Cosmic Rulers

TL;DR

The authors develop a general, covariant, gauge-invariant formalism for six observable distortions of ideal standard rulers in a perturbed FRW universe, valid on the full sky and organized into two scalars ($\\mathcal{C}, \\mathcal{M}$), a two-component vector ($\\mathcal{B}_i$), and two tensor components (the spin-2 shear, $\\gamma_{\\pm2}$). They show how these observables decompose into $E$ and $B$ modes, with scalar perturbations unable to produce linear $B$-modes in the vector sector and tensor modes capable of sourcing them, enabling a potential probe of primordial gravitational waves via large-scale structure. Explicit, gauge-invariant expressions are derived for the six quantities in both conformal-Newtonian and synchronous-comoving gauges, including the magnification and shear with their line-of-sight integrals (Sachs–Wolfe, Doppler, ISW) and the metric-shear terms required for gauge invariance. The work lays out a practical path to estimators for these degrees of freedom from data (galaxy, 21cm, and CMB distortions) and discusses implications for probing tensor modes with upcoming surveys, while noting intrinsic-alignments and wide-angle effects as further considerations. Overall, the framework unifies standard-candle/ruler observables and provides new tools to access tensor information through galaxy-scale distortions and backgrounds.

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".

Figures (3)

• Figure 1: Illustration of the apparent and actual standard ruler. Photons arrive out of the observed directions $\hat{\mathbf{n}},\,\hat{\mathbf{n}}'$ and with observed redshifts $\tilde{z},\,\tilde{z}'$. The apparent positions are indicated by $\tilde{x}^\mu,\,\tilde{x}'^\mu$, while the true positions are at $x^\mu,\,x'^\mu$, perturbed by the displacements $\Delta x^\mu,\,\Delta x'^\mu$ (whose magnitude is greatly exaggerated here). $\tilde{r}$ is the apparent size of the ruler, while $r_0$ is the true ruler.
• Figure 2: Illustration of the distortion of standard rulers due to the longitudinal (2-)scalar $\mathcal{C}$, (2-)vector $\mathcal{B}$, and transverse components, magnification $\mathcal{M}$ and shear $\gamma$. The first row shows the projection onto the sky plane, while the second (third) row show the projection onto the line-of-sight and $x_\perp^1$ ($x_\perp^2$) axes, respectively. In case of $\mathcal{B}$ and $\gamma$, we only show one of the two components. See also Fig. 3 in sachs:1961.
• Figure 3: Angular power spectra of the different standard ruler perturbations produced by a standard scale-invariant power spectrum of curvature perturbations: $\mathcal{C}$, $E$-mode of $\mathcal{B}_i$, $E$-mode of the shear, and magnification $\mathcal{M}$. All quantities are calculated for a non-evolving ruler and a sharp source redshift of $\tilde{z} = 2$. For comparison, the thin dotted line shows the angular power spectrum at $z=2$ of the matter density field in synchronous-comoving gauge. Note that all quantities shown here, except for $\delta_m^{\rm sc}$, are gauge-invariant and (in principle) observable.