Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On finding gravitational waves from anisotropies of the Cosmic Microwave Background

Yiran Wang

TL;DR

The paper studies an inverse problem for cosmological gravitational waves by modeling TT perturbations of a perturbed FLRW space and using a light-ray transform to encode ISW-induced CMB signatures. It develops a microlocal inversion framework based on novel back-projections and graph-type Fourier integral operators to separately recover the $\times$ and $+$ tensor polarizations from the observable $Xu$. The work proves unique determination on a local region and provides explicit parametrices and elliptic inversions on key Lagrangians, yielding localized stability for recovering initial data. This advances CMB tomography by enabling tensor perturbation reconstruction from ISW data, complementing polarization-based searches for primordial gravitational waves.

Abstract

The integrated Sachs-Wolfe (ISW) effect describes how photons are gravitationally redshifted, producing anisotropies in the Cosmic Microwave Background. We study the inverse problem and show that primordial gravitational perturbations, in particular their polarizations in the transversally traceless (TT) gauge can be identified from the local observation of the ISW effect.

On finding gravitational waves from anisotropies of the Cosmic Microwave Background

TL;DR

The paper studies an inverse problem for cosmological gravitational waves by modeling TT perturbations of a perturbed FLRW space and using a light-ray transform to encode ISW-induced CMB signatures. It develops a microlocal inversion framework based on novel back-projections and graph-type Fourier integral operators to separately recover the and tensor polarizations from the observable . The work proves unique determination on a local region and provides explicit parametrices and elliptic inversions on key Lagrangians, yielding localized stability for recovering initial data. This advances CMB tomography by enabling tensor perturbation reconstruction from ISW data, complementing polarization-based searches for primordial gravitational waves.

Abstract

The integrated Sachs-Wolfe (ISW) effect describes how photons are gravitationally redshifted, producing anisotropies in the Cosmic Microwave Background. We study the inverse problem and show that primordial gravitational perturbations, in particular their polarizations in the transversally traceless (TT) gauge can be identified from the local observation of the ISW effect.

Paper Structure

This paper contains 9 sections, 3 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The physical problem
  3. The main results
  4. The outline and main idea
  5. The unique determination
  6. The analysis of the back-projection
  7. The microlocal inversion for the $\times$ polarization
  8. The analysis for the $+$ polarization
  9. Proof of the theorems

Key Result

Theorem 1.1

Let $u$ be a TT wave satisfying eq-wave with Cauchy data $f \in H^s_{\operatorname{loc}}({\mathbb R}^3), h \in H^{s-1}_{\operatorname{loc}}({\mathbb R}^3), s>2$ of the form eq-fh. Then $f, h$ on ${\mathcal{R}}_{T-1, T+1}$are uniquely determined by $Xu$ on ${\mathcal{C}}$, namely if $Xu = 0$ on ${\ma

Figures (1)

  • Figure 1: Left figure: Illustration of Theorem \ref{['thm-main0']}. The picture shows the projection of ${\mathcal{M}}$ to $t-x_1$ plane. The visible region corresponds to ${\mathcal{R}}_{T-1, T+1}$. Right figure: Illustration of Theorem \ref{['thm-main']}. The picture shows the projected view of ${\mathcal{M}}$ to $t = 0$ on the $x_2-x_3$ plane. The ring bounded by two dashed circles is ${\mathcal{R}}_{T-1, T+1}$. The shaded disk in the middle is ${\mathcal{B}}_1$ representing the projection of the set on $t = T$ where the ISW is observed. The other two shaded disks are ${\mathcal{R}}^\pm$ where singularities of the initial data can be recovered.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['thm-main']}
  • proof : Proof of Theorem \ref{['thm-main1']}