Exercise: Dark Matter as fields that evolve backward in time

We do the exercise of thinking the qualitative features of a new model in which the problems of Dark Matter (DM) and the Cosmic Microwave Background (CMB) homogeneity are apparently simultaneously solved. We consider that DM consists of fields that evolve backward in time and have a positive defined energy to provide the correct observed curvature of space. We postulate that these fields do not interact with SM fields other than by sharing the same metric, therefore physics in small curvature regimes is causal, without contradictions with available experiments. In strongly curved space causality is lost and solutions must fulfill given equations. The homogeneity problem is naturally solved by these DM fields since, evolving from our future to our past, they bring patches that are currently thought to be causally disconnected into thermal equilibrium for $t\gtrsim 0$ by the Big Bang time. This thermal equilibrium of DM fields in a strongly curved regime must be shared by the metric and our fields, therefore we see these patches with the same CMB temperature. Although we find unlikely that Nature is described by this theory, we find interesting to conclude that there are not obvious logical nor observational inconsistencies and we explore its features. Hopefully some thoughts and results in this work may be useful for other ideas.