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A Consistency Relation for the Observed Galaxy Bispectrum and the Local non-Gaussianity from Relativistic Corrections

A. Kehagias, A. Moradinezhad Dizgah, J. Noreña, H. Perrier, A. Riotto

TL;DR

The paper derives a non-perturbative consistency relation for the observed galaxy bispectrum in the squeezed limit by treating long-wavelength modes as adiabatic gauge modes in a relativistic framework. The authors show that the long mode can be absorbed into a coordinate transformation, reducing the problem to frame changes that couple through projection effects like redshift distortions and lensing, and they express the squeezed bispectrum in multipole space using angular power spectra. A key result is an explicit multipole-space relation that holds even when short modes are non-linear, allowing a clean separation of relativistic corrections from local primordial non-Gaussianity; they also quantify the effective GR-induced f_NL, finding f^{GR}_{NL} ≈ -3.0 at z ≈ 1.5 with no magnification bias. These findings provide a robust diagnostic for upcoming large-scale surveys to extract primordial signals without misinterpreting GR effects as local non-Gaussianity.

Abstract

We obtain a consistency relation for the observed three-point correlator of galaxies. It includes relativistic effects and it is valid in the squeezed limit. Furthermore, the consistency relation is non-perturbative and can be used at arbitrarily small scales for the short modes. Our results are also useful to compute the non-linear relativistic corrections which induce a signal in the observations that might be misinterpreted as primordial non-Gaussianity with a local shape. We estimate the effective local non-Gaussianity parameter from the relativistic corrections. The exact value depends on the redshift and the magnification bias. At redshift of $ z = 1$, in the absence of magnification bias, we get $\,\, f^{\rm GR}_{\rm NL} \simeq - 3.7 $.

A Consistency Relation for the Observed Galaxy Bispectrum and the Local non-Gaussianity from Relativistic Corrections

TL;DR

The paper derives a non-perturbative consistency relation for the observed galaxy bispectrum in the squeezed limit by treating long-wavelength modes as adiabatic gauge modes in a relativistic framework. The authors show that the long mode can be absorbed into a coordinate transformation, reducing the problem to frame changes that couple through projection effects like redshift distortions and lensing, and they express the squeezed bispectrum in multipole space using angular power spectra. A key result is an explicit multipole-space relation that holds even when short modes are non-linear, allowing a clean separation of relativistic corrections from local primordial non-Gaussianity; they also quantify the effective GR-induced f_NL, finding f^{GR}_{NL} ≈ -3.0 at z ≈ 1.5 with no magnification bias. These findings provide a robust diagnostic for upcoming large-scale surveys to extract primordial signals without misinterpreting GR effects as local non-Gaussianity.

Abstract

We obtain a consistency relation for the observed three-point correlator of galaxies. It includes relativistic effects and it is valid in the squeezed limit. Furthermore, the consistency relation is non-perturbative and can be used at arbitrarily small scales for the short modes. Our results are also useful to compute the non-linear relativistic corrections which induce a signal in the observations that might be misinterpreted as primordial non-Gaussianity with a local shape. We estimate the effective local non-Gaussianity parameter from the relativistic corrections. The exact value depends on the redshift and the magnification bias. At redshift of , in the absence of magnification bias, we get .

Paper Structure

This paper contains 11 sections, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Adiabatic modes and residual gauge symmetry
  3. Observed galaxy overdensity in the presence of long mode
  4. Galaxy number density
  5. Long-short couplings
  6. Validity checks
  7. Spherical Harmonic Decomposition
  8. The distant observer approximation
  9. Second-order limit
  10. Effective local non-Gaussianity parameter from non-linear general relativity corrections
  11. Conclusions

Figures (2)

  • Figure 1: Reduced observed bispectrum in the squeezed limit at equal-time (z=0.55) on linear scales. Note that the consistency relation is also valid when the short modes $\ell_2, \ell_3$ are in the non-linear regime. The vertical line indicates the linear scale up to which the numerical code is valid.
  • Figure 2: Effective $f^{\rm GR}_{\rm NL}$ induced by the relativistic effects computed in this paper. The three different lines correspond to different magnification bias parameters. We have assumed $e = 3/a$ for all cases. This estimate is valid for $z\gtrsim 1$. At lower redshifts, integration over the redshift bins can change this result.