Introduction

High energy physics

String cosmology relates two fields of research that historically have evolved separately: high energy physics and cosmology.

One the one hand, in particle physics, we have an extremely accurate theory since the 1970's: the Standard Model. It has passed all experimental tests with flying colours. Only the Higgs particle, which plays a crucial role in the generation of fermion masses, has so far eluded observation. Nevertheless, for theoretical reasons it is believed that the Standard Model is only an effective theory, which will require modifications at higher energies. The proposed extensions include grand unified theories (extending the gauge group), supersymmetry (relating fermions and bosons) and extra dimensions. It is strongly hoped that the LHC will shed light on these matters.

 

Figure 1: The building blocks of particle physics.

Cosmology

In cosmology, on the other hand, the situation is radically different. Instead of theory, experiments have been the driving force over the last decade. Precision data on the cosmic microwave background (CMB), supernovae (SNe) and the large scale structure (BAO) have shown that the present universe consists of a mix of well-known, less well-known and unknown ingredients, collected in the Concordance Model of cosmology. A mere 4% of the energy density of the universe is composed of Standard Model baryons, while an additional 23% consists of cold dark matter. Finally, as illustrated in figure 1, a massive 73% is the mysterious dark energy that drives the surprising present-day acceleration, first observed in 1998.

 


Figure 2: The universe's energy and matter density fractions $ \Omega_\Lambda $ and $ \Omega_m $.

 

Furthermore, experimental confirmation has been found for inflation, a hypothetical period of accelerated expansion in the early universe that would explain why the universe is flat, homogeneous and isotropic to such a large extent. Inflationary parameters are now being measured with increasing accuracy. There are many experiments running or about to start that will further test and constrain the present cosmological models. ESA's Planck satellite, launched in May 2009, features prominently amongst these. It will measure the CMB in great detail and is hoped to detect primordial gravitational waves.

The situation in cosmology also differs from that in particle physics in an other respect. In the Standard Model the main objections are of theoretical nature, concerning aesthetics and naturalness, while in cosmology the underlying mechanisms responsible for inflation and dark matter and energy are completely unknown. At present we are lacking theoretical guidelines, such as the relevance of gauge theories for elementary particles. For this reason cosmology has a stronger need for input from a fundamental theory of Nature such as string theory, which unifies general relativity and quantum field theory and can explain these phenomena from first principles.

String theory

String theory first surfaced around 1970 as a possible model for the strong force. It lost this battle to QCD, partly because of the appearance of a spin-2 particle in its spectrum. However, it was subsequently realised that precisely this spin-2 feature allows it to describe quantum gravity. As a bonus there are also spin-1 gauge bosons, and hence the theory could unify all forces of Nature: weak, strong, E-M and gravity (figure 3). In addition, many ingredients that have been proposed as extensions of the Standard Model, such as grand unification, supersymmetry and extra dimensions, have a natural place in string theory.


Figure 3: Do strings tie together the four forces?

 

To avoid certain quantum inconsistencies, string theory is required to live in a ten-dimensional space-time. So how does one make contact between the ten-dimensional string theories and our four-dimensional world? This goes via the route of compactifications, where the extra dimensions are taken to be very small. For this reason they will be invisible at low energies: the effective description is four-dimensional.

In string theory, the most popular compactification manifolds are referred to as Calabi-Yau (CY) manifolds, consisting of six dimensions curled up in a very specific way. These manifolds are the simplest route to minimally supersymmetric theories in 4D, which are phenomenologically highly attractive. There are a large number of such manifolds with different topologies. Furthermore, given the topology, a manifold can be continuously deformed. The latter data are encoded in scalar fields, the so-called moduli. The moduli of Calabi-Yau manifolds naturally split up into two groups: the complex structure moduli, parametrising the shape, and the Kähler moduli, governing the size of the manifold.

Moduli stabilisation

It is an important asset of string theory, a tentative theory of quantum gravity and therefore including the spin-2 graviton, that it also leads to additional forces mediated by spin-1 and spin-0 particles. Spin-1 particles have a natural place in Nature as the force carriers of the Standard Model. The spin-0 particles are associated to the moduli scalar fields. However, these have never been observed, e.g. in fifth force experiments. Accordingly, to be consistent with observations, the mass of these particles must satisfy a lower bound.

Simple Calabi-Yau compactifications give spin-0 particles with zero mass. It is possible that by incorporating some additional effect, for instance a more general compactification, that these particles acquire a mass. In terms of the moduli, this corresponds to a non-trivial scalar potential with a minimum. In this way all moduli can be stabilised in the minimum, as explained in figure 3. Moreover, moduli stabilisation is necessary to extract predictions from string theory, as the 4D physical quantities depend on the value of the moduli. Finally, scalar potentials are crucial to incorporate some of the most successful features of cosmology, such as inflation and dark energy, and particle physics, such as the Higgs mechanism and supersymmetry breaking. It is for these reasons that the possible scalar potentials that can be generated by string theory are of the utmost importance to both cosmology and particle physics.

Figure 4: Moduli stabilisation leading to a stable vacuum. The two horizontal axes correspond to the values of the moduli, while the vertical axis depicts the scalar potential energy. Minima of the scalar potential give rise to vacua. These correspond to different compactifications of string theory, illustrated by the donut shapes.

 

During the last decade, important steps have been made in moduli stabilisation. Calabi-Yau compactifications have been extended by including so-called gauge fluxes. These are non-vanishing field strengths of the spin-1 gauge potentials of string theory in the Calabi-Yau manifold, similar to 4D electric and magnetic fluxes. It was demonstrated in 2001 that fluxes and branes can be included in a particular flavour of string theory (type IIB) in such a way that all complex structure moduli are stabilised [GKP]. The necessary ingredients are ‘imaginary self-dual’ NS-NS and R-R three-form fluxes, D3 and D7-branes and O3-planes. This means that the Kähler moduli are arbitrary. It was subsequently realised in 2003 that all moduli can be stabilised by including non-perturbative effects [KKLT]. In contrast, in IIA string theory, the inclusion of fluxes was shown in 2005 to allow for a stabilisation of all moduli [DGKT]. Here the ingredients are different types of fluxes and D6-branes and O6-planes. All vacua are supersymmetric and have a negative cosmological constant, however.

Goals of our research

The key question is of course to find a vacuum that could correspond to our world; if string theory is correct, we should be able to stabilise the moduli in such a way that both the Standard Model and the Concordance Model come out. In this way one could make predictions for (near future) observable quantities of modern cosmology, such as the amplitude and slope of the scalar and tensor power spectra, or the dark energy equation of state parameter and its evolution in time.

It is exactly this relation between cosmology and fundamental physics that we want to investigate in our group. The proposed research aims to explore the possibilities of embedding the current cosmological paradigm into string theory. In other words, our goal is to assess to what extent string theory is compatible with modern cosmology: can it accommodate dark energy and/or inflation, and explain the underlying fundamental physics? These are the big questions about the universe that one would expect a fundamental theory like string theory to be able to address. Given the wealth of recent and upcoming precision data on both the early universe and its present state, this is a very promising avenue to test string theory experimentally.