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FACULTY OF SCIENCE Compactifying all spatial dimensions of the universe Quentin Decant Thesis supervisor: Thesis submitted for the degree of prof. dr. Thomas Van Riet Master of Science of Physics, option Research Academic year 2017 – 2018 © Copyright KU Leuven Without written permission of the thesis supervisor and the author it is forbidden to reproduce or adapt in any form or by any means any part of this publication. Requests for obtaining the right to reproduce or utilize parts of this publication should be addressed to Faculteit Wetenschappen, Geel Huis, Kasteelpark Arenberg 11 bus 2100, B-3001 Leuven (Heverlee), +32-16-321401. A written permission of the thesis supervisor is also required to use the meth- ods, products, schematics and programs described in this work for industrial or commercial use, and for submitting this publication in scientific contests. Preface This thesis would not have been what it is now without the direct or indirect help of other people. I use this text to thank them. To start I would like to thank prof. dr. Thomas Van Riet for giving me the opportunity to write a thesis on such an exciting and interesting subject. I learned a lot from you this year, and the fact that you had the patience for allowing me to walk into your office anytime to ask questions was very much appreciated. I would like to express my gratitude to some of my fellow master students: David, Arno, Sam, Kristof, Kesha, for the good times at the office. My special thanks go to Vincent and Rob, with whom I had a lot of interesting and helpful discussion about physics. All the other people at the institute are also thanked, I really felt at home at the institute. The coffee breaks at four o’clock were really fun and a good relaxation moment. I also would like to thank my friends for all the good times we had together, not only this year, but the past five years. You made my student time very fun and memorable. I would like to express my deep gratitude to Anne, not only for the correcting of typos in my thesis. This thesis would simply not have been possible without all your support this past year. To conclude I am the most thankful to my parents. You always made sure that I had everything I needed to succeed. You continued to believe in me no matter what, and the freedom that you gave me due to this trust made me the person who I am now. Quentin Decant i Scientific summary This thesis introduces a new approach towards solving the cosmological constant problem. This is a very subtle issue that plagues the high energy physics community for decades now. The problem is a naturalness problem which we formulated as follows: the enormous difference between the Hubble scale and the length scale of possible extra dimensions, is something very unnatural. These hypothetical extra dimensions are a fundamental concept in modern theoretical physics and are an essential ingredient in superstring theory, the most studied theory of quantum gravity. Understanding the reason that the dimensions we live are so much bigger compared to the extra dimensions is therefore an important and fundamental question. We try to understand this unnatural situation by starting from a more natural situ- ation and evolve via a dynamical mechanism to the unnatural one. More concretely we want to start from a situation were all the spatial dimensions of our universe are treated equally. We take our natural situation to be that all dimensions are compact because we know from observations that the size of the universe has grown in time. To obtain such an initial condition for our universe we try to find one dimensional vacua of string theory, such that the scalar potential obtained from compactification admits only a meta-stable vacuum. This can be obtained by having a scalar potential which has only a local minimum for the volume modulus, no global one. In this case, classically the system will stay in this local minimum. However, quantum mechanically, the volume modulus will tunnel out of this meta-stable state, resulting in a spontaneous decompactification of some of the dimensions. This then gives rise to a universe which has some very large dimensions, and where the others remain compact. Even if there are no extra dimensions this mechanism could maybe provide initial conditions for inflation. The mistake of associating this mechanism to old inflation could easily be made, however they are not the same thing. In old inflation a four dimensional scalar field tunnels out of false vacuum state, we consider a one-dimensional field in a one-dimensional vacuum, which is a different thing. This means that the issues of old inflation do not automatically apply to our model. Under the approximation of smeared orientifold planes we derive that one- dimensional vacua do exist in string theory, and it is very likely that performing the calculation with localised planes results in the same conclusion. A simple compacti- fication of these solutions yields an unfixed modulus, and is therefore not suited as a realisation of our mechanism. We did not investigate this further, but changed our approach. String theory and other theories with extra dimensions need to have phenomeno- logical acceptable vacua, otherwise they can not describe the world we live in. On this fact the following reasoning is based. If we can compactify a four dimensional theory to one dimension, and this new theory admits a meta-stable vacuum, then the decompactification can lead to the theory we started from. Therefore we took four dimensional gravity combined with electromagnetism, axions and a cosmological constant, and investigated if we can compactify this to a theory admitting a one ii SCIENTIFIC SUMMARY dimensional vacuum. This led us to a theorem excluding stable one-dimensional vacua obtained by compactifications using the above mentioned ingredients,. This turned out not to be a bad thing. We found that meta-stable pockets can appear in the scalar potential. Dependent on the quantum corrections in our model, they are classicaly stable and yield compact periodically blowing-up and deflating universes. However these pockets are only local minima, so these models can decompactify. The most interesting thing about these pockets is that they are only allowed under certain conditions. Namely the cosmological constant and the curva- ture have to be strictly positive, and there must exist background electromagnetic fluxes. Using these conditions we tried some compactifications, but none were suc- cessful. However we investigated only a tiny region in the space of all possible compactifications, therefore the mechanism proposed in this thesis remains very promising and should be investigated further. iii Summary in layman’s terms The Standard Model of particle physics is our best tested scientific theory, however it does not incorporate gravity. Therefore a lot of physicist are searching for new theories extending the Standard Model, and the best candidate at the time of writing is superstring theory. For this theory to be mathematically consistent extra dimensions, besides the three spatial dimensions we are used to, need to be introduced. Because we did not observe any extra dimensions yet, those have to be extremely small such that they are hidden from us. This idea of extra super tiny dimensions seems very strange and unnatural. Our observable universe is a ball with a radius of about thirteen billion light years, the size of the extra dimensions has to be much smaller than a billionth of the size of an atom otherwise we would have detected them. If these extra dimensions are really there, why is there such a difference between these and the dimensions we live in? This is the question which drives the research preformed in this thesis. The most natural situation for the universe to be in is the situation were all dimensions are treated on an equal footing, hence we expect that all dimensions are either large or small. We know that our universe grew in size after the big bang, and thus we expect that around that time all dimensions were small. For this reason we take the natural situation to be that all dimensions are small. Therefore we want to find solutions to our equations which describe a universe were all dimensions are compact, but were some can spontaneously grow large. If this can be found, we discovered a potential mechanism which takes us from a natural situation, to the unnatural situation we find ourselves in. This could potentially explain why there is such a difference between the dimensions we live in, and those that are hidden from us. How do we achieve this? We reversed the above reasoning and started from a universe like ours, one which has three large dimensions and were the others are compact. We then try to fold up the three remaining dimensions, to obtain the situation were all the dimensions are of comparable size. If these three dimensions can unfold, then they will probably unfold to the universe we used to construct this model, and by this demonstrate that you can evolve from a natural to an unnatural situation. We discovered that under certain assumptions the mechanism we propose is possible. What is interesting is that for it to be possible, the universes we fold up have to meet certain conditions. These are that the three large dimensions have to be positively curved, that we need to be able to fill the universe with electromagnetic fields, and that they have to posses a positive cosmological constant. These are all things which are possible in our universe, hence this is encouraging.
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