Cosmic Rulers

TL;DR

This paper derives gauge-invariant, full-sky expressions for the six observable distortion modes of standard rulers in a perturbed universe, facilitating analysis of weak lensing and cosmological correlations.

Contribution

It provides the first comprehensive, gauge-invariant, full-sky formalism for all six distortion modes of standard rulers, including scalar, vector, and tensor components.

Findings

Derived gauge-invariant expressions for all six distortion modes.

Classified modes into scalars, vectors, and tensors with rotational properties.

Opened new avenues for probing tensor modes via scalar mode analysis.

Abstract

We derive general covariant expressions for the six independent observable modes of distortion of ideal standard rulers in a perturbed Friedmann-Robertson-Walker spacetime. Our expressions are gauge-invariant and valid on the full sky. These six modes are most naturally classified in terms of their rotational properties on the sphere, yielding two scalars, two vector (spin-1), and two tensor (spin-2) components. One scalar corresponds to the magnification, while the spin-2 components correspond to the shear. The vector components allow for a polar/axial decomposition analogous to the E/B-decomposition for the shear. Scalar modes do not contribute to the axial (B-)vector, opening a new avenue to probing tensor modes. Our results apply, but are not limited to, the distortion of correlation functions (of the CMB, 21cm emission, or galaxies) as well as to weak lensing shear and magnification, all of which can be seen as methods relying on "standard rulers".