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An inflatable balloon with magnetic field lines drawn on it with a marker is a simple model of how a weak magnetic field behaves in an expanding universe. While the balloon is small, the lines lie obediently on the rubber and don't hinder its stretching. That's roughly how a field fills ready-made space without arguing with its geometry.

But if the marker itself started to inflate the rubber, the picture would be different. That's exactly what happens with a very strong magnetic field: its energy becomes so significant that it starts to dominate space itself. In the new work, physicists considered both cases for de Sitter universes (empty space expanding with acceleration) and anti-de Sitter (empty space with negative curvature, like a saddle). They solved the weak field using Maxwell's equations on a fixed background, like a drawing on an already inflated balloon. For the strong field, they had to include the back reaction: the field's energy deforms space, and Einstein's equations start working together with Maxwell's equations.

The result turned out to be elegant: a strong magnetic field transforms a de Sitter or anti-de Sitter universe into a magnetic analog of Melvin's universe — a solution where all of space is permeated by a uniform magnetic field and therefore has cylindrical symmetry. The authors obtained it using a transformation similar to the Harrison method, which was previously used to charge black holes. They had to carefully transform not only the geometry but also the pressure and energy density of a hypothetical fluid filling space. Thus, the balloon with drawn lines became a balloon that the lines themselves inflate, and its shape is no longer quite spherical.