The construction of the \(p\)-adic local Langlands correspondence for \(\mathrm{GL}_2(\mathbf{Q}_p)\) uses in an essential way Fontaine's theory of cyclotomic \((\varphi ,\Gamma )\)-modules. Here cyclotomic means that \(\Gamma = \mathrm {Gal}(\mathbf{Q}_p(\mu_{p^\infty})/\mathbf{Q}_p)\) is the Galois group of the cyclotomic extension of \(\mathbf Q_p\). In order to generalize the \(p\)-adic local Langlands correspondence to \(\mathrm{GL}_{2}(L)\), where \(L\) is a finite extension of \(\mathbf{Q}_p\), it seems necessary to have at our disposal a theory of Lubin-Tate \((\varphi ,\Gamma )\)-modules. Such a generalization has been carried out, to some extent, by working over the \(p\)-adic open unit disk, endowed with the action of the endomorphisms of a Lubin-Tate group. The main idea of this article is to carry out a Lubin-Tate generalization of the theory of cyclotomic \((\varphi ,\Gamma )\)-modules in a different fashion. Instead of the \(p\)-adic open unit disk, the authors work over a character variety that parameterizes the locally \(L\)-analytic characters on \(o_L\). They study \((\varphi ,\Gamma )\)-modules in this setting and relate some of them to what was known previously.