Physical Space Time

Within the framework of general relativity, the dynamics of the physical space-time is actually related with the history and evolution of the Universe. The mathematical description of space-time does allow for a wide range of scenarios; however, recent developments in observational cosmology do indicate that our Universe is well described by a flat Robertson-Walker metric, meaning that the energy density of the Universe is fairly close to the critical one, pc = 3H^/8nG ~ 10-29g/cm3, where H0 ~ 73 km s-1Mpc-1 is the Hubble expansion parameter at present. Furthermore, CMBR, Supernova, and large-scale structure data are consistent with each other if and only if the Universe is dominated by a smoothly distributed energy that does not manifest itself in the electromagnetic spectrum - dark energy. Moreover, it is found that the large scale structure of the Universe, as well as the dynamics of galaxies, requires matter that like dark energy, does not manifest electromagnetically - dark matter. More exactly, in the cosmic budget of energy, dark energy corresponds to about 73% of the critical density, while dark matter to about 23% and baryonic matter, the matter that we are made of, to only about 4% [30].

Actually, the dominance of dark energy at the present does have deep implications for the evolution of space-time. For instance, if dark energy remains the dominant component in the energy budget in the future, then geometry is no longer the determinant factor in the destiny of the Universe. As is well known, in a Universe where dark energy is subdominant, flat, and hyperbolic geometries give origin to infinity universes in the future; in opposition, a spherical universe does eventually recollapse and undergoes a Big Crunch in a finite time. If however, dark energy is the dominant component, the fate of the Universe is determined by the way it evolves. If its energy density is decreasing, the Universe will eventually be dominated by matter and its destiny is again ruled by its geometry as described above. If, on the other hand, the energy density remains constant, then the Universe expansion will continue to grow and the Universe will be quite diluted of matter. That is to say that, in the remote future the Universe will correspond to a dS space with a future horizon. This means that the world will have features similar to an isolated thermal cavity with finite temperature and entropy. A more drastic fate is expected if the energy density of dark energy continues to grow. This growth will eventually cause a Big Rip, that is, the growing velocity of the space-time expansion will eventually disrupt its very fabric and all known structures will be ripped off.

Actually, an ever accelerating Universe might not be compatible with some fundamental physical theories. For instance, an eternally accelerating Universe poses a challenge for string theory, at least in its present formulation, as it requires that its asymptotic states are asymptotically free, which is inconsistent with a space-time that exhibits future horizons [31-33]. Furthermore, it is pointed out that theories with a stable supersymmetric vacuum cannot relax into a zero-energy ground state if the accelerating dynamics is guided by a single scalar field [31,32]. This suggests that the accelerated expansion might be driven by at least two scalar fields. It is interesting that some two-field models allow for solutions with an exit from a period of accelerated expansion, implying that decelerated expansion is resumed (see e.g. Ref. [34]). Hence, a logical way out of this problem is to argue that the dS space is unstable. This might also occur, for instance, due to quantum tunneling, if the cosmological constant is not too small.

Another significant feature about our Universe is that only if it has undergone a period of quite rapid and accelerated expansion in its early history, one can understand why its spatial section is so close to flat and why it is so homogeneous and isotropic on large scale [35-37]. This inflationary phase of accelerated expansion, a tiny fraction of a second after the big bang, about 10~35 seconds, corresponds to a period where the geometry of the Universe is described by a dS space. It is quite remarkable that a rather brief period of inflation, a quite generic behavior of most of the anisotropic Bianchi-type spaces [1], Kantowski-Sachs spaces [39] and inhomogeneous spaces [40] dominated by a cosmological constant, drives a microscopic Universe into a large one, whose features closely resemble ours. Moreover, in this process, small quantum fluctuations of the field responsible for inflation, the inflaton, are amplified to macroscopic sizes and are ultimately responsible for the formation of large-scale structure (see e.g. Ref. [41] for an extensive discussion). It is a great achievement of modern cosmology that the broad lines of this mechanism are corroborated by the observed features of the CMBR, such as its main peak, whose position is consistent with the size of the scalar density fluctuations that first reentered the horizon, as well as the nearly scale invariant and Gaussian nature of these fluctuations.

However, in what concerns space-time, the stock of surprises arising from physics is far from over. Indeed, recent developments on the understanding of string theory have led to speculations that may be regarded as somewhat disturbing for those who believe that the laws of nature can be described by an action, which encompasses the relevant underlying fundamental symmetries, and from which an unique vacuum arises and the spectrum of elementary objects, particles, can be found. These view has been recently challenged by a quite radical set of ideas. The genesis of these can be traced from the understanding that the initial outlook concerning the original five distinct string theories was not quite correct. It is now understood that there is instead a continuum of theories, that includes M-theory, interpolating the original five string theories. One rather speaks of different solutions of a master theory than of different theories. The space of these solutions is often referred to as the moduli space of supersymmetric vacua or supermoduli-space. These moduli are fields, and their variation allows moving in the supermoduli-space. The moduli vary as one moves in the space-time, as moduli have their own equations of motion.

