Planck 2013 results. XXVI. Background geometry and topology of the Universe

TL;DR

This paper uses Planck 2013 CMB data to test whether the Universe has non-trivial topology or anisotropic geometry. It implements both topology-based (circles in the sky and pixel/harmonic likelihood) and Bianchi VII$_h$ analyses, across flat, spherical, and hyperbolic geometries, accounting for orientation and masking. The results show no evidence for a topology whose fundamental domain intersects the last scattering surface and no physical Bianchi VII$_h$ cosmology is favored; the limits place strong lower bounds on fundamental-domain sizes relative to the last-scattering distance, with a decoupled Bianchi template offering only phenomenological hints. Planck polarization is anticipated to further constrain these global properties and potentially extend sensitivity to large-scale topology.

Abstract

Planck CMB temperature maps allow detection of large-scale departures from homogeneity and isotropy. We search for topology with a fundamental domain nearly intersecting the last scattering surface (comoving distance $χ_r$). For most topologies studied the likelihood maximized over orientation shows some preference for multi-connected models just larger than $χ_r$. This effect is also present in simulated realizations of isotropic maps and we interpret it as the alignment of mild anisotropic correlations with chance features in a single realization; such a feature can also exist, in milder form, when the likelihood is marginalized over orientations. Thus marginalized, the limits on the radius $R_i$ of the largest sphere inscribed in a topological domain (at log-likelihood-ratio -5) are: in a flat Universe, $R_i>0.9χ_r$ for the cubic torus (cf. $R_i>0.9χ_r$ at 99% CL for a matched-circles search); $R_i>0.7χ_r$ for the chimney; $R_i>0.5χ_r$ for the slab; in a positively curved Universe, $R_i>1.0χ_r$ for the dodecahedron; $R_i>1.0χ_r$ for the truncated cube; $R_i>0.9χ_r$ for the octahedron. Similar limits apply to alternate topologies. We perform a Bayesian search for an anisotropic Bianchi VII$_h$ geometry. In a non-physical setting where the Bianchi parameters are decoupled from cosmology, Planck data favour a Bianchi component with a Bayes factor of at least 1.5 units of log-evidence: a Bianchi pattern is efficient at accounting for some large-scale anomalies in Planck data. However, the cosmological parameters are in strong disagreement with those found from CMB anisotropy data alone. In the physically motivated setting where the Bianchi parameters are fitted simultaneously with standard cosmological parameters, we find no evidence for a Bianchi VII$_h$ cosmology and constrain the vorticity of such models: $(ω/H)_0<8\times10^{-10}$ (95% CL). [Abridged]

Figures (22)

• Figure 1: The top row shows the correlation structure (i.e., a single row of the correlation matrix) of a simply-connected universe with isotropic correlations. For subsequent rows, the left and middle column show positively curved multiply-connected spaces (left: dedocahedral, middle: octahedral) and the right column shows equal sided tori. The upper row of three maps corresponds to the case when the size of the fundamental domain is of the size of the diameter to the last scattering surface and hence the first evidence for large angle excess correlation appears. Subsequent rows correspond to decreasing fundamental domain size with respect to the last scattering diameter, with parameters roughly chosen to maintain the same ratio between the models.
• Figure 2: Random realisations of temperature maps for the models in Fig. \ref{['fig:pixcorr']}. The maps are smoothed with a Gaussian filter with full-width-half-maximum ${\rm FWHM}=640^{\prime}$^$$.
• Figure 3: Simulated deterministic CMB temperature contributions in Bianchi VII$_{h}$ cosmologies for varying $x$ and $\Omega_\mathrm{tot}$ (left-to-right $\Omega_\mathrm{tot} \in \{0.10, 0.50, 0.95\}$; top-to-bottom $x\in\{0.1, 0.3, 0.7, 1.5, 6.0 \}$). In these maps the swirl pattern typical of Bianchi-induced temperature fluctuations is rotated from the South pole to the Galactic centre for illustrational purposes.
• Figure 4: A simulated map of the CMB sky in a universe with a $T[2,2,2]$ toroidal topology. The dark circles show the locations of the same slice through the last scattering surface seen on opposite sides of the sky. They correspond to matched circles with radius $\alpha\simeq 24^\circ$.
• Figure 5: An example of the $S_{\rm max}^{-}$ statistic as a function of circle radius $\alpha$ for a simulated CMB map (shown in Fig. \ref{['fig:map_cmb_t222']}) of a universe with the topology of a cubic 3-torus with dimensions $L = 2H_0^{-1}$ (solid line). The dash-dotted line show the false detection level established such that fewer than 1% out of 300 Monte Carlo simulations of the CMB map, smoothed and masked in the same way as the data, would yield a false event.