Improving the Lagrangian perturbative solution for a cosmic fluid: Applying Shanks transformation

We study the behavior of Lagrangian perturbative solutions. For a spherical void model, the higher order the Lagrangian perturbation we consider, the worse the approximation becomes in late-time evolution. In particular, if we stop to improve until an even order is reached, the perturbative solution describes the contraction of the void. To solve this problem, we consider improving the perturbative solution using Shanks transformation, which accelerates the convergence of the sequence. After the transformation, we find that the accuracy of higher-order perturbation is recovered and the perturbative solution is refined well. Then we show that this improvement method can apply for a $\Lambda \mathrm{CDM}$ model and improved the power spectrum of the density field.

Figure 1

The evolution of the spherical model for positive fluctuation. The thick solid line shows evolution by exact solution. The fine solid line, the dashed-dotted line, and the dashed line show evolution by first-, fifth-, and 11th-order Lagrangian perturbation, respectively. In this case, the higher-order Lagrangian approximation gives an accurate description. In this figure, the scale factor is normalized by the time of a caustic formation in a first-order perturbation.

Figure 2

The evolution of the spherical model for negative fluctuation. In this figure, the scale factor is normalized as in Fig. 1. The thick solid line shows the evolution by the exact solution. The fine solid line, the dashed-dotted line, and the dashed line shows the evolution by first-, fifth-, and 11th-order Lagrangian perturbation, respectively. In this case, the higher-order Lagrangian approximation deviates from the exact solution at a late time. In other words, ZA gives the best description for the late-time evolution of a void.

Figure 3

The evolution of the spherical model for negative fluctuation. In this figure, the scale factor is normalized as in Fig. 1. We apply Shanks transformation to the Lagrangian perturbative solution and improve its accuracy. The thick solid line shows the evolution by the exact solution. The fine solid line shows the behavior of original 11th-order perturbative solution. The dashed-dotted line, and the dashed line shows the evolution by once transformed and twice transformed perturbative solutions, respectively. After 3 times transformation, because the difference between exact solution and perturbative solution becomes quite small. Therefore, if we also draw the curve of 3 times transformed perturbative solutions, we cannot find the difference on the figure. In this case, by application of Shanks transformation, the Lagrangian approximation improves its accuracy. In other words, we can describes late-time evolution of voids well.

Figure 4

The evolution of the spherical model for negative fluctuation. In this figure, the scale factor is normalized as in Fig. 1. We apply Padé approximation to the Lagrangian perturbative solution and improve its accuracy. The thick solid line shows the evolution by the exact solution. The fine solid line shows the behavior of original 11th-order perturbative solution. The dashed-dotted line, and the dashed line shows the evolution by the case of $(M,N)=(1,10)$ and (3,8), respectively. When $N$ is greatly different from $M$, the approximation is not improved well. When we choose $(M,N)=(5,6)$, the difference between exact solution and perturbative solution becomes quite small. Using Padé approximation, we can improve the perturbative solution well.

Figure 5

The power spectrum of the density field in $N$-body simulation. Because of a nonlinear effect in small structures, the large-$k$ components in the power spectrum grow remarkably.

Figure 6

The power spectrum of the density field in Lagrangian third-order approximation. Because of the shell-crossing, the spectrum distorted from the large-$k$ components to the small-$k$ components gradually. At $z=0$, the spectrum is really differs greatly that in $N$-body simulation.

Figure 7

The power spectrum of the density field in Shanks transformation. Because of the improvement of the nonlinear effect, the distortion of the spectrum is improved well.