Kinetic equation for Lifshitz scalar

Abstract

Employing the method of Wigner functions on curved spaces, we study classical kinetic (Boltzmann-like) equations of distribution functions for a real scalar field with the Lifshitz scaling. In particular, we derive the kinetic equation for z = 2 on general curved spaces and for z = 3 on spatially flat spaces under the projectability condition N = N (t), where z is the dynamical critical exponent and N is the lapse function. We then conjecture a form of the kinetic equation for a real scalar field with a general dispersion relation in general curved geometries satisfying the projectability condition, in which all the information about the nontrivial dispersion relation is included in the group velocity and which correctly reproduces the equations for the z = 2 and z = 3 cases as well as the relativistic case. The method and equations developed in the present paper are expected to be useful for developments of cosmology in the context of Hořava-Lifshitz gravity. I. INTRODUCTIONGravitational phenomena observed to the present are well described by general relativity (GR). However, at short distances quantum gravity effects become significant and GR loses its predictability. Among various approaches to quantum gravity, Hořava-Lifshitz (HL) gravity is unique in the sense that it is a local field theory of gravity which is perturbatively renormalizable at least at the power-counting level and which is free from the Ostrogradsky ghost [1]. The theory is rendered power-counting renormalizable by the scaling anisotropic in space and time (the so-called Lifshitz scaling) at high energy,and recovers the usual isotropic scaling at low energy. Here, z is a constant often called the dynamical critical exponent and b is an arbitrary constant. The renormalizability beyond the power-counting level has recently been proven in the minimal setup called the projectable theory, where the lapse function depends only on time [2, 3]. As for observational constraints on parameters, see, e.g., Refs. [4,5] for the nonprojectable theory and Sec. 3.2 of Ref.[6] for the projectable theory. Since HL gravity is a candidate for quantum gravity and thus can presumably describe gravity at very short distances, it is natural to seek its implications to the early universe cosmology [6]. In the early universe or at high energies, it is expected from theoretical consistency that not only the gravity sector but also the matter sector should exhibit the Lifshitz scaling. A field in the matter sector should then obey a dispersion relation of the form ω 2 ∼ (k 2 /a 2 ) z /M 2(z−1) in the early universe, where k is the comoving momentum, a is the scale factor of the universe, M is the (momentum) scale characterizing the Lifshitz scaling, and z is the dynamical critical exponent. The energy of each high energy particle should decay as ω ∝ 1/a z as the universe expands. As a result the energy density of a gas of Lifshitz particles should decay as ρ z ∝ 1/a z+3 [7]. In reality, we need to consider the transition from a high value of z to lower values, say, from...