Gravitational Clustering of Ricci Holographic Dark Matter and Need for a Lagrangian

Establish whether holographic dark matter derived from the Ricci cutoff, defined by the energy density relation rho_HDM = 3 c^2 L^{-2} with L = (α H^2 + β Ḣ)^{-1/2} and β = α/2, clusters gravitationally as observed; to resolve this, develop an underlying Lagrangian for the holographic model to determine the perturbation dynamics and physical sound speed.

Background

In the proposed model, dark matter arises from a holographic energy density using the Granda–Oliveros (Ricci) cutoff with parameters chosen so that one term scales as a^-3, mimicking matter. While matching the background evolution is necessary, successful dark matter must also exhibit gravitational clustering consistent with structure formation.

The authors note that the adiabatic sound speed in their model is zero, but emphasize that the relevant physical sound speed c_s governing clustering cannot be determined without an underlying Lagrangian; previous studies often insert c_s by hand, which the authors deem inadequate. They state explicitly that resolving whether HDM clusters as observed requires constructing a Lagrangian for the holographic model.