Superstrings,
Black Holes, Branes, and Cosmology
Superstring theory is one of the most promising candidates for the ultimate theory of the quantum gravitational field, incorporating the hope that quantum gravity is unified with the other fundamental interactions [1,2].
The essence of string theories is based on the hypothesis that fundamental objects are not pointlike particles but one-dimensional entities whose structure becomes evident only at very short distances (around 10-33 cm). Superstrings, which exist in a spacetime with ten dimensions, six of which are believed to be compactified to an unobservable small size. Elementary particles are identified by the vibrational modes of the strings. Supergravity theories can arise from string theories as effective field theories in the long-wavelength limit.
Five distinct mathematically consistent string theories are known. However, it has become clear that these theories are actually related by duality relations. Possibly, they can be interpreted as different realisations of a deeper new theory called M-theory. M-theory is believed to be an eleven-dimensional theory whose fundamental objects are not superstrings but, rather, higher dimensional entities such as two-dimensional and five-dimensional supermembranes. The low-energy limit of M-theory is given by supergravity in eleven dimensions [2]. Again, seven dimensions are thought to be curled up to an unobservable small size.
A natural arena where the ideas mentioned above can be tested, at least at a phenomenological level, is given by early universe cosmology and black hole physics. Indeed, both black hole theory and cosmology should be re-examined in the context of Supergravity/Superstring theory [3-6]:
· The description of black holes in string theory points to very important results because it may provide the answer to well-known problems, for instance the origin of the black hole entropy and of the information paradox [2,7,8];
· Furthermore, string theories may constitute an important research program in early cosmology since the corresponding field equations have obviously a different structure from Einstein equations. Thus short-distance modifications of general relativity due to superstring/supergravity theories would affect the dynamics of the early universe [4-6,9-12] and be crucial in order to understand long standing problems. More precisely, symmetries of string theory provide an alternative picture with the aditional feature that superstrings and M-theory are naturally formulated in a higher dimensional spacetime. So it seems natural to query how extra dimensions are affecting the physics of the four-dimensional world.
The aim of the research project to carry on at the Astrophysics and Cosmology Group of the Physics Department of UBI (GATC-DF/UBI) is therefore to give a contribution to the ongoing discussion of both superstring/supergravity theories. In particular, focusing on black hole/early universe physics by investigating related issues in the context of effective theories derived from M-theory, superstring and/or supergravity theories.
The underlying motivations for this specific research proposal rely in the mutual benefit that may follow from the interplay of techniques and results peculiar of superstring/supergravity and black hole/early cosmology research programs. Let us illustrate in detail some ideas that we plan to realise in the framework of this research project.
Ø Quantum Cosmological Implications of Supersymmetry and String Theory
The presence of a Quantum Gravity theory is mandatory if one wants to thoroughly address long-standing and fundamental questions such as ``Why is our Universe as it is’’ and ``How did it evolve to its present form’’. En route towards a theory of quantum gravity many significant results have been obtained in the last 15 years or so (see ref. [5,13,14,15] for a review).
Further advances in our understanding about the nature of quantum gravity can be explored within a specific methodology that is directly based on two celebrated approaches to quantum gravity: quantum cosmology and string theory [4-6,9-12]. This essential aspect brings about innovative concepts and instruments, which are new in comparison with other approaches regarding quantum gravity and cosmology. In more precise terms, we will therefore focus this section of our research in quantum cosmological models derived from supergravity and string theory inspired actions.
Regarding string theory, it basically describes particles and interactions by way of the oscillation modes of different types of strings. It is now widely accepted that all five 10-dimensional superstring theories are equivalent and related by a particular type of symmetry transformation [2] designated as dualities. At a cosmological scenario, these may correspond to (for the scale factor of a cosmological model) and (for the corresponding scalar field), playing thereby an important role: it maps an expanding onto a contracting cosmological solution, leaving the string action invariant [9-11]. Moreover, it provides a promising inflationary scenario (driven by the dilaton field) in a quite natural way, in what as been designated as a Pre Big-Bang scenario [10]. The basic assumption is that the Universe starts at a flat, empty, string vacuum state and then evolves accelerating (kinetically driven by the inflation) towards a state of increasing curvature and typically a non-perturbative regime. A transition (yet to be fully clarified) should then occur, leading the universe to the Standard (Post Big Bang) phase of evolution.
