Can the clustered dark matter and the smooth dark energy arise from the same scalar field?

Cosmological observations suggest the existence of two different kinds of energy densities dominating at small $(\ensuremath{\lesssim}500\mathrm{Mpc})$ and large $(\ensuremath{\gtrsim}1000\mathrm{Mpc})$ scales. The dark matter component, which dominates at small scales, contributes ${\ensuremath{\Omega}}_{m}\ensuremath{\approx}0.35$ and has an equation of state $p=0,$ while the dark energy component, which dominates at large scales, contributes ${\ensuremath{\Omega}}_{V}\ensuremath{\approx}0.65$ and has an equation of state $p\ensuremath{\simeq}\ensuremath{-}\ensuremath{\rho}.$ It is usual to postulate weakly interacting massive particles (WIMPs) for the first component and some form of scalar field or cosmological constant for the second component. We explore the possibility of a scalar field with a Lagrangian $L=\ensuremath{-}V(\ensuremath{\varphi})\sqrt{1\ensuremath{-}{\ensuremath{\partial}}^{i}\ensuremath{\varphi}{\ensuremath{\partial}}_{i}\ensuremath{\varphi}}$ acting as both clustered dark matter and smoother dark energy and having a scale-dependent equation of state. This model predicts a relation between the ratio $r={\ensuremath{\rho}}_{V}/{\ensuremath{\rho}}_{\mathrm{DM}}$ of the energy densities of the two dark components and an expansion rate n of the universe [with $a(t)\ensuremath{\propto}{t}^{n}]$ in the form $n=(2/3)(1+r).$ For $r\ensuremath{\approx}2,$ we get $n\ensuremath{\approx}2$ which is consistent with observations.

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