Distinguishing between void models and dark energy with cosmic parallax and redshift drift

Two recently proposed techniques, involving the measurement of the cosmic parallax and redshift drift, provide novel ways of directly probing (over a time span of several years) the background metric of the universe and therefore shed light on the dark-energy conundrum. The former makes use of upcoming high-precision astrometry measurements to either observe or put tight constraints on cosmological anisotropy for off-center observers, while the latter employs high-precision spectroscopy to give an independent test of the present acceleration of the universe. In this paper, we show that both methods can break the degeneracy between Lemaître-Tolman-Bondi void models and more traditional dark-energy theories. Using the near-future observational missions Gaia and CODEX we show that this distinction might be made with high confidence levels in the course of a decade.

Figure 1

Overview, notation and conventions of the cosmic parallax effect. Note that for clarity purposes we assumed here that the points $C$, $O_1$, $a_1$, $b_1$ all lie on the same plane. By symmetry, points $O_2$, $a_2$, $b_2$ remain on this plane as well. Comoving coordinates $r$ and $r_{\mathrm{obs}}$ correspond to physical coordinates $X$ and $X_{\mathrm{obs}}$.

Figure 2

$H_{\parallel }$ and $H_{\perp }$ for Model I (solid), Model II (dashed curve) and the cGBH model (red, long-dashed) in units of $100\mathrm{km}/(\mathrm{s}\mathrm{Mpc})$, as a function of the physical distance $X$. Note that both Hubble parameters differ only around the void-transition region.

Figure 3

${\Omega }_{m0}$ for Model I (solid), Model II (dashed curve) and the cGBH model (red, long-dashed) as a function of the physical distance $X$. Note that the definition for ${\Omega }_{m0}$ we use differ from the one in 3, 33, 34 and we do not get their characteristic overdensity bump (or shell) surrounding the void.

Figure 4

${\Delta }_t\gamma$ for two sources at the same shell, at $z=1$, for Model I (full lines), Model II (dashed), the cGBH model (red, long-dashed lines) and the FRW-like estimate (dotted). The lines correspond to a separation of 90° in the sky between the sources. The off-center distance is assumed to be 30 Mpc.

Figure 5

${\Delta }_t\gamma$ for two sources at the same shell but separated by 90° as a function of redshift assuming a 30 Mpc off-center distance. The dark, brown lines correspond to the cosmic parallax in Models I (full lines) and II (dashed); the red long-dashed lines to the cGBH model; the light, blue dotted lines represent $1/40$ of the aberration-induced signal (see text), which does not depend on redshift; the dark dotted lines stand for the parallax induced by our own peculiar velocity (assumed to be $400\mathrm{km}/\mathrm{s}$). Since all effects are dipolar, the curves plotted here are proportional to the amplitude of such dipoles. The actual amount of noise depend on the angle between the center of the void and the directions of acceleration and peculiar velocity of the measuring instrument. Notice that as expected, in Model II the CP is zero inside the void.

Figure 6

The Sandage effect for a source at $z=1$ for an observer 30 Mpc away from the center as a function of the angle $\xi$ for both Models I (full) and II (dashed lines). Note that the fractional fluctuation of the redshift drift in the sky is less than 5%, and one can therefore assume isotropy to good precision when computing this effect on void models.

Figure 7

The annual redshift drift for different models assuming an observer at the center. The upper, blue solid lines represent the $\Lambda \mathrm{CDM}$ model. The green, dashed line corresponds to a self-accelerating DGP model with ${\Omega }_{r_c}=0.13$. The dot-dashed lines stand for the 3 void models considered here: the dark brown (indistinguishable) lines are for Models I and II, while the red line just above correspond to the cGBH model. The bottommost line corresponds to an universe with only matter in a FRW metric (the CDM model). Note that the void models predict a curve which is in between CDM and $\Lambda \mathrm{CDM}$ but closer to the former.

Figure 8

Redshift drift for different dark-energy models for a total mission duration of 5 (top), 10 (middle) and 15 (bottom plot) years and CODEX forecast error bars (the time-span between each measurement is $2/3$ of that—see text). In each plot, the upper 3, solid lines represent wCDM models for $w=-1.25$ (uppermost), $w=-1$ (second) and $w=-0.75$ (third uppermost). The green, dashed line corresponds to a self-accelerating DGP model with ${\Omega }_{r_c}=0.13$. The three bottommost, dot-dashed lines stand for the 3 void models considered here: the dark brown (indistinguishable) lines are for Models I and II, while the red line just above correspond to the cGBH model. Note that a $4\sigma$ separation between voids and $\Lambda \mathrm{CDM}$ can be achieved in a decade.

Figure 9

Target Gaia astrometric accuracy (dark, full lines) and projected quasar distribution (light, dashed) as a function of magnitude in the $G$ and $I$ band, respectively. Also plotted is the quasar distribution obtained using SDSS DR7 (red, dot-dashed), plotted against its own $G$ band magnitude. A simple weighted average gives the typical positional precision of Gaia on quasars: either $102\mu \mathrm{as}$ (projected) or $82\mu \mathrm{as}$ (SDSS-like distribution).

Figure 10

${\Delta }_t\gamma$ for two sources at the same shell for both Models I (full) and II (dashed lines) and for a time span of 10 (top), 20 (middle), and 30 (bottom plot) years, together with Gaia forecast statistical error bars. Although the nominal Gaia duration is only 6 years, a mission extension allow for smaller errors. Here we are not considering the two main systematics identified in the text. The lines correspond to the average cosmic parallax effect over the whole sky which is given by (25). Note that the CP quickly becomes the best probe of present anisotropy and, therefore, of the combination of distance and velocity towards the center of a void.