  The cosmological constant (CC) problem is the biggest enigma of theoretical
physics ever. In recent times, it has been rephrased as the dark energy problem
in order to encompass a wider spectrum of possibilities. It is, in any case, a
polyhedric puzzle with many faces, including the cosmic coincidence problem,
i.e. why the density of matter is presently so close to the CC density.
However, the oldest, toughest and most intriguing face of this polyhedron is
the big CC problem, namely why the measured value of the CC at present is so
small as compared to any typical density scale existing in high energy physics,
especially taking into account the many phase transitions that our Universe has
undergone since the early times, including inflation. In this letter, we
propose to extend the field equations of General Relativity by including a
class of invariant terms that automatically relax the value of the CC
irrespective of the initial size of the vacuum energy in the early epochs. We
show that, at late times, the Universe enters an eternal de Sitter stage
mimicking a tiny positive cosmological constant. Thus, these models could solve
the big CC problem and have also a bearing on the cosmic coincidence problem.
Remarkably, they mimic the LCDM model to a large extent, but they still leave
some characteristic imprints that should be testable in the next generation of
experiments.
