Cosmology and String Theory
Recent Developments: Brane Cosmology Scenarios
In contrast, in the strong coupling regime, higher-dimensional branes become light and can be produced copiously, driving the Universe to a phase of brane-domination, typical of M-theory. Such a phase, triggered by the growth of the coupling predicted in the context of pre-Big-Bang models, could be the outcome of an epoch of pre-Big-Bang inﬂation, and could characterize the transition to the post-Big-Bang regime. But let us proceed step by step.
The ﬁrst point to be stressed is that M-theory, as a theory of branes, suggests a new and revolutionary interpretation of the extra dimensions present in our Universe and required for the uniﬁcation of the fundamental interactions. In fact, according to M-theory, the charges associated with the electromagnetic, strong, and weak interactions, as well as the ﬁelds of forces generated by these charges, may turn out to be strictly conﬁned on a brane (as occurs, for instance, in the model studied by Horawa and Witten). This has suggested what is known as the braneworld scenario, based on the assumption that our three-dimensional world could be just a three-brane, embedded in an external eleven- dimensional space-time (also called bulk space-time), and that at least one of the extra space-like dimensions orthogonal to the brane – instead of being compactiﬁed and compressed to a very small (Planckian) volume – could have a very large extension (even inﬁnite, in principle). Indeed, if all the interactions can only propagate on the brane, it becomes physically impossible to “feel” the extra orthogonal dimensions, no matter how “large” they are.
There is, however, an important exception to the conﬁnement of the interactions on the brane. What applies to the gauge inter- actions – generated by the charges situated on the ends of open strings – does not necessarily apply to the gravitational interaction, a universal interaction associated with closed strings, and free to propagate in all spatial directions, even those orthogonal to the brane (see Fig. 10.2). Thus, using the gravitational force, we could in principle detect the presence of the extra dimensions (if they are large enough), in spite of our intrinsic nature as three-dimensional creatures, physically unable to get out of the brane. In fact, the gravitational force would spread along all available spatial directions, and its intensity on the brane would become weaker than predicted by the classical laws of Newton and Einstein. So how can one reconcile the possible existence of large extra dimensions orthogonal to the brane with the absence of any observed deviation from the laws of four-dimensional theories of gravity?
A possible answer to the above question has been provided recently by interesting work by two theoretical physicists, Lisa Randall and Raman Sundrum. They have shown that the long- range component of the gravitational force can indeed be conﬁned on the braneworld in which we are living, being free to propagate only along three preferred spatial directions, provided that the extra dimensions external to the brane are characterized by an appropriate curved geometry. In that case there is no need to require the extra dimensions to form a small and compact manifold, as in the conventional Kaluza–Klein scenario. In fact, thanks to the action of the external curvature, the long-range gravitational ﬁeld produced by a mass located on the brane is unable to come out of the brane itself, quite irrespectively of the size of the extra dimension (it is as if they were absent).
For a naive visualization of this effect, we may think of a thin metal plate which is fully embedded in a can ﬁlled with a very viscous paint, and then drawn out. Just as the plate remains covered by a coat of paint, in the same way the gravitational ﬁeld tends to be “glued” to our brane.
According to the Randall–Sundrum model, however, the conﬁnement of the gravitational ﬁeld on the brane is a one hundred percent effective mechanism only for the massless, long-range com- ponent of the gravitational interaction. Just as in the case of the painted plate some heavier drops of paint can break off and drip away from the plate, in the same way the “heavier” components of the gravitational interaction (associated with a new type of massive graviton) could “drop away” from the brane, spreading out in the surrounding dimensions. If we could observe, and measure, such a small “leakage” of gravity, we could directly probe the existence of the extra spatial dimensions.
However, as stressed by the above analogy where the falling drops are the heavier ones, only the heavier components of the gravitational ﬁeld, composed of massive particles, can propagate outside the brane. The exchange of these particles certainly in- duces corrections to the usual form of the gravitational forces, but such corrections have a short range. Summing up the contributions of all massive particles it turns out, in particular, that the corrections to the usual Newtonian potential between two gravitating bodies are proportional to the square of the curvature radius of the extra-dimensional space. If such a radius is sufﬁciently small (i.e., if the curvature is sufﬁciently large), then those gravitational corrections cannot be detected by present experiments, and the extra dimensions may remain invisible.
