Periodic Reporting for period 4 - Qosmology (Quantum Effects in Early Universe Cosmology)
Okres sprawozdawczy: 2023-03-01 do 2024-03-31
It is notoriously difficult to reconcile the two most basic frameworks of the fundamental laws of nature, namely general relativity and quantum theory. The first describes gravity and how it influences space and time—you could say it relates to the large—while the second characterises the behaviour of all known elementary particles—relating to the small. In order to understand the big bang, they will have to be combined.
Why is it so hard to unify gravity and quantum theory? To illustrate this, it helps to recall the bizarreness of the quantum world. According to quantum theory, nature explores all possibilities to determine the probability with which an event occurs. So when a ball is thrown, absolutely all conceivable trajectories are considered, but only a few weigh in significantly. (In the case of the ball, in fact, only the familiar classical trajectory is relevant.) When applied to space-time, however, it gets tricky. All possible space-time evolutions must be considered, but the trouble is that on tiny scales, the fabric of space-time tends to become very messy.
Faced with those difficulties, James Hartle and Stephen Hawking put forth their "no-boundary" proposal. Their highly elegant idea was that one should not consider all possible space-time evolutions but only those that have a smooth initial space-time geometry. You can picture this by imagining that the early universe would have been rounded off like the surface of a ball, not only in space but also in time. Time would have had no edge—hence the name of the proposal. This idea has two highly desirable consequences. The first is that it might actually allow one to calculate things, as the geometries are forced to be smooth and, thus tractable, initially. The second is that by providing a theory of what effectively replaces the big bang, we would know the most likely starting point of the universe. Still, until recently, it remained difficult to calculate the true consequences of this idea. Despite the simplification of having to deal only with these "no-boundary" geometries, it is not easy to calculate how all the different space-time evolutions sum up, or how to determine which ones are the most important.
About seven years ago, my collaborators Job Feldbrugge and Neil Turok, both of the Perimeter Institute in Canada, and I realised that there exists a mathematical framework, called Picard-Lefschetz theory, which has been continuously developed by mathematicians over the last 100 years and that is perfectly suited to this kind of calculation. It is only over the last decade that physicists have become aware of its existence. Using these methods, we encountered a surprise. When we imposed the condition that the universe starts from zero size, the resulting universes inevitably developed large fluctuations. In other words, the geometries that develop strong irregularities contributed the most to the final answer. This implies that one would get a highly crumpled universe popping out of nothing and presumably collapsing again right away, rather than expanding into the vast universe we know.
This was the puzzling situation we found ourselves in at the beginning of this project. The main aim of the project was to see if either the no-boundary idea must be reformulated so as to work, or to look for alternative descriptions of the big bang and the emergence of space and time. In the end, we did both.
Why is it so hard to unify gravity and quantum theory? To illustrate this, it helps to recall the bizarreness of the quantum world. According to quantum theory, nature explores all possibilities to determine the probability with which an event occurs. So when a ball is thrown, absolutely all conceivable trajectories are considered, but only a few weigh in significantly. (In the case of the ball, in fact, only the familiar classical trajectory is relevant.) When applied to space-time, however, it gets tricky. All possible space-time evolutions must be considered, but the trouble is that on tiny scales, the fabric of space-time tends to become very messy.
Faced with those difficulties, James Hartle and Stephen Hawking put forth their "no-boundary" proposal. Their highly elegant idea was that one should not consider all possible space-time evolutions but only those that have a smooth initial space-time geometry. You can picture this by imagining that the early universe would have been rounded off like the surface of a ball, not only in space but also in time. Time would have had no edge—hence the name of the proposal. This idea has two highly desirable consequences. The first is that it might actually allow one to calculate things, as the geometries are forced to be smooth and, thus tractable, initially. The second is that by providing a theory of what effectively replaces the big bang, we would know the most likely starting point of the universe. Still, until recently, it remained difficult to calculate the true consequences of this idea. Despite the simplification of having to deal only with these "no-boundary" geometries, it is not easy to calculate how all the different space-time evolutions sum up, or how to determine which ones are the most important.
About seven years ago, my collaborators Job Feldbrugge and Neil Turok, both of the Perimeter Institute in Canada, and I realised that there exists a mathematical framework, called Picard-Lefschetz theory, which has been continuously developed by mathematicians over the last 100 years and that is perfectly suited to this kind of calculation. It is only over the last decade that physicists have become aware of its existence. Using these methods, we encountered a surprise. When we imposed the condition that the universe starts from zero size, the resulting universes inevitably developed large fluctuations. In other words, the geometries that develop strong irregularities contributed the most to the final answer. This implies that one would get a highly crumpled universe popping out of nothing and presumably collapsing again right away, rather than expanding into the vast universe we know.
