Perturbation theory challenge for cosmological parameters estimation: Matter power spectrum in real space

We study the accuracy with which cosmological parameters can be determined from a real space power spectrum of matter density contrast at weakly nonlinear scales using analytical approaches. From power spectra measured in $N$-body simulations and using the Markov chain Monte Carlo technique, the best-fitting cosmological input parameters are determined with several analytical methods as a theoretical template, such as the standard perturbation theory, the regularized perturbation theory, and the effective field theory. We show that at redshift 1, all two-loop level calculations can fit the measured power spectrum down to scales $k\sim 0.2h{\mathrm{Mpc}}^{-1}$, and cosmological parameters are successfully estimated in an unbiased way. Introducing the figure of bias (FoB) and figure of merit (FoM) parameter, we determine the validity range of those models and then evaluate their relative performances. With one free parameter, namely the damping scale, the regularized perturbation theory is found to be able to provide the largest FoM parameter while keeping the FoB in the acceptance range.

Figure 1

Predictions of the power spectrum for each analytical method with fiducial cosmological parameters. For RegPT+ and IR-resummed EFT models, nuisance parameters are determined by the least-squares method using the power spectrum up to $k_{\max }=0.27h{\mathrm{Mpc}}^{-1}$. The arrow shows the maximum wave number in the least-squares method. Note that spectra with these two models for wave numbers larger than $k_{\max }$ are shown as dashed lines.

Figure 2

Power spectra with each analytical method with best-fit parameters with $k_{\max }=0.27h{\mathrm{Mpc}}^{-1}$. Note that spectra for wave numbers larger than $k_{\max }$ are shown as dashed lines and the arrow shows $k_{\max }$. The best-fit cosmological parameters and nuisance parameters are estimated from MCMC chains.

Figure 3

Best-fitting power spectra for RegPT at the two-loop level with best-fit parameters with different maximum wave numbers from $0.15h{\mathrm{Mpc}}^{-1}$ to $0.30h{\mathrm{Mpc}}^{-1}$. The arrows show corresponding maximum wave numbers $k_{\max }$. Note that spectra for wave numbers larger than $k_{\max }$ are shown as dashed lines. The best-fit cosmological parameters are estimated from MCMC chains.

Figure 4

Best-fitting power spectra for SPT at the two-loop level with best-fit parameters with different maximum wave numbers from $0.15h{\mathrm{Mpc}}^{-1}$ to $0.30h{\mathrm{Mpc}}^{-1}$. The arrows show corresponding maximum wave numbers $k_{\max }$. Note that spectra for wave numbers larger than $k_{\max }$ are shown as dashed lines. The best-fit cosmological parameters are estimated from MCMC chains.

Figure 5

Best-fitting power spectra for RegPT+ at the two-loop level with best-fit parameters with different maximum wave numbers from $0.15h{\mathrm{Mpc}}^{-1}$ to $0.36h{\mathrm{Mpc}}^{-1}$. The arrows show corresponding maximum wave numbers $k_{\max }$. Note that spectra for wave numbers larger than $k_{\max }$ are shown as dashed lines. The best-fit cosmological parameters are estimated from MCMC chains.

Figure 6

Best-fitting power spectra for IR-resummed EFT at the two-loop level with best-fit parameters with different maximum wave numbers from $0.15h{\mathrm{Mpc}}^{-1}$ to $0.36h{\mathrm{Mpc}}^{-1}$. The arrows show corresponding maximum wave numbers $k_{\max }$. Note that spectra for wave numbers larger than $k_{\max }$ are shown as dashed lines. The best-fit cosmological parameters are estimated from MCMC chains.

Figure 7

Parameter confidence regions with RegPT at the two-loop level and $k_{\max }=0.21h{\mathrm{Mpc}}^{-1}$. The light, normal, and dark blue lines correspond to the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ limits, respectively. The median and 16% and 84% percentiles are also shown on the top of each histogram. The black dashed lines correspond to 16% and 84% percentiles. The red lines show the input parameters.

Figure 8

Parameter confidence regions with SPT at the two-loop level and $k_{\max }=0.21h{\mathrm{Mpc}}^{-1}$. The light, normal, and dark blue lines correspond to the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ limits, respectively. The median and 16% and 84% percentiles are also shown on the top of each histogram. The black dashed lines correspond to 16% and 84% percentiles. The red lines show the input parameters.

