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Non-Perturbative Effects of Geometry in Wide-Angle Redshift Distortions

Peter Papai, Istvan Szapudi

TL;DR

This work addresses inaccuracies in linear redshift-space two-point functions for wide-angle surveys by reinstating a non-perturbative geometric term in the Jacobian, previously neglected. It employs the Szapudi (2004) tripolar-harmonic formalism to derive explicit, finite expressions for the redshift-space correlation function that include wide-angle effects. New expansion coefficients (e.g., B^{101}, B^{011}, B^{121}, B^{211}, B^{123}, B^{213}, B^{110}, B^{112}) capture the geometric contributions, substantially altering predictions. Comparison with the Hubble Volume N-body simulation shows excellent agreement when integral-constraint corrections are accounted, validating the enhanced theory for tens-of-Mpc scales and enabling more accurate analyses of wide-angle redshift data. The results provide practical, explicit formulas for use in Karhunen–Loève analyses and cosmological parameter inference, with clear paths for generalization.

Abstract

We use the formalism of Szapudi(2004} to derive full explicit expressions for the linear two-point correlation function, including redshift space distortions and large angle effects. We take into account a non-perturbative geometric term in the Jacobian, which is still linear in terms of the dynamics. This term had been identified previously (Kaiser 1987,Hamilton and Culhane 1996), but has been neglected in all subsequent explicit calculations of the linear redshift space two-point correlation function. Our results represent a significant correction to previous explicit expressions and are in excellent agreement with our measurements in the Hubble Volume Simulation.

Non-Perturbative Effects of Geometry in Wide-Angle Redshift Distortions

TL;DR

This work addresses inaccuracies in linear redshift-space two-point functions for wide-angle surveys by reinstating a non-perturbative geometric term in the Jacobian, previously neglected. It employs the Szapudi (2004) tripolar-harmonic formalism to derive explicit, finite expressions for the redshift-space correlation function that include wide-angle effects. New expansion coefficients (e.g., B^{101}, B^{011}, B^{121}, B^{211}, B^{123}, B^{213}, B^{110}, B^{112}) capture the geometric contributions, substantially altering predictions. Comparison with the Hubble Volume N-body simulation shows excellent agreement when integral-constraint corrections are accounted, validating the enhanced theory for tens-of-Mpc scales and enabling more accurate analyses of wide-angle redshift data. The results provide practical, explicit formulas for use in Karhunen–Loève analyses and cosmological parameter inference, with clear paths for generalization.

Abstract

We use the formalism of Szapudi(2004} to derive full explicit expressions for the linear two-point correlation function, including redshift space distortions and large angle effects. We take into account a non-perturbative geometric term in the Jacobian, which is still linear in terms of the dynamics. This term had been identified previously (Kaiser 1987,Hamilton and Culhane 1996), but has been neglected in all subsequent explicit calculations of the linear redshift space two-point correlation function. Our results represent a significant correction to previous explicit expressions and are in excellent agreement with our measurements in the Hubble Volume Simulation.

Paper Structure

This paper contains 4 sections, 10 equations, 1 figure.

Table of Contents

  1. INTRODUCTION
  2. REDSHIFT DISTORTION OF THE TWO-POINT CORRELATION FUNCTION
  3. DISCUSSION AND SUMMARY
  4. ACKNOWLEDGMENT

Figures (1)

  • Figure 1: (Left) The measurement of the correlation function without redshift distortion of the Hubble volume simulation (symbols) compared with linear theory(dashed and solid lines). The error bars were estimated from $9^3$ subvolumes of th Hubble volume. Shifting the theory by 0.00081 downward, motivated by the integral constraint, provides an excellent fit to the data. (Right) Redshift distorted correlation function of the Hubble volume simulation (symbols) at constant opening angle (0.71 radian) and while the ratio of the distances of the particles in the pair are kept fixed (at 1.57). The error bars were estimated as before. The lines indicate the linear theories with (this paper) and without the geometric terms. The solid line is the corrected theory with a downshift of 0.0016. The integral constraint correction is expected to be larger since the average of the two point function is larger.