Abstract
We present a simple (microscopic) model in which bulk viscosity plays a role in explaining the present acceleration of the universe. The effect of bulk viscosity on the Friedmann equations is to turn the pressure into an “effective” pressure containing the bulk viscosity. For a sufficiently large bulk viscosity, the effective pressure becomes negative and could mimic a dark energy equation of state. Our microscopic model includes self-interacting spin-zero particles (for which the bulk viscosity is known) that are added to the usual energy content of the universe. We study both background equations and linear perturbations in this model. We show that a dark energy behavior is obtained for reasonable values of the two parameters of the model (i.e. the mass and coupling of the spin-zero particles) and that linear perturbations are well-behaved. There is no apparent fine tuning involved. We also discuss the conditions under which hydrodynamics holds, in particular that the spin-zero particles must be in local equilibrium today for viscous effects to be important.
Highlights
Background evolution3.1 Implementation in Cosmic Linear Anisotropy Solving System1 (CLASS)The pressure, energy density and bulk viscosity are all functions of the dimensionless inverse temperature x = m0/Ts (c.f. eqs. 2.7–2.9): ps(x) m40 2π2 K2(x) x2, ρs(x) K2′ (x) x, ζ=m 4 λ4 mth x mth m0eκ1 eκ2(mth/m0)x, where primes denote derivatives with respect to x and the subscript s relates to the scalar particles
We should instead bear in mind that the viscosity of the fluid remains negligible in an adiabatic transformation, so we must define the sound speed for the scalar fluid using the equilibrium pressure: c2s p′s(x) ρ′s(x)
In summary we present a model of cosmological evolution with an additional scalar field
Summary
Hydrodynamics is an effective theory that describes the evolution of “fluid cells”, where the equations of motion are given by exact local conservation of energy-momentum (we implicitly assume that there are no other conserved charges in the following). Macroscopic quantities such as pressure p, energy density ρ and velocity v are smooth inside a fluid cell and local thermal equilibrium is maintained. The effect of dissipation due to microscopic interactions is characterized by transport coefficients (shear and bulk viscosities). Momentum flows between neighboring fluid cells due to microscopic interactions and is driven by velocity gradients. The most general energy-momentum tensor linear in velocity gradients that satisfies the second law of thermodynamics for all fluid configurations is [33]: Tvμiνscous = Tpμeνrfect − ζ (gμν + U μU ν ) Dγ U γ ,. In the absence of bulk viscosity, the fluid relaxes instantaneously and pressure and density are related by the equation of state. The bulk viscosity coefficient ζ depends on microscopic interactions and must be computed from a more fundamental theory
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