As far as quantum cosmological methods applied to the early universe are concerned, the physical state, , of the universe is identified as a wave function instead of a classical space-time solution. The almost totally of models that have been considered [5,14,15] have all but a finite number of degrees of freedom ``frozen’’. This is achieved by restricting the fields to be homogeneous and such models are known as (finite dimensional) minisuperpaces.
By combining the main features of the string and quantum cosmological approaches we get a pertinent as well as active framework to analyse the very early stages of our universe. Some advances have been recently achieved, where members of this project have played quite a noteworthy role [4-6,9-12,16-22]. In particular, it has been suggested in recent years or so that the presence of dualities in quantum cosmological models induce the existence of quantum states that have invariance under Supersymmetry (SUSY) [9,11,22].
The presence of SUSY invariance in a description of the very early universe represents an element of the uppermost value. In fact, SUSY plays a crucial role in Supergravity (that is, local SUSY) and Superstring theory by inducing the cancellation of divergences that would otherwise be present in plain quantum gravity theories. Moreover, some supergravity theories represent a ``square-root’’ of Einstein gravity: to determine physical states it is sufficient to employ the Lorentz and Supersymmetry invariances. Instead of dealing with the second-order differential equation H = 0 we may just have to solve a system of coupled first-order differential equations --- each one similar to a Dirac-like equation. This scenario is designated as Supersymmetric Quantum Cosmology (SQC). Most of the previous research in SQC has been aimed at finding quantum states and overcome consistency problems (see ref. [5]).
We thus propose to advance the current understanding of quantum cosmology by investigating thoroughly the following issues within the framework of superstrings and supergravity applied to cosmology:
(1) Quantum Cosmological Models in the presence of String Dualities
(1.1) Examine how the presence of duality transformations induces the existence of SUSY invariance. Previous studies involving FRW models (that is, homogeneous and isotropic) have suggested the existence of N=2 SUSY [9,22]. We will extend this assertion towards a considerable larger range by analysing a broader class of homogeneous cosmologies (that is, Bianchi models, which are not isotropic). If this could be proven it would point to deeper structures of symmetry that could be present in a full quantum gravity theory.
(1.2) Investigate (specific) minisuperspace models arising from the bosonic sector of string effective actions with potentials derived from the requirement of S and T duality [10]. Namely, by employing inhomogenous perturbations of the metric and dilaton fields. In that context, new quantum states, which would have a physical significance regarding (a) a period of evolution from String (Pre Big Bang)/Supergravity cosmological physics towards a semi-classical stage, together with (b) identifying the existence of any quantum state associated with dilaton driven inflation and to structure formation, may be found.
(1.3) Quite recently, numerical simulations regarding inhomogenous cosmological models in string cosmology [23] have been investigated. However, they have not provided a full agreement or a consistent scenario. Our purpose is to further investigate these models but within the more general scenario of scalar-tensor theories, where the dilaton kinetic term in the action is multiplied by a constant factor . These include low-energy string actions as a particular case (= -1), as well as a cosmological constant. Examining such a general scenario where string theory fits, will clearly assist us in identifying the reasons(s) for those apparent differences. Moreover, it will provide a relevant picture of the dynamical processes that are involved.
(1.4) Another line of research that we aim to analyse is the recent cosmological scenario within the Horava-Witten formulation of M-theory [24]. This is an active and most innovative area of research where a fundamental description for the origin and subsequent evolution of the very early Universe ought to be addressed. In particular, when dealing with a framework where cosmological solutions entail as a direct consequence precise models for particle physics.
(2) Supersymmetric Quantum Cosmology (SQC)
(2.1) Address the same Bianchi models indicated in (1.1) but derived directly from N=1 supergravity, where a fermionic matter sector is explicitly present, that is, in a SQC context [4,5,20,21]. This an urgent and pertinent issue: previous results with other models and different matter contents have pointed instead to N=4 SUSY [5,20,22]. Our purpose is then to compare these results with (1.1), discussing the physical reasons for any possible differences. In particular, regarding the level of SUSY and its relation with the existence of dualities: how does duality relates to supersymmetry?
