Determining the metric of the Cosmos: Stability, accuracy, and consistency

The ultimate application of Einstein’s field equations is to empirically determine the geometry of the Universe from its matter content, rather than simply assuming the Universe can be represented by a homogeneous model on all scales. Choosing a Lemaître-Tolman-Bondi model as the most convenient inhomogeneous model for the early stages of development, a data reduction procedure was recently validated using perfect test data. Here, we simulate observational uncertainties and improve the previous numerical scheme to ensure that it will be usable with real data as soon as observational surveys are sufficiently deep and complete. Two regions require special treatment—the origin and the maximum in the areal radius. To minimize numerical errors near the origin, we use a Lemaître-Tolman-Bondi series expansion to provide the initial values for integrating the differential equations. We also use an improved method to match the numerical integration to the series expansion that bridges the region near the maximum in the areal radius. Because the mass enclosed within the maximum obeys a specific relationship, we show that it is possible to correct for a fixed systematic error in either the distance scale or the redshift-space mass density, such that the integrated values are consistent with the data at the maximum.

Figure 1

Comparison of the relative errors in the bang time $t_B$ versus $z$ using the old (crosses) and new (asterisks) methods. The new method fits a LTB series expansion to the first 50 points, before starting the numerical integration, thus allowing the solution to be inhomogeneous while avoiding the numerical difficulties near the origin. This plot is for an inhomogeneous universe with origin Hubble and deceleration parameters $h_0=0.72$ and $q_0=0.6$, and with ${\Omega }_{\Lambda }=0$.

Figure 2

Comparison of the relative errors in $W$ (top) and $M$ (bottom) versus $z$ in the region before ${\widehat{R}}_m$, using the old (small crosses) and new (large diamonds) methods for matching from the numerical values to the near-maximum series expansion values. These plots are for a FLRW universe with $h_0=0.72$, $q_0=0.22$, and ${\Omega }_{\Lambda }=0$, with statistical errors added to the data to simulate observational errors. Simply matching $M$ can lead to a large jump in $W$ and a divergence in $M$, whereas the new method leads to small jumps in both $M$ and $W$ that always bring them toward the correct values.

Figure 3

Relative $\varphi$ error versus $z$, with the estimated error (large points) plotted alongside the actual error (small points) for comparison. Gaussian deviates are added to the data, and the estimated uncertainty calculated, along with the actual error from the ideal data. It can be seen that the error in $\varphi$ grows before the maximum as it becomes increasingly unstable, and then decreases after the maximum as it becomes stable again. This plot is for an inhomogeneous universe with origin parameters $h_0=0.72$, $q_0=0.6$, and ${\Omega }_{\Lambda }=0$.

Figure 4

Systematic errors in the data lead to an inconsistency when attempting to join numerical and near-maximum series regions, leading to jumps in $\varphi$, $W$, $M$, and $t_B$. This is illustrated for a systematic error in $\widehat{R}$ only. The functions $\varphi$ (top left), $r$ (top right), $W$ (middle left), $M$ (middle right), and $t_B$ (bottom) versus $z$ are plotted with the exact data (thick line) and with various corrections of the systematic error (thin lines). Each subsequent curve represents a 5% difference in the correction of the $\widehat{R}$ data. The plots are for a FLRW universe with $h_0=0.72$, $q_0=0.22$, and ${\Omega }_{\Lambda }=0$.

Figure 5

Similar to Fig. 4, but with a systematic error added to the $4\pi \mu n$ data instead of the $\widehat{R}$ data. The jumps due to various corrections of the systematic error in $4\pi \mu n$ are illustrated for the functions $\varphi$ (top left), $r$ (top right), $W$ (middle left), $M$ (middle right), and $t_B$ (bottom) versus $z$, with the exact data (thick line) for comparison. Each subsequent curve (thin lines) represents a 5% difference in the correction of the $4\pi \mu n$ data. The plots are for a FLRW universe with $h_0=0.72$, $q_0=0.22$, and ${\Omega }_{\Lambda }=0$.

Figure 6

Corrected curves with systematic errors added to both the $\widehat{R}$ and $4\pi \mu n$ data ($+10\%$ and $-10\%$ respectively). The plots show the exact data (dashes) and the corrected curves obtained by matching the curves for the $\widehat{R}$ data (dots) or $4\pi \mu n$ data (solid curve) to eliminate jumps. It is apparent the jumps may be removed by correcting either the $\widehat{R}$ or the $4\pi \mu n$ data. It is not possible to determine which correction should be made, but either result is much more self-consistent than with no correction. This is illustrated for the functions $\varphi$ (top left), $r$ (top right), $W$ (middle left), $M$ (middle right), and $t_B$ (bottom) versus $z$. The plots of $r$, $M$, and $t_B$ clearly end up overcorrected by 10% when a correction is made to one variable. These plots are for an inhomogeneous universe with origin parameters $h_0=0.72$, $q_0=0.6$, and ${\Omega }_{\Lambda }=0$: a strong inhomogeneity is present around $z=0.2:0.3$.