  The heavy-fermion superconductor CeCoIn5 is the first material, where
different experimental probes show strong evidence pointing to the realization
of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state. The inhomogeneous
superconducting FFLO state with a periodically modulated order parameter was
predicted to appear in Pauli-limited, sufficiently clean type-II
superconductors already more than 40 years ago. On the other hand, CeCoIn5 is
supposed to be close to a magnetic quantum critical point (QCP) showing strong
antiferromagnetic (AFM) spin fluctuations (SF) at atmospheric pressure. We
studied the evolution of the FFLO phase away from the influence of the strong
AFM-SF by heat capacity experiments under pressure (0 GPa <= P <= 1.5 GPa, 0 T
<= mu_0 H <= 14 T, and 100 mK <= T <= 4 K). Our results prove the stability of
the the FFLO phase under pressure. It even expands, while the Pauli-limiting
becomes weaker and the AFM-SF are suppressed. This shows the intriguing
influence of the AFM-SF on the FFLO state.
