A stone thrown into a pond creates ripples that spread far. But if you throw it into thick jelly, the waves die out almost immediately. That's roughly how an additional gravity field behaves under different mathematical descriptions.
The authors built a unified approach to calculate how such a field manifests itself in the Solar System. They took three things into account: the field's coupling to ordinary gravity, its own energy, and the presence of a hidden potential. The calculations were done for two 'sets of rules'—the metric formalism and the Palatini formalism. In the first case, the geometry of spacetime is rigidly tied to the field, like rails to a train. In the second, they're more like a bicycle and a road: connected, but each retains its freedom.
A comparison with data from the Cassini probe (which measured the delay of radio signals passing near the Sun) showed that within Palatini's framework, the field could be much stronger and still go unnoticed. In the more familiar version of the theory, even a small admixture of the field would immediately show up as deviations in orbits. For one particular case, the so-called $f(\hat{R})$ gravity, the difference turned out to be dramatic: in the Palatini formalism, the external field is indistinguishable from Einstein's, whereas in the metric one, it leaves a noticeable trace.
The Cassini probe measured radio signal delays with precision down to billionths of a second — enough to test the theory of gravity with an error of less than one hundred-thousandth.