Figure 9

Parameter confidence regions with RegPT+ at the two-loop level and $k_{\max }=0.21h{\mathrm{Mpc}}^{-1}$. The light, normal, and dark blue lines correspond to the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ limits, respectively. The median and 16% and 84% percentiles are also shown on the top of each histogram. The black dashed lines correspond to 16% and 84% percentiles. The red lines show the input parameters.

Figure 10

Parameter confidence regions with IR-resummed EFT at the two-loop level and $k_{\max }=0.21h{\mathrm{Mpc}}^{-1}$. The light, normal, and dark blue lines correspond to the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ limits, respectively. The median and 16% and 84% percentiles are also shown on the top of each histogram. The black dashed lines correspond to 16% and 84% percentiles. The red lines show the input parameters.

Figure 11

Parameter confidence regions with RESPRESSO at the two-loop level and $k_{\max }=0.21h{\mathrm{Mpc}}^{-1}$. The light, normal, and dark blue lines correspond to the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ limits, respectively. The median and 16% and 84% percentiles are also shown on the top of each histogram. The black dashed lines correspond to 16% and 84% percentiles. The red lines show the input parameters.

Figure 12

Medians of cosmological parameters estimated from MCMC chains with errors, shown as the fractional ratios with respect to the fiducial values. The lower (upper) limits of error bars correspond to 16% (84%) percentile.

Figure 13

Medians of nuisance parameters, ${\sigma }_{\mathrm{d}}$ for RegPT+ and ${\alpha }_1$, ${\alpha }_2$, and $\mathrm{\Sigma }$ for IR-resummed EFT, estimated from MCMC chains. The lower (upper) limits of error bars correspond to 16% (84%) percentile.

Figure 14

Figure of bias for different models estimated from MCMC chains. The black dashed lines show the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ critical values 1.88, 2.83, and 3.76, respectively. The open (filled) symbols represent that the figure of bias exceeds (falls below) the $1\text{−}\sigma$ critical value.

Figure 15

Figure of merit for different models estimated from MCMC chains. The result with RESPRESSO is shown as a dashed line since it is different from other methods in the sense that it relies on the simulation-aided approach, not completely the analytical prescription. The cyan line shows figure of merit from Fisher matrix with RESPRESSO. The open (filled) symbols represent that the corresponding figure of bias exceeds (falls below) the $1\text{−}\sigma$ critical value (see Fig. 14).

Figure 16

Correlation coefficients for RegPT at the two-loop level. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.

Figure 17

Correlation coefficients for SPT at the two-loop level. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.

Figure 18

Correlation coefficients for RegPT+ at the two-loop level. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.

Figure 19

Correlation coefficients for IR-resummed EFT at the two-loop level. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.

Figure 20

Correlation coefficients for RESPRESSO. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.

Figure 21

Medians of cosmological parameters estimated from MCMC chains with errors as the fractional ratios with respect to the fiducial values. All of estimates are based on one-loop level calculations. The lower (upper) limits of error bars correspond to 16% (84%) percentile.

Figure 22

Medians of nuisance parameters, ${\sigma }_{\mathrm{d}}$ for RegPT+, ${\alpha }_1$ and $\mathrm{\Sigma }$ for IR-resummed EFT, and $c_{s(1)}/k_{\mathrm{NL}}$ for EFT, estimated from MCMC chains. The lower (upper) limits of error bars correspond to 16% (84%) percentile.

Figure 23

Figure of bias for different models at the one-loop level estimated from MCMC chains. The black dashed line shows the $1\text{−}\sigma$, $2\text{−}\sigma$, and $3\text{−}\sigma$ critical values 1.88, 2.83, and 3.76. The open (filled) symbols represent that the figure of bias exceeds (falls below) the $1\text{−}\sigma$ critical value.

Figure 24

Figure of merit for different models at the one-loop level estimated from MCMC chains. The cyan line shows figure of merit from Fisher matrix with RESPRESSO. The open (filled) symbols represent that the corresponding figure of bias exceeds (falls below) the $1\text{−}\sigma$ critical value (see Fig. 23).

Figure 25

Correlation coefficients for IR-resummed EFT model at the one-loop level. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.

Figure 26

Correlation coefficients for EFT at the one-loop level. The upper left (lower right) triangle shows results with $k_{\max }=0.18(0.24)h{\mathrm{Mpc}}^{-1}$. The red (blue) parts correspond to positive (negative) correlations.