Concerning this point, we should recall that, according to re- cent experimental results, there is no observed deviation from the predictions of standard Newtonian gravity down to distances of about 0.1 millimeters.2 This imposes an upper limit on the curvature radius of the extra dimensions, and thus an upper limit on the parameter that controls the strength of the gravitational interaction in the space external to the brane. Within the Randall–Sundrum model, the gravitational coupling constant in the external bulk manifold is determined by the product of the Newtonian constant G times the curvature radius of the bulk geometry. Hence, the bulk coupling constant may be larger than the usual, four-dimensional Newtonian constant, but not “too much” larger, if the size of the curvature radius is bounded.
The possibility of a bulk space with a large gravitational coupling parameter is indeed one of the main physical motivations at the heart of all higher-dimensional models with “large” extra dimensions; and not only if such dimensions are inﬁnitely ex- tended, as in the braneworld scenario of Randall and Sundrum, but also if they are compact, as in previous models proposed by Ignatios Antoniadis, Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali. In fact, in our four-dimensional physical world, the strength of the gravitational force appears to be much weaker than the strength of all other forces active at the nuclear and subnuclear level. This is at the origin of the so-called hierarchy problem: Why is there such a difference of strength among the fundamental forces of nature?
In models with large extra dimensions, the coupling strength of the bulk gravitational interaction can be much larger than in the case of four-dimensional gravity, and in principle similar to that of the other interactions, thus resolving (or alleviating) the hierarchy problem. The weakness of the gravitational forces that we experience in four dimensions would in this case be explained as a consequence of the higher-dimensional structure of our Universe, and the particular geometry characterizing the extra dimensions.
It is fair to say that such higher-dimensional scenarios are based on a number of assumptions, and are constrained by various phenomenological consequences. For instance, in order to generate the appropriate curvature of the extra-dimensional geometry, so as to conﬁne four-dimensional gravity and possibly explain its hierarchical weakness, the Randall–Sundrum model requires the presence of a negative cosmological constant in the bulk space external to the brane. In other words, one needs a negative energy density for the higher-dimensional vacuum, a rather unconventional property which leads to a bulk geometry described by the anti-de Sitter metric (associated with a constant but negative space-time curvature).
Among the new phenomenological consequences, we should mention, for instance, the fact that the brane (which is assumed to be static and rigidly ﬁxed at a given position to a ﬁrst approximation) could oscillate in the higher-dimensional space, thus generating massive scalar waves, possible sources of additional short-range corrections to the effective gravitational interaction on the brane.
If we make all necessary assumptions, and accept their possible phenomenological consequences, we may nevertheless formulate consistent models describing our world as a three-brane embedded in a higher-dimensional space-time. In this framework, we may develop a new approach to cosmology and a new perspective for the evolution of the primordial epochs. In particular, we may ask what happens if there is more than one brane, and if they can interact and collide with one another.
A possible answer to the last question has been provided recently by the collaboration of a group of astrophysicists and theoretical physicists, Justine Khoury and Paul Steinhardt at Princeton, Burt Ovrut at Philadelphia, and Neil Turok at Cambridge. They have suggested that the collision of two (or more) branes might indeed simulate the Big Bang marking the beginning of standard cosmological evolution. The resulting cosmological scenario was termed “ekpyrotic”, a name suggesting how our present Universe may emerge “from the ﬁre” (i.e., from the outburst of radiation) produced by the collision of two branes. This scenario tries to ex- plain the presence of the cosmic radiation background and of its temperature anisotropies through the process of brane collision, without resorting to a phase of standard inﬂationary expansion.
According to the ekpyrotic scenario, the Universe must there- fore contain at least two three-branes, one (the “visible” brane) corresponding to our physical world, and another (the “invisible” brane) parallel to the ﬁrst one, located at a certain distance along an extra dimension orthogonal to the branes. The scenario is in- spired by the eleven-dimensional M-theory model of Horawa and Witten, and the two four-dimensional world-volumes spanned by the three-branes are domain walls representing the boundaries of a ﬁve-dimensional bulk-manifold (the remaining six spatial dimensions are assumed to be compactiﬁed on a much smaller scale). In the initial conﬁguration the two branes are ﬂat, parallel, and static, with no matter or radiation present in either brane.
