Cosmic rulers

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 cosmic microwave background, 21-cm emission, or galaxies) as well as to weak lensing shear and magnification, all of which can be seen as methods relying on “standard rulers.”

Figure 1

Illustration of the apparent and actual standard ruler. Photons arrive out of the observed directions $\widehat{\mathbf{n}}$, ${\widehat{\mathbf{n}}}^{\prime }$ and with observed redshifts $\tilde{z}$, ${\tilde{z}}^{\prime }$. The apparent positions are indicated by ${\tilde{x}}^{\mu }$, ${\tilde{x}}^{\prime \mu }$, while the true positions are at $x^{\mu }$, $x^{\prime \mu }$, perturbed by the displacements $\Delta x^{\mu }$, $\Delta x^{\prime \mu }$ (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 shows the projection onto the line-of-sight and $x_{\perp }^1$ ($x_{\perp }^2$) axis, respectively. In the cases of $\mathcal{B}$ and $\gamma$, we only show one of the two components. See also Fig. 3 in Ref. 12.

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 nonevolving 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 the sc gauge. Note that all quantities shown here, except for ${\delta }_m^{\mathrm{sc}}$, are gauge invariant and (in principle) observable.