Detectability of nontrivial topologies

We study how the uncertainty in the cosmological parameters impacts on the detection of topological signals, focussing on three cubic torus universes and using three tests: the information content, the $S/N$ statistic, and the Bayesian evidence. We find, within the concordance cosmological model, that 3D torus universes with a size of $\sim 29{\mathrm{Gpc}}^3$ or larger cannot be detected. For the toroidal models that can be detected, the detection significance is primarily influenced by ${\Omega }_{\Lambda }$, which enters both in the noise amplitude due to the Integrated Sachs-Wolfe effect and in the size of the causal horizon which limits the accessible fundamental domain. On large angular scales $\ell <40$, only ${\Omega }_{\Lambda }$ significantly alters the detection for all three estimators considered here.

Figure 1

The information content as a function of the size of the fundamental domain and of ${\ell }_{\max }$: The left panel shows $H$ and right panel the increase per $\ell$, $\Delta H$, for (from the top downwards) T[2, 2, 2] [solid (red) line], T[4, 4, 4] [dashed (blue) line], and T[6, 6, 6] [dotted (green) line].

Figure 2

$S/N$ for (from the top downwards) T[2, 2, 2] [solid (red) line], T[4, 4, 4] [dashed (blue) line], and T[6, 6, 6] [dotted (green) line].

Figure 3

Evidence for (from the top downwards) T[2, 2, 2] [solid (red) line], T[4, 4, 4] [dashed (blue) line], and T[6, 6, 6] [dotted (green) line] relative to an infinite universe, for the full-sky reconstructed $a_{\ell m}$ of 17.

Figure 4

Diameter of the particle horizon, $D=2a_0{\eta }_0{H}_0/c$, for a flat universe and as a function of ${\Omega }_{\Lambda }$. This corresponds roughly to the maximal fundamental domain size $X$ that we can hope to detect.

Figure 5

Influence of the ISW effect on the information content: The dashed (blue) line shows the $H$ for the T[4, 4, 4] model while the solid (red) line shows the information content for the same model but without ISW effect. The difference is very important.

Figure 6

Influence of reionization on the information content: The dashed (blue) line shows the $H$ for the T[4, 4, 4] model while the solid (red) line shows the information content for the same model but without reionization. The difference is negligible.

Figure 7

For T[4, 4, 4], the effect of varying $n_S$: The top panel shows the information content, for the fiducial model [solid (black) line, $n_S=0.950$] as well as $n_S=0.916$ [dashed (red) line], and $n_S=0.982$ [dotted (blue) line]. The middle panel ($S/N$) and bottom panel (evidence) show how the detection changes when testing simulated maps against the fiducial model. The (black) solid line uses a map derived from the fiducial model, while the (red) dashed and (blue) dotted lines are for the smaller and larger $n_S$, respectively.

Figure 8

The effect of varying $h$ for T[4, 4, 4]: The top panel shows the information content, for the fiducial model [solid (black) line, $h=0.724$] as well as $h=0.678$ [dashed (red) line], and $h=0.770$ [dotted (blue) line]. The middle panel ($S/N$) and bottom panel (evidence) show how the detection changes when testing simulated maps against the fiducial model. The (black) solid line uses a map derived from the fiducial model, while the (red) dashed and (blue) dotted lines are for the smaller and larger $h$, respectively.

Figure 9

The effect of varying ${\Omega }_{\Lambda }$ for T[4, 4, 4]: The top panel shows the information content, for the fiducial model [solid (black) line, ${\Omega }_{\Lambda }=0.75$] as well as ${\Omega }_{\Lambda }=0.80$ [dashed (red) line], ${\Omega }_{\Lambda }=0.90$ [dotted-dashed (magenta) line], ${\Omega }_{\Lambda }=0.7$ [dotted (blue) line], and ${\Omega }_{\Lambda }=0.50$ [long-dashed (cyan) line]. The middle panel ($S/N$) and bottom panel (evidence) show how the detection changes when testing simulated maps against the fiducial model. The (black) solid line uses a map derived from the fiducial model. The line styles of the other models are the same as those of the information content (top panel).