Jeans instability in a universe with dissipation

The problem of Jeans gravitational instability is investigated for static and expanding universes within the context of the five and thirteen field theories which account for viscous and thermal effects. For the five-field theory a general dispersion relation has been derived with the help of relevant linearized perturbation equations, showing that the shear viscosity parameter alters the propagating modes for large and small wavelengths. The behavior of density and temperature contrasts are analyzed for the hard-sphere model in detail. In the small wavelengths regime, increasing the amount of shear viscosity into the system forces the harmonic perturbations to damp faster, however, in the opposite limit larger values of shear viscosity lead to smaller values of density and temperature contrasts. We also consider the hyperbolic case associated with the thirteen-field theory which involves two related parameters, namely the shear viscosity and the collision frequency, the last one is due to the production terms which appear in the Grad method. The dispersion relation becomes a polynomial in the frequency with two orders higher in relation to the five-field theory, indicating that the effects associated with the shear viscosity and heat flux are nontrivial. The profile of Jeans mass in terms of the temperature and number density is explored by contrasting with several data of molecular clouds. Regarding the dynamical evolution of the density, temperature, stress and heat flux contrasts for a universe dominated by pressureless matter, we obtain also damped harmonic waves for small wavelengths. In the case of large wavelengths, the density and temperature contrasts grow with time (due to the Jeans mechanism) while the stress and heat flux contrasts heavily decay with time. For an expanding universe, the Jeans mass and Jeans length are obtained and their physical consequences are explored.

Figure 1

Behavior of the dimensionless dispersion relations in terms of ${\xi }_J={\lambda }_J/\lambda$ for two different cases, namely, ${\mu }^{*}=0$ and ${\mu }_{*}={10}^{-5}$.

Figure 2

Time evolution of the density contrast for small wavelengths with $\alpha =1$ and ${\xi }_J=100$.

Figure 3

Time evolution of the density contrast and temperature contrast for large wavelengths with $\alpha =1$ and ${\xi }_J=1/2$.

Figure 4

Upper level: Time evolution of the density (straight line), temperature (dashed line), stress (dotted line), and heat flux (dot-dashed line) as damped harmonic waves for small wavelengths ${\xi }_J=100$ and dimensionless shear viscosity ${\mu }_{*}={10}^{-5}$. Lower level: Time evolution of the density (straight line), temperature (dashed line), stress (dotted line), and heat flux (dot-dashed line) for large wavelengths ${\xi }_J=0.5$ and dimensionless shear viscosity ${\mu }_{*}={10}^{-5}$.