String Cosmology and Flux Compactifications
Right before the turn of the millenium some cosmological observations have revealed that our universe contains dark energy. This source of energy/matter in the universe satisfies an anomalous equation of state with respect to ordinary matter or radiation and it corresponds to the vacuum energy present in our universe. Dark energy can be accommodated inside General Relativity by including an extra term to the Einstein equations which is often called the cosmological constant and is normally denoted by Λ. Combined measurements coming from supernovae [1, 2], the Cosmic Microwave Background (CMB) radiation [3, 4] and the Baryonic Acoustic Oscillations (BAO) [5, 6] concluded that we live in a universe with a positive and small cosmological constant and gave rise to what we call nowadays the concordance model of cosmology. The energy/matter content giving the best fit is depicted in figure 1.
Figure 1: The concordance model of cosmology predicts that our universe has a positive cosmological constant. In a recent phase of the history of the universe the vacuum energy took over and became the dominant energy content.
The cosmological constant drives an accelerated expansion of the universe which is described by de Sitter spacetime. This suggests that, after dark energy started to dominate, our universe started approaching a de Sitter vacuum rather than a Minkowski one. This implies that the suitable string compactifications for phenomenological purposes should give rise to de Sitter vacua. One can show that plain Calabi-Yau compactifications present the unfortunate feature of producing a large amount of massless scalar fields (a.k.a. moduli). Hence, in order to reproduce de Sitter vacua, one should go beyond these well-known compactifications.
An extra motivation for considering accelerated expanding universes in string theory is that of embedding inflationary models within string theory. Inflation describes a phase of accelerated expansion of the universe right after the big bang. This was proposed to explain an almost perfect homogeneity and isotropy relating regions in the sky which had never been in causal contact with each other throughout the history. Inflationary models are described by a quasi-de Sitter phase driven by a scalar field called the inflaton.
The above issues provide two challenges for string theory compactifications related to de Sitter. The first one is finding de Sitter vacua in order to describe the late-time accelerating phase we are approaching now. The second one is embedding inflation in string theory by providing examples of compactifications in which quasi-de Sitter phases are possible with a very flat potential for the inflaton. These approaches in string theory result in what is often called string cosmology and they have been extensively followed in several directions in the last decade.
Concentrating for a moment on inflation, it is a particularly striking fact that string theory suggests some preferred classes of inflationary models, in which, for instance, no detectable tensor modes are present in the spectrum of cosmological perturbations of the early universe. This information, which is encoded in the CMB, can still be detected now, and is the result of frozen quantum fluctuations grown to observable size in the present universe. Precision measurements on the CMB carried out in the last decade by WMAP [7-9] already provided very precious data, although the existence of tensor perturbations still remains an open question. There is a possibility that the PLANCK satellite, which is currently collecting data, might tell us more about this. Such an experimental input would be a valuable opportunity for constraining models of inflation, among which there are stringy inflationary proposals.
Coming now back to the search for de Sitter vacua in string theory, right after the experimental detection of the cosmological constant, the existence of a huge “zoo” of vacua [10, 11] (about 10500 !!) was conjectured on the basis of statistical analysis. This enormous amount of different string vacua is often referred to as the landscape (see what sketched in figure 2). However, there has been more recently a lot of debate on this after the many failed attempts of finding classical (i.e. at tree level) de Sitter solutions from string theory compactifications.
Figure 2: The shape and size of the extra dimensions in String Theory are treated as scalar excitations that become massive when sitting at local minima of a scalar potential.
Going beyond the search for classical solutions in string theory, people have considered the possibility of stabilising the moduli in an anti-de Sitter vacuum by means of quantum non-perturbative effects  and subsequently providing an uplifting to de Sitter by means of several mechanisms. In ref.  such an uplifting was provided by additional extended sources breaking supersymmetry explicitly. Nevertheless, this mechanism completely ignores the backreaction of such sources and some recent analyses indicate that it might cause the arising of a singularity and possibly related instabilities. In ref.  the possibility of D-term uplifting was considered. However, later in refs [14, 15] the inconsistency of this construction was pointed out due to the violation of gauge invariance occurring in a supergravity model with D-terms and yet vanishing F-terms. In ref.  a valid proposal is given to overcome this inconsistency. The third possible type of uplifting mechanism is F-term uplifting, which was worked out e.g. in ref. .
The current state of the art concerning models of inflation within String Theory compatible with PLANCK data is discussed in ref. .
String Theory is a 10D formulation of unified interactions. Given this, it is very interesting to study its backgrounds with four large dimensions and six compact ones. In order to obtain maximally symmetric solutions other than the trivial Minkowski one, we need to use gauge fluxes, curvature and branes in order to stabilise the scalar excitations encoding the geometric information about the internal manifold. See e.g. the situation depicted in figure 3.
Figure 3: An example of an internal manifold with non-trivial geometry and topology, where gauge fluxes and branes can wrap cycles of the internal manifold.
This is what could be referred to as a top-down approach, which has the advantage of being explicit and constructive, but it is limited to the supergravity regime, where calculations are under perturbative control. A parallel but somehow related research line has regarded supergravity models as lower-dimensional effective descriptions coming from flux compactifications. In this way, one can hope of capturing some extra information concerning string dualities (see figure 4) and strong-coupling effects which are missed out by the 10D supergravity approximation.
Figure 4: The chain of dualities that relate all the different formulations of string theory amongst them. Each of them taken in its natural perturbative regime may be seen as a different perturbative corner of a more fundamental 11D theory called M-theory.
In this context a lot of work has been done in the case of flux backgrounds preserving minimal supersymmetry in four dimensions. Some work has been done also in the context of compactifications preserving larger amount of supersymmetry. A very welcome ingredient (or even crucial in the case of (half-)maximal supergravities) for obtaining de Sitter solutions turns out to be given by generalised fluxes . These objects appear as deformation parameters in the lower-dimensional effective description even though they do not have a clear higher-dimensional interpretation. Their appearance was first conjectured in ref.  based on duality covariance arguments.
In the context of extended supergravities (N=4 or N=8), generalised fluxes can be interpreted as gaugings. Such deformations in extended supergravities are classified and comprised in a universal duality-covariant formulation making use of the so-called embedding tensor formalism [21, 22]. Studying such more constrained situations might shed a light upon the role of string dualities in flux compactifications, with the final goal of learning something about strong-coupling effects in the top-down approach.
For a nice review of Flux Compactifications and its relation to gauged supergravities, we respectively advise refs. [23, 24].
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