(2.2) Analyse the retrieval of semi-classical features and origin of structure formation in SQC [5,6]. Is it possible to identify a consistent quantum to classical transitions in SQC? How does this compare to plain gravitational theories with matter fields but no SUSY? That is, which additional feature(s) does the presence of SUSY invariance brings about in a quantum mechanical description of the very early universe? Is there an imprint of an early SUSY quantum epoch into the observed universe?
(2.3) Examine very recent results concerning the possibility of quantum creation of open FRW universes (i.e., with negative spatial curvature) [25]. Such theoretical scenarios could provide a consistent justification regarding some cosmological observations, even though the evidence that we live in a spatially open universe has become less strong. However, there is no widely accepted theoretical quantum description for them. Furthermore, there persist sharp confronting issues to be settled [25]. Our objective is to investigate quantum open universes in the presence of complex scalar fields derived from N=1 supergravity [5,16,19], since all quantum models studied so far have employed real scalar fields within the theory of General Relativity.
Ø Black Holes in Quantum Gravity Theories
The quantisation of black holes will provide a key to the construction of a full, nonperturbative, theory of quantum gravity. Although still far from this final goal, recent years have seen significant progress in this respect;
· First, within canonical gravity, the quantisation of the eternal spherically symmetric black hole has been understood, while the quantisation of black holes resulting from collapsing stars was understood within a consistent semiclassical quantisation scheme[26-40];
· Second, progress in superstring theory [1] led to a statistical mechanical description of extremal and near-extremal black holes in terms of D-brane states [1,2].
In general terms, our purpose is to both understand more thoroughly the semiclassical expansion scheme (derived from canonical quantisation) for black holes as well as connecting these results with methods and results derived within a string framework.
More precisely, we will compare and examine contrasting features regarding canonical quantization and string theory, as far as black hole thermodynamics is concerned. On the one hand, results obtained within canonical quantum gravity (i.e., from a wave function for the black hole system) have shown how thermodynamical quantities (entropy, temperature) can acquire a geometrical interpretation from quantum black holes [26-40]. On the other hand, string theory has given a novel statistical context to black hole thermodynamics. Namely, from such concepts as supersymmetric (BPS) solitonic states and extended objects designated as D-branes [1,2]. One particular urgent and issue we want to investigate is why and how some thermodynamical quantities (e.g., temperature) differ if we employ a canonical gravity framework or string theory. Other important challenging and actual problems comprise analysing the above issues within black strings and infer if the mass of black holes can be quantified.
In addition, we shall study the so-called black strings [33] and try to apply recent results about the interpretation of black hole entropy from the canonical approach to this case [26-40]. This should yield a deeper understanding of the entropy as arising from counting string states [1,2].
Moreover, we shall study in detail the question whether the area of quantum black holes is quantised in Planck units. We hope thereby to put earlier heuristic derivations of this quantisation on a firmer footing. In essence, we want to get some insight into the final phase of black hole evaporation from canonical gravity.
Finally, we will further investigate the quantisation of the Schwarzschild black-hole in the apparent horizon [3,39,40]. The relevant feature is that at the apparent horizon the Hamiltonian and difeomorphism (momentum) constraints become proportional. Hence, the solution of the problem may become substantially simple. This scenario is also most adequate to examine dynamical evaporating black holes, where the mass is a function of a time coordinate and there is as matter source flux of outgoing radiation. We thus plan to examine (a) which vacuum states are preferred within this framework and (b) investigate which consequences it brings about for black-holes in Einstein-yang-Mills theories.
The investigation programme thoroughly described above constitutes the intended sequel regarding the last six or seven years of research work conducted by the Principal Investigator. The overwhelming majority of this research has been conducted at the DAMTP, University of Cambridge, where Post-Doctoral experience as well as scientific research on SQC and string theory were acquired. Our research proposal thus represents the resolute intention of young and newly graduated scientists to conduct and consolidate these new areas of research work in Portugal. In particular, it will sustain previously initiated lines of investigation, strengthening collaboration with other research centres as well. Consequently, it will definitely contribute towards enlarging and promoting innovative investigation issues in Portugal for future prospective researchers.