An international team of researchers including Dr Sebastian von Hausegger, a Carlsberg Fellow at Oxford’s Rudolf Peierls Centre for Theoretical Physics, used insights from physics, mathematics, and computer science to create a novel method that traces the movement of galaxies back in time. The group’s most recent work on ‘cosmological reconstruction’ was led by Dr von Hausegger and is now published in *Physical Review Letters*.

### Where do galaxies come from?

How did galaxies move through space and time to arrive at where they are today? In principle, researchers understand this: long in the past, matter was evenly distributed across all of space; only tiny deviations in its uniformity could be seen in this landscape. Gravity notices these small over-densities and as they gradually attract more matter they grow increasingly larger. Eventually, they form stars and galaxies. These move through space according to the law of gravitation. One can even run computer simulations to show how.

However, Dr von Hausegger argues: ‘Simply understanding how all this works in principle is not enough. We want to know, not how any set of galaxies moved, but how the very galaxies that we observe today have moved from where they started. Doing so would make possible a multitude of new research.’ This latest work has done exactly this – it ‘reconstructs’ the past trajectories of galaxies.

### Turning back time

The answer to the question of where today’s galaxies came from lies in recognising that the matter collected in each galaxy must have travelled from a corresponding region of space in the smooth landscape of the early universe. ‘Think of space being cut up into distinct cells like a three-dimensional mosaic,’ continues Dr von Hausegger. ‘Each cell is thought of containing the right amount of matter to create a galaxy. Over cosmic time, each cell will move and collapse to arrive at the location of its corresponding galaxy.’

To solve this problem one needs a law that assigns a specific region of space in the early universe to a specific galaxy — or, vice versa, a method that assigns every galaxy today to a corresponding cell in space back in time. The solution lay in a field of mathematics called optimal transport theory.

Optimal transport theory generally describes the movement of probability distributions through some space according to some plan, much like the movement of matter through space according to the law of gravitation. Thanks to a link between gravitation and optimal transport theory, this assignment problem has a unique solution and co-author Dr Roya Mohayaee from the Institut d’Astrophysique de Paris, was among the first to rephrase the reconstruction problem in terms of optimal transport theory back in 2002. However, at that time it was not possible to formulate this method very efficiently, among other things, due to computational limitations.

### Turning to computer science

Serendipitously, advances in computer science, specifically in the field of numerical geometry, proved themselves exceptionally useful in this project. The authors’ calculations showed that the cells that divide up space into chunks of matter are so-called Laguerre cells — each of the grey objects shown in the figure is one such Laguerre cell. Collectively, they are referred to as a Laguerre diagram, a higher-order version of the perhaps better-known Voronoi diagram.

Dr Bruno Lévy, director of the Inria Nancy-Grand Est research centre and co-author of the article, is an expert in numerical geometry. He notes: ‘We created a common language of physics, mathematics, and computer science. This language revealed surprisingly aesthetic and symmetric mathematical structures in the cosmological setting. In particular, finding the connection to Laguerre diagrams instead of resorting to previous methods made our approach more efficient by orders of magnitude.’ His team already had some experience with such methods in the field of fluid dynamics and he recalls, ‘when I first started I would have never thought that these methods could also be useful in cosmology!’

### Seeing patterns in the landscape of galaxies

In their article, the team highlights the power of their method by measuring a faint pattern in the smooth landscape of matter, the baryonic acoustic oscillations. The smooth landscape and its tiny ripples are known from observations of the cosmic microwave background. Observed from far away, the landscape reveals a pattern — certain arrangements in the hills and valleys — which encodes the most fundamental laws of the universe. Hence it is important to understand these arrangements in all their detail and how they evolve across time. In fact, it should be possible to find the same pattern, the structure of the early-universe landscape, also in late-universe observations of collections of galaxies. However, once too much matter accumulates and the density becomes too large, this pattern begins to blur. Therefore, observed today, the distribution of galaxies does not show this pattern as clearly.

With their algorithm, the authors were able to push matter out of these over-dense regions to once again reveal the pattern — it has essentially been reconstructed. ‘Measuring this feeble pattern was a stringent test of our method,’ confirms Dr von Hausegger. ‘And, having withstood this scrutiny, the opportunities for astrophysics are great. We are really excited as many so-far unexplored challenges can now be tackled for the first time.’

### Other applications of optimal transport theory

After his Carlsberg Fellowship, Dr von Hausegger will move on to hold a Beecroft Fellowship in Cosmology within the Astrophysics sub-department. ‘I will be applying this method to data from large upcoming astronomical surveys, to retrace galaxies’ movement, and to infer their present-day velocities,’ he continues. ‘For the first time, this will result in large-scale velocity catalogues, which can be used to study the isotropy of the universe — another hot topic of my research — or inform other cosmological studies, such as those of the kinetic Sunyaev-Zeldovich effect.

‘The relation of optimal transport theory to fluid dynamics is what propelled this project as it is possible to describe the motion of matter in our cosmos by the fluid equations,’ concludes Dr von Hausegger. ‘Yet the universality of optimal transport and the versatility of numerical geometry find fruitful research grounds in other fields as well, be it biophysics or general relativity. Catalysed by this work, it is only a matter of time that multi-disciplinary research will lead to more breakthrough discoveries.’

Accurate baryon acoustic oscillations reconstruction via semidiscrete optimal transport, Sebastian von Hausegger et al, *Phys Rev Lett*, 128 May 2022