Can cosmic parallax distinguish between anisotropic cosmologies?

In an anisotropic universe, observers not positioned at a point of special symmetry should observe cosmic parallax—the relative angular motion of test galaxies over cosmic time. It was recently argued that the nonobservance of this effect in upcoming precision astrometry missions such as GAIA may be used to place strong bounds on the position of off-center observers in a void-model universe described by the Lemaitre-Tolman-Bondi metric. We consider the analogous effect in anisotropic cosmological models described by an axisymmetric homogeneous Bianchi type I metric and discuss whether any observation of cosmic parallax would distinguish between different anisotropic evolutions.

Figure 1

In the LTB model the observer is located at a distance $r_0$ from the center of the void along the axis of symmetry. Each point along the geodesic is described by the spatial coordinates $(\theta (\lambda ),r(\lambda ))$ or equivalently by $x^i(\lambda )$.

Figure 2

Initial angle ${\gamma }_0$ defined by the velocity vectors at observing time for two sources.

Figure 3

Location of sources as seen by an observer at the measuring event. The vertical axis points along the axis of symmetry. We are considering sources at equal redshifts of $z=1$ and in a plane defined by a fixed value of the polar angle $\varphi$.

Figure 4

A sequence of cosmic parallax signals for different values of $\Delta t$ and fixed $ϵ={10}^{-6}$. Top row, left to right: $\Delta t=20\mathrm{yrs}$, $\Delta t=80\mathrm{yrs}$. Bottom row, left to right: $\Delta t=120\mathrm{yrs}$, $\Delta t=200\mathrm{yrs}$. The signal-to-noise ratio becomes smaller as $\Delta t$ becomes smaller.

Figure 5

Cosmic parallax as a function of $\Delta t$. Here $ϵ={10}^{-6}$ and sources are at $z=1$, separation angle ($\Delta \theta$) of 90 degrees.

Figure 6

A sequence of cosmic parallax signals for different values of $ϵ$ at $\Delta t=10\mathrm{yrs}$. Top row, left to right: $ϵ=6\times {10}^{-6}$, $ϵ=1.6\times {10}^{-5}$. Bottom row, left to right: $ϵ=3.6\times {10}^{-5}$, $ϵ=7.6\times {10}^{-5}$.

Figure 7

Cosmic parallax as a function of $ϵ$ for a fixed value of $\Delta t=10\mathrm{yrs}$.

Figure 8

Estimate of the cosmic parallax for $\Delta t=10\mathrm{years}$, $ϵ={10}^{-6}$, and for sources at $z=1$ with a separation angle of 90 degrees, plotted as a function of the angle between the trailing source and the symmetry axis (i.e., the angle between the leading source and the symmetry axis is advanced by 90 degrees). The two sets of plotted data correspond to interpolated values found using 100 trials for $\Delta t$ spread from 5–500 years, and interpolated values found using 20 trials in $ϵ$ in the ${10}^{-6}–{10}^{-4}$ range, respectively.