New method for initial density reconstruction

A theoretically interesting and practically important question in cosmology is the reconstruction of the initial density distribution provided a late-time density field. This is a long-standing question with a revived interest recently, especially in the context of optimally extracting the baryonic acoustic oscillation (BAO) signals from observed galaxy distributions. We present a new efficient method to carry out this reconstruction, which is based on numerical solutions to the nonlinear partial differential equation that governs the mapping between the initial Lagrangian and final Eulerian coordinates of particles in evolved density fields. This is motivated by numerical simulations of the quartic Galileon gravity model, which has similar equations that can be solved effectively by multigrid Gauss-Seidel relaxation. The method is based on mass conservation, and does not assume any specific cosmological model. Our test shows that it has a performance comparable to that of state-of-the-art algorithms that were very recently put forward in the literature, with the reconstructed density field over $\sim 80\%$ (50%) correlated with the initial condition at $k\lesssim 0.6h/\mathrm{Mpc}$ ($1.0h/\mathrm{Mpc}$). With an example, we demonstrate that this method can significantly improve the accuracy of BAO reconstruction.

Figure 1

Method test using a cubic particle distribution. The final density field is a uniform distribution of $128^3$ particles inside a $10\mathrm{Mpc}/h$-per-side cubic box placed at the center of the simulation box of size $128\mathrm{Mpc}/h$, with the faces of the two boxes parallel to each other. The color map is the $\theta$ field computed by the new reconstruction method, with the color bar on the right indicating the values of $\theta$. The horizontal and vertical gray lines are respectively lines with equal Lagrangian $q_y$ and $q_x$ coordinates. Note that only gray lines within the central blue box (which is zoomed in the lower right corner of the figure) are meaningful—particles at the corners of this blue box are at the corners of the simulation box in the initial (reconstructed) distribution. As mentioned in the abstract, the reconstruction method is based solely on mass conservation, with no information about how the density field has evolved. Only $64\times 64$ (out of $128\times 128$) lines are shown here for a clear view, and the plot is a $0.64\mathrm{Mpc}/h$-thick slice perpendicular to the $z$-axis at the middle of the box. Note that the $q$-grid here is regularly spaced except the region near the edge, where numerical errors occurred.

Figure 2

Similar to Fig. 1, but now the reconstruction is performed starting from a proper $N$-body simulation snapshot at $z=0$. The blue dots are the particles inside the slice shown at their Eulerian coordinates at $z=0$. Note how the $q$-grid distorts following the distribution of matter particles and becomes concentrated (expanded) in overdense (underdense) regions.

Figure 3

The correlation coefficient $r(k)$ between the initial and final (green solid line), initial and reconstructed (red dotted), and final and reconstructed (blue dashed) density fields, which show that the reconstruction successfully recovers information in the initial density field that gets lost due to structure formation.

Figure 4

Normalized histograms of the initial ($z=49$; red), final ($z=0$; green), and reconstructed (blue) density field values, and $\delta =\rho /{\rho }_0-1$ is the overdensity. The initial density field has been linearly extrapolated to $z=0$ by multiplying with the linear growth factor $D_{+}\approx 35.8$. All density fields have been smoothed with a $2h^{-1}\mathrm{Mpc}$ Gaussian filter.

Figure 5

Comparison of the BAO signals from the initial (red dashed line), final (green dot-dashed), and reconstructed (blue solid) density fields. As we can see, for the reconstructed field the amplitude and positions of the peaks are in good agreement with those of the initial linear density field even after $k\approx 0.3h/\mathrm{Mpc}$, while in the nonlinearly evolved density field the peak features are degraded substantially. This demonstrates that the reconstruction can greatly improve the accuracy of BAO measurements.