Implementing spectra response function approaches for fast calculation of power spectra and bispectra

Perturbation theory of large-scale structures of the Universe at next-to-leading order and next-to-next-to-leading order provides us with predictions of cosmological statistics at subpercent level in the mildly nonlinear regime. Its use to infer cosmological parameters from spectroscopic surveys, however, is hampered by the computational cost of making predictions for a large number of parameters. In order to reduce the running time of the codes, we present a fast scheme in the context of the regularized perturbation theory approach and applied it to power spectra at 2-loop level and bispectra at 1-loop level, including the impact of binning. This method utilizes a Taylor expansion of the power spectrum as a functional of the linear power spectrum around fiducial points at which costly direct evaluation of perturbative diagrams is performed and tabulated. The computation of the predicted spectra for arbitrary cosmological parameters then requires only one-dimensional integrals that can be done within a few minutes. It makes this method suitable for Markov chain Monte-Carlo analyses for cosmological parameter inference.

Figure 1

The fiducial and target cosmological parameters. The arrow from each target cosmology indicates the nearest fiducial cosmology and its color corresponds to the distance [Eq. (22)].

Figure 2

The reconstructed density auto-, density-velocity cross-, velocity autopower spectra at the redshift $z=1$. In the upper panels, we show power spectra of 10 target models computed with the fast scheme and the color corresponds to the distance between the target and the nearest fiducial models.

Figure 3

The reconstructed and direct bispectra for equilateral configuration with all combinations of density and velocity at the redshift $z=1$. In the upper panels, we show power spectra of 10 target models computed with the fast scheme and the color corresponds to the distance between the target and the nearest fiducial models.

Figure 4

The reconstructed and direct density bispectra for isosceles configurations $(k^{\prime },k,k)$ with $k^{\prime }/(h{\mathrm{Mpc}}^{-1})=0.015$, 0.035, 0.055, 0.075, 0.095 at the redshift $z=1$. In the upper panels, we show power spectra of 10 target models computed with the fast scheme and the color corresponds to the distance between the target and the nearest fiducial models.

Figure 5

The reconstructed and direct density bispectra for isosceles configurations $(k^{\prime },k^{\prime },k)$ with $k^{\prime }/(h{\mathrm{Mpc}}^{-1})=0.075$, 0.095, 0.115, 0.135, 0.155 at the redshift $z=1$. In the upper panels, we show power spectra of 10 target models computed with the fast scheme and the color corresponds to the distance between the target and the nearest fiducial models.

Figure 6

Comparison with power spectra measured from $N$-body simulation, linear power spectrum, predictions based on SPT and RegPT, and the Halofit fitting formula at the redshift $z=0.901$. In the lower panel, the ratios in terms of Halofit are shown.

Figure 7

Comparison for density bispectra for equilateral configuration: measurement from $N$-body simulations, tree level bispectrum, unbinned bispectra based on SPT and RegPT, binned bispectra with the fast method, and the BiHalofit fitting formula at the redshift $z=0.901$. In the lower panel, the ratios in terms of BiHalofit are shown.

Figure 8

Comparison of density bispectra for isosceles configurations $(k^{\prime },k,k)$ with $k^{\prime }/(h{\mathrm{Mpc}}^{-1})=0.015$, 0.035, 0.055, 0.075, 0.095: tree level bispectrum, unbinned bispectra based on SPT and RegPT, binned RegPT bispectra with the fast method, and the BiHalofit fitting formula at the redshift $z=0.901$. In the lower panels, the ratios in terms of BiHalofit are shown and dashed lines correspond to the flattened triangle $k^{\prime }=k/2$.

Figure 9

Comparison of density bispectra for isosceles configurations $(k^{\prime },k^{\prime },k)$ with $k^{\prime }/(h{\mathrm{Mpc}}^{-1})=0.075$, 0.095, 0.115, 0.135, 0.155: tree level bispectrum, unbinned bispectra based on SPT and RegPT, binned RegPT bispectra with the fast method, and the BiHalofit fitting formula at the redshift $z=0.901$. In the lower panels, the ratios in terms of BiHalofit are shown and dashed lines correspond to the flattened triangle $k^{\prime }=2k$.

Figure 10

Power spectra based on RegPT with UV cutoff scales of $k_{\mathrm{\Lambda }}=k,k/2,k/4,k/6,k/8,k/10$ at the redshift $z=0.901$. The results of $N$-body simulation and Halofit are also shown. The fiducial result ($k_{\mathrm{\Lambda }}=k/2$) is shown in a solid line and other results are shown in dashed lines.

Figure 11

Bispectra for equilateral configurations based on RegPT with UV cutoff scales of $k_{\mathrm{\Lambda }}=k,k/2,k/4,k/6,k/8,k/10$ at the redshift $z=0.901$. The results of $N$-body simulation and BiHalofit are also shown. The fiducial result ($k_{\mathrm{\Lambda }}=k/6$) is shown in a solid line and other results are shown in dashed lines.

Figure 12

Bispectra for isosceles configurations $(k^{\prime },k,k)$ based on RegPT with UV cutoff scales of $k_{\mathrm{\Lambda }}=k,k/2,k/4,k/6,k/8,k/10$ at the redshift $z=0.901$. The results of $N$-body simulation and BiHalofit are also shown. The fiducial result ($k_{\mathrm{\Lambda }}=k/6$) is shown in solid line and other results are shown in dashed lines.

Figure 13

Bispectra for isosceles configurations $(k^{\prime },k^{\prime },k)$ based on RegPT with UV cutoff scales of $k_{\mathrm{\Lambda }}=k,k/2,k/4,k/6,k/8,k/10$ at the redshift $z=0.901$. The results of $N$-body simulation and BiHalofit are also shown. The fiducial result ($k_{\mathrm{\Lambda }}=k/6$) is shown in solid line and other results are shown in dashed lines.

Figure 14

Same as Fig. 2 but for $z=3$, 2, 0.5, 0.

Figure 15

Same as Fig. 3 but for $z=3$, 2, 0.5, 0.

Figure 16

Same as Fig. 4 but for $z=3$, 2, 0.5, 0.

Figure 17

Same as Fig. 5 but for $z=3$, 2, 0.5, 0.

Figure 18

Same as Fig. 6 but for $z=3.13$, 2.12, 0.521, 0.

Figure 19

Same as Fig. 7 but for $z=3.13$, 2.12, 0.521, 0.

Figure 20

Same as Fig. 8 but for $z=3.13$, 2.12, 0.521, 0.

Figure 21

Same as Fig. 9 but for $z=3.13$, 2.12, 0.521, 0.