  We review on a chiral power counting for in-medium chiral perturbation theory
with nucleons and pions as explicit degrees of freedom coupled to external
sources. It allows for a systematic expansion including both local and
pion-mediated inter-nucleon interactions. One can identify from this power
counting classes of non-perturbative diagrams that require resummation. A
non-perturbative method based on Unitary Chiral Perturbation Theory was also
developed for performing those resummations. This power counting and
non-perturbative techniques were firstly applied to calculate the pion
self-energy, the pion-decay constants and the quark condensate in nuclear
matter up-to-and-including next-to-leading order (NLO) contributions. The
cancellation of the contributions at NLO to the pion self-energy and decay
constants from in-medium nucleon-nucleon (NN) interactions was derived. Some
NLO NN contributions survive for the quark condensate due to the quark-mass
dependence of the pion mass. Next, we discuss the calculation of the energy
density in the nuclear medium by employing the in-medium NN scattering
amplitudes. For symmetric and neutron matter it reproduces in good agreement,
and without fine tuning, calculations from realistic NN potentials with a model
for the three-nucleon interaction. These results are applied to derive the
equation of state (EOS) for neutron stars and obtain an upper limit for a
neutron mass slightly above 2 solar masses. Our results also fulfill other
constraints from the detection of the gravitational waves in the event
GW170817, like the upper bound on the maximal mass of a neutron star and the
allowed interval for the radius of a 1.4-solar-mass neutron star. The knowledge
of the neutron-matter EOS is also employed to give an upper bound of the
gravitational constant within the strong gravitational field of a 2 solar-mass
neutron star.
