Consider the homotopical category of rational dg-modules. I suppose this ought to present the rationalization of the homotopy theory of parameterized spectra. Has such "rational parameterized stable homotopy theory" been worked out a little, anywhere?

Here by the homotopical category of dg-modules I mean the obvious thing: the category of pairs $(A,N)$ consisting of a dg-algebra $A$ over $\mathbb{Q}$ and a dg-module $N$ over $A$, with homomorphisms the evident pairs of morphisms, and with weak equivalences those pairs where both components are quasi-isomorphism. This category has been studied a lot by Agustí Roig, who considers minimal dg-modules as tools for computing the rational cohomology of topological fibrations, disregarding rational homotopy groups, see below.

Similarly, parameterized spectra are pairs $(X,E)$ consisting of a space $X$ and a spectrum object $E$ in the slice of the classical homotopy theory over $X$.

In two subcases the above equivalence is well known:

On the one hand, classical rational homotopy theory says that if we restrict to zero spectra inside all parameterized spectra, hence to the classical homotopy theory of spaces, then its rationalization is equivalent to the opposite of rational dg-algebras (on simply connected objects of finite type at least), hence to the sub-category of dg-modules, as above, on the zero modules.

At the other extreme, if we restrict attention to the trivial base space, then the homotopy theory of parameterized spectra reduces to that of plain spectra, rationalization yields $H \mathbb{Q}$-module spectra, and theorem 5.1.6 in Schwede-Shipley 03 says that these are equivalently modeled by the homotopy theory of rational chain complexes, which are the dg-modules over the trivial dg-algebra.

Finally, for any fixed base space we may apply a relative version of this argument, following the general considerations in Schwede 97.

This makes it plausible that the homotopy theory of rational dg-modules is generally equivalent to the rationalization of parameterized spectra, at least for simply-connected finite-type base objects and with homomorphisms of $H \mathbb{Q}$-module spectra understood.

For example on page 2 of Roig-Saralegi 00 the authors give a minimal model, in dg-modules over the minimal Sullivan dg-algebra for the 3-sphere, of the rationalization of the map

$$ S^4 /^h S^1 \to S^3 $$

(where $S^1$ acts canonically on $S^4 \simeq S(\mathbb{R} \oplus \mathbb{H})$ and where the slash means the homotopy quotient, in contrast to the naive quotient, which is $S^3$). I am after statements such as that this minimal dg-module is (if indeed it is) a rational presentation of the $S^3$-parameterized spectrum $\Sigma^\infty_{S^3}(S^4 /^h S^1)$ (the fiberwise suspension spectrum of this projection).