Chaotic Dynamics of the Big Bang | Coffee and Theorems | Science

Inside the Universe, peculiarities space-time manifest themselves as cosmic anomalies. Examples include black holes or the Big Bang, points at which current physical models collapse and reality becomes a mysterious enigma. Its existence is expected according to the mathematical predictions of the theorems formulated by Roger Penrose in the 1960s and which were recognized by the Nobel Prize in 2020. However, little is known about its dynamics, that is, how space-time behaves near singularities.

Modern cosmology believes that a good approximation of the present universe on a large scale is given by the solution of Albert Einstein’s theory of relativity, found by Alexander Friedmann in 1922. According to Friedmann’s solution, space is not stationary, but expands with time, which was confirmed experimentally by Edwin Hubble in 1929. From the expansion of the universe comes the existence of a singularity: if we turn back time and travel into the past, space will collapse into a point (the Big Bang).

In his resolution, Friedmann assumed that space was homogeneous, meaning it behaved the same at all points, and isotropic, meaning it behaved the same in all directions. However, this might not have been the case at the Big Bang, and therefore Friedmann’s solutions could not explain what happens near this singularity.

Measurements of the cosmic background radiation show that shortly after the Big Bang, the universe expanded almost equally in all spatial directions. But a slight asymmetry in the isotropy could lead to different behavior near the singularity than that observed by Friedmann’s solutions. In particular, unlike what happens in these cases, spatial dimensions could play different roles in configuring our fate.

To answer these questions, Vladimir Belinsky, Isaak Khalatnikov, and Evgeny Lifshitz proposed in the 1970s that shortly after the universe’s explosive birth, it went through a chaotic phase of development. Chaos in this cosmic context refers to astonishing complexity, not disorder. According to this hypothesis, called BKL, chaotic fluctuations suggest mysterious patterns and complex mathematical structures that eventually shaped our current cosmos. More than 50 years later, this mathematical question is still far from answered.

The BKL hypothesis suggests that singularities generally have three characteristics. First, they are local, i.e., particles disconnect from each other, and each of them independently evolves towards a singularity. Thus, Einstein’s equations become ordinary differential equations.

Second, singularities are vacuum-dominated, meaning that for most types of matter, their influence on the dynamics of spacetime geometry near the singularity is negligible. In the words of John Wheeler, “matter doesn’t matter” near a singularity.

Finally, singularities are oscillatory and chaotic. At the same time, Charles Misner proposed a model for analyzing these chaotic oscillations, called mixmaster –in reference to an electric kitchen mixer for making dough–. This model describes a cosmological dance in which each spatial direction becomes an expansion and contraction, just like making pizza dough: you iteratively knead, stretch, and fold the dough, changing direction slightly each time the procedure is repeated. In both cosmology and pizza making, a slight modification of the initial conditions can lead to very complex and confusing results.

There are still many unanswered questions in this cosmic narrative of the birth of the Universe. It remains very difficult to obtain experimental data and confirm theories of gravity, especially in the region of extreme gravitational fields. Thus, in the absence of direct observations, a reliable mathematical framework becomes a crucial guide to plausible and meaningful theories. The Penrose singularity theorems and the BKL hypothesis open up a rich panorama of cosmic evolution that continues to arouse wonder and scientific curiosity, offering a different perspective on the complex dynamics of the past, present and future of the Universe.

Filippo Lappisi He is a Marie Curie Fellow (Una4Career) in Complutense University of Madrid

Coffee and theorems — is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest developments in the discipline, share the points of contact between mathematics and other social and cultural expressions, and remember those who have marked their development and who knew how to turn coffee into theorems. The title recalls the definition of the Hungarian mathematician Alfréd Rényi: “A mathematician is a machine that turns coffee into theorems.”

Editing and coordination: Agate Timon Garcia-LongoriaShe is the coordinator. Department of Mathematical Culture of the Institute of Mathematical Sciences (ICMAT)

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