skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Multiple soft limits of cosmological correlation functions

Abstract

We derive novel identities satisfied by inflationary correlation functions in the limit where two external momenta are taken to be small. We derive these statements in two ways: using background-wave arguments and as Ward identities following from the fixed-time path integral. Interestingly, these identities allow us to constrain some of the O(q{sup 2}) components of the soft limit, in contrast to their single-soft analogues. We provide several nontrivial checks of our identities both in the context of resonant non-Gaussianities and in small sound speed models. Additionally, we extend the relation at lowest order in external momenta to arbitrarily many soft legs, and comment on the many-soft extension at higher orders in the soft momentum. Finally, we consider how higher soft limits lead to identities satisfied by correlation functions in large-scale structure.

Authors:
 [1];  [2];  [3]
  1. Enrico Fermi Institute and Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637 (United States)
  2. Center for Particle Cosmology, Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA 19104 (United States)
  3. SISSA, via Bonomea 265, 34136, Trieste (Italy)
Publication Date:
OSTI Identifier:
22382033
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Cosmology and Astroparticle Physics; Journal Volume: 2015; Journal Issue: 01; Other Information: Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 79 ASTROPHYSICS, COSMOLOGY AND ASTRONOMY; CORRELATION FUNCTIONS; PATH INTEGRALS; SOUND WAVES; VELOCITY; WARD IDENTITY

Citation Formats

Joyce, Austin, Khoury, Justin, and Simonović, Marko, E-mail: ajoy@uchicago.edu, E-mail: jkhoury@sas.upenn.edu, E-mail: msimonov@sissa.it. Multiple soft limits of cosmological correlation functions. United States: N. p., 2015. Web. doi:10.1088/1475-7516/2015/01/012.
Joyce, Austin, Khoury, Justin, & Simonović, Marko, E-mail: ajoy@uchicago.edu, E-mail: jkhoury@sas.upenn.edu, E-mail: msimonov@sissa.it. Multiple soft limits of cosmological correlation functions. United States. doi:10.1088/1475-7516/2015/01/012.
Joyce, Austin, Khoury, Justin, and Simonović, Marko, E-mail: ajoy@uchicago.edu, E-mail: jkhoury@sas.upenn.edu, E-mail: msimonov@sissa.it. Thu . "Multiple soft limits of cosmological correlation functions". United States. doi:10.1088/1475-7516/2015/01/012.
@article{osti_22382033,
title = {Multiple soft limits of cosmological correlation functions},
author = {Joyce, Austin and Khoury, Justin and Simonović, Marko, E-mail: ajoy@uchicago.edu, E-mail: jkhoury@sas.upenn.edu, E-mail: msimonov@sissa.it},
abstractNote = {We derive novel identities satisfied by inflationary correlation functions in the limit where two external momenta are taken to be small. We derive these statements in two ways: using background-wave arguments and as Ward identities following from the fixed-time path integral. Interestingly, these identities allow us to constrain some of the O(q{sup 2}) components of the soft limit, in contrast to their single-soft analogues. We provide several nontrivial checks of our identities both in the context of resonant non-Gaussianities and in small sound speed models. Additionally, we extend the relation at lowest order in external momenta to arbitrarily many soft legs, and comment on the many-soft extension at higher orders in the soft momentum. Finally, we consider how higher soft limits lead to identities satisfied by correlation functions in large-scale structure.},
doi = {10.1088/1475-7516/2015/01/012},
journal = {Journal of Cosmology and Astroparticle Physics},
number = 01,
volume = 2015,
place = {United States},
year = {Thu Jan 01 00:00:00 EST 2015},
month = {Thu Jan 01 00:00:00 EST 2015}
}
  • Soft limits of inflationary correlation functions are both observationally relevant and theoretically robust. Various theorems can be proven about them that are insensitive to detailed model-building assumptions. In this paper, we re-derive several of these theorems in a universal way. Our method makes manifest why soft limits are such an interesting probe of the spectrum of additional light fields during inflation. We illustrate these abstract results with a detailed case study of the soft limits of quasi-single-field inflation.
  • Correlation functions of two long-wavelength modes with several short-wavelength modes are shown to be related to lower order correlation functions, using the background wave method, and independently, by exploiting symmetries of the wavefunction of the Universe. These soft identities follow from the non-linear extension of the adiabatic modes of Weinberg, and their generalization by Hinterbichler et al. The extension is shown to be unique. A few checks of the identities are presented.
  • Gravitational lensing affects observed cosmological correlation functions because observed images do not coincide with true source locations. We treat this universal effect in a general way here, deriving a single formula that can be used to determine to what extent this effect distorts any correlation function. We then apply the general formula to the correlation functions of galaxies, the 21-cm radiation field, and the cosmic microwave background.
  • Knowledge of N-point correlation functions for all N allows one to invert and obtain the probability distribution of mass fluctuations in a fixed volume. I apply this to the hierarchical sequence of higher order correlations with dimensionless amplitudes suggested by the BBGKY equations. The resulting distribution is significantly non-Gaussian, even for quite small mean square fluctuations. The qualitative and to some degree quantitative results are to a large degree independent of the exact sequence of amplitudes. An ensemble of such models compared with N-body simulations fails in detail to account for the low-density frequency distribution. Subject headings: cosmology-galaxies: clustering-numerical methods
  • Motivated by recent results on lognormal statistics showing that the moment hierarchy of a lognormal variable completely fails at capturing its information content in the large variance regime, in this work we discuss the inadequacy of the hierarchy of correlation functions to describe a correlated lognormal field, which provides a roughly accurate description of the nonlinear cosmological matter density field. We present families of fields having the same hierarchy of correlation functions than the lognormal field at all orders. This explicitly demonstrates the little studied though known fact that the correlation function hierarchy never provides a complete description of amore » lognormal field, and that it fails to capture information in the nonlinear regime, where other simple observables are left totally unconstrained. We discuss why perturbative, Edgeworth-like approaches to statistics in the nonlinear regime, common in cosmology, can never reproduce or predict that effect, and why it is, however, generic for tailed fields, hinting at a breakdown of the perturbation theory based on the field fluctuations. We make a rough but successful quantitative connection to N-body simulations results that showed that the spectrum of the log-density field carries more information than the spectrum of the field entering the nonlinear regime.« less