Natural modification of quantum uncertainty, modified gravity, and cosmology

A common approach in physics and mathematics is to extend and modify theories and frameworks by considering what is often described as a `natural' extension or modification by including higher-order terms or by introducing other non-linearities. We show that such an approach must be taken with care as physical models can be connected in indirect ways. What looks like a natural approach in one setting will likely not be natural in another. We use the flat Friedmann-Lemaitre-Robertson-Walker equations of cosmology to connect the generalized uncertainty principle to modified theories of gravity. A simple additional term in one setting leads to enormous complications in the other. We identify Born-Infeld models as the only ones which appear natural in both settings.