Integrated perturbation theory and one-loop power spectra of biased tracers

General and explicit predictions from the integrated perturbation theory (iPT) for power spectra and correlation functions of biased tracers are derived and presented in the one-loop approximation. The iPT is a general framework of the nonlinear perturbation theory of cosmological density fields in the presence of nonlocal bias, redshift-space distortions, and primordial non-Gaussianity. Analytic formulas of auto and cross power spectra of nonlocally biased tracers in both real and redshift spaces are derived and the results are comprehensively summarized. The main difference from previous formulas derived by the present author is to include the effects of generally nonlocal Lagrangian bias and primordial non-Gaussianity, and the derivation method of the new formula is fundamentally different from the previous one. Relations to recent work on improved methods of nonlinear perturbation theory in the literature are clarified and discussed.

Figure 1

The diagrammatic representation of the two-point propagator with a partially resummed vertex up to one-loop contributions.

Figure 2

The diagrammatic representation of the three-point propagator with a partially resummed vertex at the tree-level contribution.

Figure 3

The diagrammatic representation of the power spectrum up to one-loop approximation.

Figure 4

The correlation functions in real space. The prediction of the one-loop iPT is compared with numerical simulations. The results of mass autocorrelation, ${\xi }_{\mathrm{mm}}$, halo autocorrelation, ${\xi }_{\mathrm{hh}}$, and mass-halo cross correlation, ${\xi }_{\mathrm{mh}}$, are compared in the above panel. Dashed lines represent the predictions of linear theory, solid lines represent those of the one-loop iPT, and symbols with error bars represent the results of numerical simulations. In the bottom panel, scale-dependent bias parameters, which are defined by $\sqrt{{\xi }_{\mathrm{hh}}/{\xi }_{\mathrm{mm}}}$ for autocorrelations and ${\xi }_{\mathrm{mh}}/{\xi }_{\mathrm{mm}}$ for cross correlations, are plotted. Predictions of the iPT are given by a solid line for autocorrelations and a dotted line for cross correlations. These two lines are almost overlapped and indistinguishable. The horizontal dashed line corresponds to the prediction of linear theory with a constant bias factor.

Figure 5

Diagrammatic representation of the resummation scheme of CLPT [52]. The original CLPT does not include the effects of nonlocal bias and can be easily extended to include them by applying the formalism of the iPT and the resummation of these types of diagrams.

Figure 6

Ingredients of the displacement correlator. All the diagrams up to one-loop approximation are shown. These diagrams are resummed in CLPT.

Figure 7

Connected diagrams with three wavy lines up to tree-level approximation. These diagrams are not resummed in CLPT.

Figure 8

Diagrammatic rules of iPT: dynamics and biasing.

Figure 9

Diagrammatic rules of iPT: primordial spectra.

Figure 10

Examples of an external vertex which is allowed (left) and not allowed (right) in the iPT.