There are two possible versions of the ekpyrotic scenario. Ac- cording to the ﬁrst version the bulk manifold contains a third ﬂoating brane, which is present from the beginning or which is formed spontaneously (at some subsequent time) thanks to the effects of quantum ﬂuctuations. This bulk brane is attracted from the visible brane, and starts to move slowly towards it, moving faster and faster as the two branes get closer. The bulk brane is not perfectly ﬂat like the other branes, but its geometry has small ripples due to the microscopic ﬂuctuations of quantum ﬁelds in vacuum. When the two branes eventually collide, the kinetic energy of the bulk brane is fully transformed into matter and radiation, producing an outgoing ﬂux of particles at very high temperature, giving rise to the the Big Bang. The visible brane, excited by the collision, heats up, gets curved, and eventually expands, reproducing our current Universe (see Fig. 10.3), while the small oscillations of the bulk brane would be the origin of the small anisotropies that we are currently observing in the cosmic background radiation.
The second version of the ekpyrotic scenario is conceptually very similar to the ﬁrst, with the difference that the colliding branes are now the two boundary branes. In that case the distances be- tween the colliding branes coincides with the size of the extra “orthogonal” dimension, which is actually the eleventh dimension of the M-theory model, controlling the strength of all couplings. The size of this dimension shrinks to zero during the pre-Big-Bang phase preceding the collision, and then bounces back to increase again when the two boundary branes separate before the collision.
With an appropriate form of the effective potential controlling the dynamics of the interbrane distance, the branes could keep separating and colliding an (almost) inﬁnite number of times, thus implementing the so-called cyclic scenario (proposed by Neil Turok and Paul Steinhardt soon after the formulation of the ekpyrotic scenario).
It is important to stress that there is a signiﬁcant difference between the pre-Big-Bang phase of the ekpyrotic (or cyclic) scenario and that of the self-dual models described in Chap. 3. In spite of the fact that, in both cases, the initial conﬁguration is ﬂat, cold, and empty, the dilaton and the associated string coupling are indeed decreasing (with the interbrane distance) during the ekpyrotic phase preceding the brane collision – instead of growing, as in pre-Big-Bang models suggested by the duality symmetry. As a consequence, there is a possible technical simpliﬁcation in the ekpyrotic scenario, due to the fact that the collision and bouncing of the branes occurs in the perturbative regime, where the string coupling becomes negligible. The initial state of the ekpyrotic scenario, on the other hand, has settled down in the strong coupling regime. Hence, such an initial state has to be “prepared” in some way. For instance (as already anticipated) by a previous epoch of growing dilaton, like the one typical of the self-dual pre-Big-Bang scenario.
When the coupling becomes strong, and the Universe enters the M-theory regime, there are other types of higher-dimensional objects which come into play, besides the branes marking the boundaries of the space-time manifold. In particular, there are the so-called Dirichlet branes (p-dimensional extended objects dubbed for short Dp-branes). Our four-dimensional world could just correspond to the hypersurface spanned by the evolution of a D3-brane.
These branes can interact among themselves, and their interaction can produce a phase of inﬂationary evolution (of conventional type), thus suggesting a primordial cosmological picture quite different from that of the ekpyrotic scenario.
Let us start by explaining what a Dp-brane is. To this end, we must recall that there are two types of string (open and closed), and that, when studying the propagation of an open string in a higher-dimensional space-time manifold, we have to specify what happens to the ends of the string, imposing appropriate boundary conditions.
There are two types of condition:
- Neumann boundary conditions, if the ends of the string move in such a way that there is no momentum ﬂowing through the boundaries.
- Dirichlet boundary conditions, if the ends of the string are held ﬁxed.
If an open string is propagating through a background manifold with D space-time dimensions, then the position of the ends of the string can be determined in such a way as to satisfy Neumann conditions along p + 1 space-time dimensions, and Dirichlet conditions along the remaining D − p − 1 (spacelike) orthogonal directions. In this way the ends of an open string are localized on two p-dimensional hyperplanes at ﬁxed positions (the two hyperplanes can also be coincident). Such p-dimensional hyperplanes are called Dp-branes.
It is important to stress that the ends of the open string are ﬁxed along the Dirichlet directions, but can move freely along the (orthogonal) p + 1 Neumann directions, spanning the world- hypervolume of the brane. On the other hand, the string ends can carry charges, sources of Abelian or non-Abelian gauge ﬁelds. This gives us a natural implementation of the previously mentioned braneworld scenario, in which the fundamental gauge interactions are strictly localized on a (p + 1)-dimensional hypersurface, which is only a “slice” of the higher-dimensional bulk manifold in which the Dp-brane is embedded. In particular, D3-brane could provide a model for our four-dimensional space-time.
