  Let X_d be the p-adic analytic space classifying the d-dimensional
(semisimple) p-adic Galois representations of the absolute Galois group of Q_p.
We show that the crystalline representations are Zarski-dense in many
irreducible components of X_d, including the components made of residually
irreducible representations. This extends to any dimension d previous results
of Colmez and Kisin for d = 2. For this we construct an analogue of the
infinite fern of Gouv\^ea-Mazur in this context, based on a study of analytic
families of trianguline (phi,Gamma)-modules over the Robba ring. We show in
particular the existence of a universal family of (framed, regular) trianguline
(phi,Gamma)-modules, as well as the density of the crystalline
(phi,Gamma)-modules in this family. These results may be viewed as a local
analogue of the theory of p-adic families of finite slope automorphic forms,
they are new already in dimension 2. The technical heart of the paper is a
collection of results about the Fontaine-Herr cohomology of families of
trianguline (phi,Gamma)-modules.