However, the continuum of solutions in the supermoduli-space are supersymmet-ric and have all a vanishing cosmological constant. Hence, in order to describe our world, there must exist some non-supersymmetric "islands" in the supermodulispace. It is believed that the number of these discrete vacua is huge, googles, G = 10100, or googleplexes 10G, instead of unique [42]. If the cause of the accelerated expansion of the Universe is due to a small cosmological constant, then the state of our Universe corresponds to moduli values some of the non-supersymmetric islands in the supermoduli-space. The fact that the magnitude of the cosmological constant is about 10120 smaller than its natural value Mp, where MP = 1.2 x 1019 GeV is the Planck mass, makes it highly unlikely to find such a vacuum, unless there exists a huge number of solutions with every possible value for the cosmolog-ical constant. The space of all such string theory vacua is often referred to as the landscape [43].

From the landscape proposal springs a radical scenario. In principle, vacua of the landscape do not need to correspond to actual worlds, however, very much on the contrary, it is argued that the string landscape suggests a multiuniverse. According to this proposal, the multiple vacua of string theory is associated to a vast number of "pocket universes" in a single large mega-universe. These pocket universes, like the expanding universe we observe around us, are all beyond any observational capability, as they lie beyond the cosmological horizon. In the words of Susskind, a vociferous proponent of the multiuniverse idea [44], "According to classical physics, those other worlds are forever completely sealed off from our world". Clearly, the implications of these ideas are somewhat disturbing. First, the vacuum that corresponds to our world must arise essentially form a selection procedure, to be dealt with via an-thropic or quantum cosmological considerations. Thus, it seems that somehow our existence plays an important role in the selection process. Second, the vast number of vacua in the landscape ensures the reality of our existence; one refers to this scenario as the anthropic landscape, when based on anthropic arguments. For sure, this interpretation is not free from criticism. It has been pointed out, for instance, that the impossibility of observing a multiuniverse implies that its scientific status is questionable. It is in the realm of metaphysics, rather than of physics [45]. It has also been argued that selection criteria like the anthropic landscape must be necessarily supplemented by arguments based on dynamics and symmetry, as only these lead to a real "enlightenment", the former are actually a "temptation" [46]. Indeed, Weinberg argues that the anthropic reasoning makes sense for a given constant whenever the range over which it varies is large compared with the anthropic allowed range. That is to say, it is relevant to know what constants actually "scan". The most likely include the cosmological constant, and the particle masses set by the electroweak symmetry-breaking mechanism. The possibility that the later is anthropically fixed is regarded as an interesting possibility, given that it renders an alternative solution for the hierarchy problem, such as technicolor or low-energy supersymmetry, that are not fully free of problems [47]. In any case, we feel that we cannot close this discussion without some words of caution. For instance, Polchinski has recently pointed out as the landscape picture requires a higher level of theoretical skepticism given that it suggests that science is less predictive. Furthermore, he remarks that the current scenario is tentative at best, as a nonperturbative formulation of string theory is still missing [48].

Let us close this discussion with a couple of remarks. The first concerns the possibility that the topology of the landscape is nontrivial. This hypothesis would imply that multiuniverses are not causally exclusive, meaning that within our Universe one might observe pocket subuniverses where the laws of physics are quite different from the ones we know. Since it is natural to assume the in these subuniverses the fundamental constants assume widely different values, one might expect to observe oddities such as quantum phenomena on macroscopic scales and relativistic effects at quite mundane velocities. Of course, this possibility would imply in a further loss of the ability to predict the properties of the cosmos.

Another relevant investigation on the selection of landscape vacua concerns the understanding of up to which extent the problem can be addressed in the context of the quantum cosmology formalism. As already discussed, this formalism allows for a theory of initial conditions, which seems to be particularly suitable to deal with the problem of vacua selection. It is quite interesting that this problem can be addressed as an N-body problem for the multiple scattering among the N-vacua sites of the landscape [49]. The use of the Random Matrix Theory methods shows that the phenomenon of localization on a lattice site with a well-defined vacuum energy, the so-called Anderson localization, occurs. It is found that the most probable universe with broken supersymmetry corresponds to a dS universe with a small cosmological constant. Furthermore, it is argued that the relevant question on why the Universe started in a low-entropy state can only be understood via the interplay between matter and gravitational degrees of freedom and the inclusion of dynamical back-reaction effects from massive long wavelength modes [49]. It is interesting to speculate whether these features remain valid beyond scalar field models, the case considered in Refs. [49]. Massive vector fields with global U(1) and SO(3) symmetries seem to be particularly suitable to generalize these results, given that the reduced matrix density and Wigner functional of the corresponding midisuperspace model [50] exhibit properties that closely resemble the localization process induced by the back-reaction of the massive long wavelength modes discussed in [49].

Let us describe some recent developments involving the AdS space, introduced in the previous section.

In the so-called braneworlds, one can admit two 3-branes at fixed positions along the 5th dimension, such that the bulk, the 5D spacetime is AdS, with a negative cos-mological constant, A = -3M|k2, where M5 is the 5D Planck mass and k a constant with dimension of mass. In this setup, compactification takes place on a Sx/Z2 orb-ifold symmetry. Einstein equations admit a solution that preserves Poincare invariance on the brane, and whose spatial background has a nonfactorisable geometry with an exponential warp form corresponding to the bulk space with metric, g5MN, a with a positive tension, a+, brane with metric, g+^v, sitting at z = 0, and a negative tension, a-, brane with metric, g-^v, sitting at z = zc. The standard model (SM) degrees of freedom lie presumably on the brane at z = zc.

In order to ensure a vanishing cosmological constant in 4D one chooses:

g^v being the 4D metric.

The action of the model is given by Randall and Sundrum [51]

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