Cosmic calibration: Constraints from the matter power spectrum and the cosmic microwave background

Several cosmological measurements have attained significant levels of maturity and accuracy over the past decade. Continuing this trend, future observations promise measurements of the cosmic mass distribution at an accuracy level of 1% out to spatial scales with $k\sim 10h{\mathrm{Mpc}}^{-1}$ and even smaller, entering highly nonlinear regimes of gravitational instability. In order to interpret these observations and extract useful cosmological information from them, such as the equation of state of dark energy, very costly high precision, multiphysics simulations must be performed. We have recently implemented a new statistical framework with the aim of obtaining accurate parameter constraints from combining observations with a limited number of simulations. The key idea is the replacement of the full simulator by a fast emulator with controlled error bounds. In this paper, we provide a detailed description of the methodology and extend the framework to include joint analysis of cosmic microwave background and large-scale structure measurements. Our framework is especially well suited for upcoming large-scale structure probes of dark energy such as baryon acoustic oscillations and, especially, weak lensing, where percent level accuracy on nonlinear scales is needed.

Figure 1

Twenty-eight data points mimicking a combined data set from a large-scale structure survey and CMB data. The error bars are chosen to match the power spectrum calculation in Ref. 37. The dashed line shows the original input power spectrum from ten realizations, while the gray lines show a subset of the 128 simulated power spectra, discussed in Sec. 2b.

Figure 2

Lower triangle of plots: 2-dimensional projections of a $m=128$ point, 5-level, OA design. Upper triangle: An OA-based LH design obtained by spreading out the 5 level OA design so that each 1-dimensional projection gives an equally spaced set of points along [0,1].

Figure 3

Simulations (left), mean (center), and the first five principal component bases (right) derived from the simulation output for the power spectrum.

Figure 4

Boxplots of posterior samples for each ${\rho }_{wil}$ for the mass power spectrum. The box extends from the 25th and 75th percentiles of the data, with the red (center) line denoting the median. The whiskers (black lines) of the boxplot extend out from the box to the full range of the data.

Figure 5

Changes to the posterior mean simulator predictions obtained by varying one input, while holding others at their central values, i.e. at the midpoint of their range. The light to dark lines correspond to the smallest parameter setting to the biggest, for each parameter.

Figure 6

Emulator predictions for a 64 run design. Left: emulator based on 128 runs, right: emulator based on 32 runs. The central gray region contains the middle 50% of the residuals, the wider light gray region, the middle 90%. The outliers are shown as dots. The improvement of the emulator accuracy with more training runs is evident, especially in the medium $k$-range.

Figure 7

Estimated posterior distribution of the parameters $\theta =(n,h,{\sigma }_8,{\Omega }_{\mathrm{CDM}},{\Omega }_{\mathrm{b}})$. The diagonal shows the estimated marginal posterior pdf for each parameter; the off-diagonal images give estimates of bivariate marginals; the contour lines show estimated 90% highest posterior density regions. The true values from which the data were generated are shown by the red dots.

Figure 8

Subset of 128 simulated TT power spectra (gray lines) along with the synthetic observations. The dashed line shows the actual spectrum from which the data were generated. The blue (dark gray) lines show the limits of the restricted parameter ranges from Eqs. (33).

Figure 9

Emulator performance on hold-out test. The central gray region contains the middle 50% of the residuals, the wider light gray region, the middle 90%. The outliers are shown as blue (light gray) curves. Upper plot: Residuals for the emulator built on the conservative priors given in Eqs. (8). Lower plot: Residuals for the emulator built on $3\mathrm{\text{−}}\sigma$ priors around the best-fit WMAP-III parameters. The emulator errors are reduced by an order of magnitude for the smaller parameter ranges.

Figure 10

Estimated posterior distribution following Fig. 7 for the synthetic CMB TT spectrum and including the optical depth $\tau$.

Figure 11

Estimated posterior distribution following Fig. 10 with the addition of matter power spectrum data.

Figure 12

Posterior univariate marginal density estimates for the cosmological parameters taking into account the TT only data (white) and both the TT and the matter power spectrum data (orange, solid color), showing the increased precision obtainable by systematically including multiple sources of data.

Figure 13

Residuals for the Pico emulator of the TT power spectrum, tested on 64 runs. The upper plot (in red) shows the residuals for runs which are in the $3\mathrm{\text{−}}\sigma$ confidence level around the best-fit WMAP-I model. 22 out of the 64 runs fulfill this criterion. The residuals are at the 0.5% accuracy level in this case. Note the slight downward trend in all cases. The lower plot (in blue) shows the remaining 42 runs which are in the $3\mathrm{\text{−}}\sigma$ confidence level around the best-fit parameters from WMAP. In this case the errors are much worse and the emulator is not controlled. Again a slight downward trend can be seen in all the runs.

Figure 14

Results following Fig. 13 but for the CMBwarp emulator. The errors are overall an order of magnitude bigger than for Pico, in agreement with the findings of Ref. 22.