This was the puzzling situation we found ourselves in at the beginning of this project. The main aim of the project was to see if either the no-boundary idea must be reformulated so as to work, or to look for alternative descriptions of the big bang and the emergence of space and time. In the end, we did both.
In thinking about the puzzle described above, namely that universes that start from zero size develop strong fluctuations in space and time, my student Alice Di Tucci and I realised that there is another feature of quantum mechanics that might come to the rescue: this is Heisenberg's uncertainty principle. This principle implies that for a small particle, one cannot simultaneously know its position and its velocity to arbitrary accuracy. For a small universe, this principle implies that one can either know its size, or how fast it is expanding. This means that if one specifies that the universe starts at zero size, one cannot know anything about how fast it is expanding, in fact it could be expanding horrendously fast. This turned out to be the cause of the instability of the no-boundary proposal. Ms. Di Tucci and I realised that we could instead demand that the expansion rate was such that the universe started off very smooth. This then automatically ensured that we would obtain a large universe that did not collapse due to its space-time fluctuations.
There are two consequences to this idea: the first is that in principle one now knows nothing about the initial size of the universe with certainty. However, it turned out that the universe that not only starts off smoothly but also happens to start at zero size is in fact the most likely one. This is a nice surprise, as it means that we truly have a theory of the initial conditions of the universe. The second aspect is that our condition demands that at the beginning there was only space, and no time. Thus according to this theory time must emerge from space. In other words, as the universe grows one space direction turns into a time direction. This is a rather mysterious aspect of the no-boundary proposal, which we intend to study in more detail in the future.
There are two consequences to this idea: the first is that in principle one now knows nothing about the initial size of the universe with certainty. However, it turned out that the universe that not only starts off smoothly but also happens to start at zero size is in fact the most likely one. This is a nice surprise, as it means that we truly have a theory of the initial conditions of the universe. The second aspect is that our condition demands that at the beginning there was only space, and no time. Thus according to this theory time must emerge from space. In other words, as the universe grows one space direction turns into a time direction. This is a rather mysterious aspect of the no-boundary proposal, which we intend to study in more detail in the future.
Much of the progress that we made over the past five years was technical in nature. But on the conceptual side, it is worth emphasising that we found a close analogy between calculations in cosmology and calculations related to black holes, demonstrating a close link between the two most extreme physical events known.
Our main advance was to find a good mathematical description of quantum theory applied to the big bang. Now the big question is what the result actually means for observations of the universe. We made further progress on this question by picking up the work of the mathematicians M. Kontsevitch and G. Segal, who developed a criterion for which kinds of spacetimes should be allowed in quantum theory. And here we encountered a further surprise (similar results were obtained by T. Hertog, O. Janssen and J. Karlsson): it turns out that according to this criterion, only spacetimes that initially expand in an accelerated fashion for long enough are allowed. This provides a possible explanation for why the universe is so large, and so smooth on large scales.
A final point of interest is the question whether different consistent theories of the beginning of the universe can be formulated. In collaboration with K. Stelle I developed a different possible theory of initial conditions in the context of a special theory of gravity known as quadratic gravity. That theory has the special property that it is renormalisable, which means that quantum corrections are under control. This theory enjoys a further property called asymptotic safety, which means that at very high energies it becomes scale invariant, and hence its predictions remain calculable. We noticed that demanding the theory to not result in mathematical infinities, implies that the early universe must have been very smooth and moreover that it must have undergone accelerated expansion. This provides an alternative explanation for the large scale smoothness of the universe, and it will be worthwhile investigating the consequences of this model more precisely in the future.
Our main advance was to find a good mathematical description of quantum theory applied to the big bang. Now the big question is what the result actually means for observations of the universe. We made further progress on this question by picking up the work of the mathematicians M. Kontsevitch and G. Segal, who developed a criterion for which kinds of spacetimes should be allowed in quantum theory. And here we encountered a further surprise (similar results were obtained by T. Hertog, O. Janssen and J. Karlsson): it turns out that according to this criterion, only spacetimes that initially expand in an accelerated fashion for long enough are allowed. This provides a possible explanation for why the universe is so large, and so smooth on large scales.
A final point of interest is the question whether different consistent theories of the beginning of the universe can be formulated. In collaboration with K. Stelle I developed a different possible theory of initial conditions in the context of a special theory of gravity known as quadratic gravity. That theory has the special property that it is renormalisable, which means that quantum corrections are under control. This theory enjoys a further property called asymptotic safety, which means that at very high energies it becomes scale invariant, and hence its predictions remain calculable. We noticed that demanding the theory to not result in mathematical infinities, implies that the early universe must have been very smooth and moreover that it must have undergone accelerated expansion. This provides an alternative explanation for the large scale smoothness of the universe, and it will be worthwhile investigating the consequences of this model more precisely in the future.