In the context of superstring models, on the other hand, an ex- tended object like a Dp-brane acts as source of an interaction which has a strength of gravitational intensity, and which is mediated by a totally antisymmetric tensor ﬁeld of rank p + 1. In particular, D3-branes can interact with one another not only gravitationally, but also through the exchange of a rank-four antisymmetric tensor ﬁeld. Such an interaction, unlike gravity, is repulsive for sources of the same sign (for instance, two identical branes), and attractive for sources of opposite sign (for instance, a brane–antibrane system), just like the interaction between two electric charges. In particular, for the system formed by two identical, static and parallel branes (like those appearing in the initial conﬁguration of the ekpyrotic scenario), one ﬁnds that the gravitational attraction is exactly balanced by the repulsion due to the antisymmetric-ﬁeld interaction, and the system remains static (up to the addition of other, non- perturbative interactions).
An inﬂationary model can be obtained, in the context of cosmological models based on Dirichlet branes if we consider the interaction of a D3-brane with an anti-D3-brane. In that case, there is no cancellation between the various types of force, and the net result is an attractive interaction between the branes. The coordinate parametrizing the interbrane distance behaves as a scalar ﬁeld, and its potential (generated by the forces between the branes) can in principle sustain a phase of inﬂationary expansion, as pointed out by various groups of theoretical physicists and astrophysicists (including Gia Dvali, Henry Tye, Clifford Burgess, Mahbub Majumdar, Detlef Nolte, Fernando Quevedo, Govindan Rajesh, and Ren-Jie Zhang).
Unfortunately, if the external dimensions orthogonal to the branes are ﬂat and topologically trivial, it turns out that the effective inﬂationary potential generated by the brane–antibrane interaction is unable to guarantee a successful resolution of all the standard cosmological problems. However, there are two ways out of this difﬁculty, at least in principle. A ﬁrst possibility relies on the assumption that the space transverse to the branes is compact and has the topology of an n -dimensional torus, with spatial sections of uniform radius r.
When the separation of the brane–antibrane pair is of order r, the effective potential experienced, say, by the antibrane must be estimated by including the contribution of all the topological “images” of the other brane, forming an n-dimensional lattice. The total effective interaction is then obtained by summing over all the contributions of the lattice sites occupied by the brane images. The resulting effective potential can satisfy the conditions for successful inﬂation (at least, as long as the interbrane separation remains in a range of distances of order r).
An alternative solution (which seems to be preferred, at present, in view of its ability to stabilize the size of all the additional compact dimensions present in the model) has been suggested by Shamit Kachru, Renata Kallosh, Andrei Linde, Juan Martin Maldacena, Liam McAllister, and Sandip Trivedi. Their model is based on the assumption that the section of space orthogonal to the D3-branes is curved, with a geometry of anti-de Sitter type (as in the case of the Randall–Sundrum model). One of the two branes (for instance, the antibrane D3) is frozen at a ﬁxed position. The D3-brane, on the other hand, is mobile along the orthogonal direction z, driven by the attractive force towards the antibrane, and has a time-dependent position (see Fig. 10.4). The potential generated by the interbrane interaction is a function of the (time-dependent) interbrane distance, exactly as in the the case of a ﬂat geometry.
However, the potential energy is now “distorted” by the curvature, which produces a warping of the spatial geometry along the z direction. As a consequence, the effective potential acquires a form satisfying all conditions required to implement a successful inﬂationary model.
Let us conclude this chapter by noting that, in all models of brane–antibrane inﬂation, as in the case of the ekpyrotic scenario, the initial conﬁguration settles down in the strong-coupling, M-theory regime, within an eleven-dimensional space-time ﬁlled with strings, membranes, three-branes, and so on, including the whole possible “zoo” of higher-dimensional objects in mutual interaction. As the value of the initial coupling decreases, however, those heavy higher-dimensional objects tend to disappear from the initial state. In the limiting case in which the initial coupling tends to zero, we recover the string perturbative vacuum, a state without strings or branes, where there is nothing at all in the ﬂat, cold, and inﬁnite space-time but the unavoidable microscopic quantum ﬂuctuations of the metric and the other background ﬁelds (the initial state described in Chap. 5).
The speciﬁc model of cosmological evolution thus crucially depends on our assumptions about the initial value of the coupling parameters (i.e., about the initial strength of all interactions). Different initial conditions can lead to different types of cosmological evolution, and different ways to reach the present cosmological state. Large-scale astrophysical observations, which are becoming more and more accurate, will soon be able to reconstruct the past history of our Universe, thus providing indirect information about the strings (or membranes, or three-branes, etc.) that were present at the beginning.