  In this first paper we discuss the linear theory and the background evolution
of a new class of models we dub SCDEW: Strongly Coupled DE, plus WDM. In these
models, WDM dominates today's matter density; like baryons, WDM is uncoupled.
Dark Energy is a scalar field $\Phi$; its coupling to ancillary CDM, whose
today's density is $\ll 1\, \%$, is an essential model feature. Such coupling,
in fact, allows the formation of cosmic structures, in spite of very low WDM
particle masses ($\sim 100$ eV). SCDEW models yields Cosmic Microwave
Background and linear Large Scale features substantially undistinguishable from
$\Lambda$CDM, but thanks to the very low WDM masses they strongly alleviate
$\Lambda$CDM issues on small scales, as confirmed via numerical simulations in
the II associated paper. Moreover SCDEW cosmologies significantly ease the
coincidence and fine tuning problems of $\Lambda$CDM and, by using a field
theory approach, we also outline possible links with inflationary models. We
also discuss a possible fading of the coupling at low redshifts which prevents
non linearities on the CDM component to cause computational problems. The
(possible) low-$z$ coupling suppression, its mechanism, and its consequences
are however still open questions -not necessarily problems- for SCDEW models.
The coupling intensity and the WDM particle mass, although being extra
parameters in respect to $\Lambda$CDM, are found to be substantially
constrained a priori so that, if SCDEW is the underlying cosmology, we expect
most data to fit also $\Lambda$CDM predictions.
