Reciprocity invariance of the Friedmann equation, Missing Matter and double Dark Energy

The current concordance model of cosmology is dominated by two mysterious ingredients: dark matter and dark energy. In this paper, we explore the possibility that, in fact, there exist two dark-energy components: the cosmological constant \Lambda, with equation-of-state parameter w_\Lambda=-1, and a `missing matter' component X with w_X=-2/3, which we introduce here to allow the Friedmann equation written in terms of conformal time \eta to be form-invariant under the reciprocity transformation a(\eta)\to 1/a(\eta) of the scale factor. Using recent cosmological observations, we constrain the present-day energy density of missing matter to be \Omega_{X,0}=-0.11\pm 0.14. This is consistent with the standard LCDM model, but constraints on the energy densities of all the components are considerably broadened by the introduction of missing matter; significant relative probability exists even for \Omega_{X,0}\sim 0.2, and so the presence of a missing matter component cannot be ruled out. Nonetheless, a Bayesian model selection analysis disfavours its introduction by about 1.5 log-units of evidence. Foregoing our requirement of form invariance of the Friedmann equation under the reciprocity transformation, we extend our analysis by allowing w_X to be a free parameter. For this more generic `double dark energy' model, we find w_X= -1.02\pm 0.20 and \Omega_{X,0}= 0.08\pm 0.57, which is again consistent with LCDM, although once more the posterior distributions are sufficiently broad that the existence of a second dark-energy component cannot be ruled out. Moreover, the two-dimensional posterior in the (w_X,\Omega_{X,0})-plane is strongly bimodal with both peaks offset from the standard LCDM corresponding to (-1,0), although the latter is still admissible; this bimodality is in contrast to the correctly-centred unimodal posterior obtained when analysing simulated observations from a LCDM model.