Dark matter as integration constant in Horava-Lifshitz gravity

TL;DR

This paper shows that in Horava-Lifshitz gravity with the projectability condition, the Hamiltonian constraint is global rather than local, allowing an integration-constant term to appear in the IR dynamics. This term, captured by a dust-like energy density $\rho^{HL}$ with $T^{HL}_{\mu\nu}=\rho^{HL} n_{\mu}n_{\nu}$, can mimic cold dark matter when integrated over space satisfies $\int d^3x \sqrt{g}\,\rho^{HL}=0$ but is nonzero within our Hubble patch. The authors derive the infrared field equations and conservation laws, showing that, in the IR limit with $\lambda=1$ and restored diffeomorphism, the dynamics reduce to general relativity plus an emergent dust component, sourced by any breaking of 4D diffeomorphism in the early universe. This provides a conceptual route for HL gravity to reproduce GR phenomenology with CDM-like behavior without introducing a new dynamical field or CDM action, highlighting the role of global constraints and initial conditions in cosmological evolution.

Abstract

In the non-relativistic theory of gravitation recently proposed by Horava, the Hamiltonian constraint is not a local equation satisfied at each spatial point but an equation integrated over a whole space. The global Hamiltonian constraint is less restrictive than its local version, and allows a richer set of solutions than in general relativity. We show that a component which behaves like pressureless dust emerges as an "integration constant" of dynamical equations and momentum constraint equations. Consequently, classical solutions to the infrared limit of Horava-Lifshitz gravity can mimic general relativity plus cold dark matter.