Strongly Coupled Dark Energy Cosmologies: preserving LCDM success and easing low scale problems I - Linear theory revisited

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.