology.

• The initial conditions problem is decoupled from the singularity problem: a solution to the former is already shaping up and looks exciting.

• A classical gravitational instability (similar to the one giving gravitational collapse and singularities in General Relativity) finds a welcome use in providing inflation.

• A quantum instability (pair creation) is able to heat up an initially cold Universe and generate a hot big bang (see A. Buonanno's and R. Brustein's talks) with the additional virtues of homogeneity and flatness.

• Last but not least: we are dealing with a highly constrained, predictive scheme, which can be tested/falsified by low-energy experiments, as will be explained in the following talks.

What is my own forecast? It is simple: probably, PBB cosmology will not reach age 10 . However, it will not be killed by Hawking, Linde, or Turner, but, more likely, by Planck . . . and I do not mean Max!

References

1 . E. Witten, The New Republic, Dec. 29, 1997. 2. S. Hawking, Proceedings of the Texas/ESO-CERN Symposium on Relativistic Astro­

physics, Cosmology, and Fundamental Physics, Brighton, 1990, eds. J .D. Barrow, L. Mes­tel and P.A. Thomas, Ann. NY Acad. Sci. 647 (1991) 315 .

3 . E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, 1990) . 4. R. Penrose, The Emperor's New Mind (Oxford University Press, 1989) , Fig. (7.19) . 5. T.R. Taylor and G. Veneziano, Phys. Lett. B213 (1988) 459. 6. T. Damour and A.M. Polyakov, Nucl. Phys. B423 (1994) 532. 7 . G. Veneziano, Phys. Lett. B265 (1991) 287; M. Gasperini and G. Veneziano, As­

tropart. Phys. 1 (1993) 317; Mod. Phys. Lett. AB (1993) 3701 ; Phys. Rev. D50 ( 1994) 2519; an updated collection of papers on the PBB scenario is available at http://www.to.infn.it/teorici/gasperin/.

8. G . Veneziano, Phys. Lett. B406 (1997) 297; A. Buonanno, K. A. Meissner, C. Ungarelli and G. Veneziano, Phys. Rev. D57 ( 1998) 2543; A. Feinstein, R. Lazkoz and M.A. Vazquez-Mozo, Closed inhomogeneous string cosmolo­gies, hep-th/9704173; J . D. Barrow and M. P. Dabrowski, Is there chaos in low-energy string cosmology?, hep-­th/9711049; J. D. Barrow and K. E. Kunze, Spherical curvature inhomogeneities in string cosmology, hep-th/9710018; K. Saygily, Hamilton-Jacobi approach to pre-big bang cosmology at long wavelengths, hep-­th/9710070.

9. V. A. Belinskii and I.M. Khalatnikov, Sov. Phys. (JETP) 36 ( 1973) 591; N. Deruelle and D. Langlois, Phys. Rev. D52 (1995) 2007; J. Parry, D. S. Salopek and J. M. Stewart, Phys. Rev. D49 (1994) 2872.

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10. J. Maharana, E. Onofri and G. Veneziano, A numerical approach to pre-big bang cosmology, gr-qc/9802001 , to appear in JHEP.

1 1 . A. Buonanno, T. Damour and G. Veneziano, to appear. 12. D. Christodoulou, Comm. P. A. Math. , 44 (1991) 339, and references therein. 13. M. Turner and E. Weinberg, Phys. Rev. D56 (1997) 4604;

N. Kaloper, A.D. Linde and R. Bousso, Pre-big bang requires the Universe to be exponen­tially large from the very beginning, hep-th/9801073.

14. M. Gasperini, M. Maggiore and G. Veneziano, Nucl. Phys. B494 ( 1997) 315; R. Brustein and R. Madden, Phys. Lett. B410 (1997) 1 10; Phys. Rev. D57 (1998) 712.