Correspondence between the adhesion model and the velocity dispersion for the cosmological fluid

Basing our discussion on the Lagrangian description of hydrodynamics, we studied the evolution of density fluctuation for nonlinear cosmological dynamics. Adhesion approximation (AA) is known as a phenomenological model that describes the nonlinear evolution of density fluctuation rather well and that does not form a caustic. In addition to this model, we have benefited from discussion of the relation between artificial viscosity in AA and velocity dispersion. Moreover, we found it useful to regard whether the velocity dispersion is isotropic produces effective pressure or viscosity terms. In this paper, we analyze plane- and spherical-symmetric cases and compare AA with Lagrangian models where pressure is given by a polytropic equation of state. From our analyses, the pressure model undergoes evolution similar to that of AA until reaching a quasinonlinear regime. Compared with the results of a numerical calculation, the linear approximation of the pressure model seems rather good until a quasinonlinear regime develops. However, because of oscillation arising from the Jeans instability, we could not produce a stable nonlinear structure.

Figure 1

The evolution of density in linear approximations (ZA, AA, and the pressure model). The viscosity parameter in AA is given by $\nu =(\pi /512{)}^2$. (a) The density of the densest point. In AA, the density remains finite forever. On the other hand, in the pressure model, if $\gamma >4/3$, the density becomes infinity. Therefore, we cannot avoid the formation of the caustic. (b) The density of the sparsest point. The growth of the void with the pressure model is fast in comparison with AA.

Figure 2

The evolution of density in AA and the pressure model (linear approximation and full-order). In the pressure model, the nonlinear effect suppresses the evolution of density fluctuation. Therefore, the linear approximation becomes worse in a strongly nonlinear regime. If $\gamma >4/3$, although we calculate a full-order model, we cannot avoid the formation of the caustic. (a) The case where $\gamma =4/3$, the density of the densest point. (b) The case where $\gamma =4/3$, the density of the sparsest point. (c) The case where $\gamma =5/3$, the density of the densest point. (d) The case where $\gamma =5/3$, the density of the sparsest point.

Figure 3

Mexican-hat-type model. The average of density fluctuation over the whole space becomes zero.

Figure 4

The evolution of the spherical-symmetric (the Mexican-hat-type) density fluctuation at $r=0$ in dust models. (a) The evolution of a density fluctuation in ZA, PZA, PPZA, the exact model, and AA in the case where $\epsilon >0$. The viscosity parameter in AA is set as $\nu =(1/512{)}^2$. The approximation is improved by higher-order perturbation. In the exact model, the caustic appears at $a\simeq 0.55$. On the other hand, the density fluctuation evolves gently in AA. (b) The same as (a) but the case where $\epsilon <0$. In PZA, the fluctuation becomes positive during evolution. Later ($a>0.6$), PPZA deviates from an exact solution greatly more than does ZA.

Figure 5

The evolution of the spherical-symmetric (Mexican-hat-type) density fluctuation at $r=0$ in the pressure models with the linear approximation. (a) The evolution of a density fluctuation in the case where $\epsilon >0$. Until a quasinonlinear regime develops, the pressure models show behavior resembling AA. However, the fluctuation oscillates in the nonlinear region. In the case where $\gamma =1$, the fluctuation oscillates little by little. Finally, the fluctuation decays and disappears. In the case where $\gamma =4/3$, the fluctuation oscillates and the caustic appears at $a\simeq 1.44$. In the case where $\gamma =5/3$, the oscillation of the fluctuation in the intermediate state grows very large. Then the caustic appears at $a\simeq 1.28$. (b) The same as (a) but here $\epsilon <0$. In the case where $\gamma =1$, finally the fluctuation decays and disappears. In the case where $\gamma =4/3$, although the fluctuation oscillates, the density asymptotically decreases. In the case where $\gamma =5/3$, the density fluctuation becomes positive during evolution because the oscillation of the fluctuation grows very large.

Figure 6

The evolution of a density fluctuation at $r=0$ in the spherical-symmetric case. These figures show the evolution in AA and the pressure models (linear approximation and the full-order calculation). These figures show the case where $\epsilon >0$. (a) The case where $\epsilon >0$ and $\gamma =1$. In the pressure model, linear approximation seems valid until $\delta \simeq 1$. After that, in linear approximation, the fluctuation oscillates violently. In a strongly nonlinear region ($\delta >10$), even if we consider full-order calculation in the pressure model, the evolution of a fluctuation similar to AA cannot be reproduced. (b) The same as (a), but here $\epsilon <0$. As in the case where $\epsilon >0$, when the fluctuation evolves fully ($\delta <-0.5$), the fluctuation begins to oscillate. Finally, the fluctuation decays and approaches $0$. (c) The same as (a), but here $\gamma =4/3$. In the pressure model with linear approximation, the fluctuation oscillates, and the caustic appears at $a\simeq 1.44$. When we consider a full-order equation, although we can delay the density divergence, we cannot avoid the formation of the caustic. (d) The same as (b), but here $\gamma =4/3$. In the linear approximation in the pressure model, although the fluctuation oscillates, the density asymptotically decreases. In the pressure model, when we consider a full-order calculation, we can realize the evolution of a fluctuation similar to that of AA. (e) The same as (a), but here $\gamma =5/3$. In the pressure model with the linear approximation, the oscillation of the fluctuation in the intermediate state grows very large. Then the caustic appears at $a\simeq 1.28$. When we consider a full-order equation, the density diverges a little to the outside at $a\simeq 1.14$, and the model fails. (f) The same as (b), but here $\gamma =5/3$. In the linear approximation in the pressure model, the density fluctuation becomes positive during evolution because the oscillation of the fluctuation grows very large. On the other hand, when we consider a full-order equation, it is different from linear approximation, the density fluctuation always remaining negative.