Curvature tension: Evidence for a closed universe

Letter

Curvature tension: Evidence for a closed universe

Will Handley

Phys. Rev. D 103, L041301 – Published 5 February 2021

Abstract

The curvature parameter tension between Planck 2018, cosmic microwave background (CMB) lensing, and baryon acoustic oscillation (BAO) data is measured using the suspiciousness statistic to be $2.5 - 3 σ$ . Conclusions regarding the spatial curvature of the Universe which stem from the combination of these data should therefore be viewed with suspicion. Without CMB lensing or BAO, Planck 2018 has a moderate preference for closed universes, with Bayesian betting odds of over $50 ∶ 1$ against a flat universe and over $2000 ∶ 1$ against an open universe.

Received 27 August 2019
Revised 4 November 2019
Accepted 19 January 2021

DOI:https://doi.org/10.1103/PhysRevD.103.L041301

Physics Subject Headings (PhySH)

Astrophysical studies of gravity Cosmic microwave background Cosmological parameters Cosmology Novae & supernovae

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Will Handley ^1,2,3,*

¹Astrophysics Group, Cavendish Laboratory, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
²Kavli Institute for Cosmology, Madingley Road, Cambridge CB3 0HA, United Kingdom
³Gonville & Caius College, Trinity Street, Cambridge CB2 1TA, United Kingdom†

^*wh260@mrao.cam.ac.uk
^†https://www.kicc.cam.ac.uk/directory/wh260

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Vol. 103, Iss. 4 — 15 February 2021

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Images

Figure 1
Three tensions from Fig. 2 plotted in the $(Ω_{K}, H_{0})$ plane using anesthetic [18]. In the first panel, while the lensing posterior alone is barely distinguishable from the prior, when combined with Planck, lensing draws the combined posterior significantly toward flatness, at a tension of $2.5 σ$ . The second panel shows baryon acoustic oscillation (BAO)’s preference for a flat universe. The BAO posterior is disconnected from the Planck posterior, at a tension of $3 σ$ . Finally, in the third panel the Planck- $S H_{0} ES$ inconsistency in the curved case is shown to be $4.5 σ$ .
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Figure 2
Parameter tensions computed using anesthetic [18]. The top line shows the severe tension between $S H_{0} ES$ and Planck detailed in [19, 20] updated for Planck 2018 data. For $Λ CDM$ Planck, lensing and BAO are all consistent. For $K Λ CDM$ , Planck and BAO are strongly inconsistent, while Planck and lensing are moderately inconsistent. The ninth and final rows indicate the triple tension from Eq. (9) and show strong mutual inconsistency between Planck, lensing and $S H_{0} ES$ and moderate mutual inconsistency between Planck, lensing and BAO. Curvature in general enhances tension but relaxes it for $Planck + lensing$ vs $S H_{0} ES$ . The large error bars for lensing vs $S H_{0} ES$ occur due to the fact their shared constrained parameters have $\tilde{d} ≪ 1$ .
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Figure 3
Model comparison between curved $K Λ CDM$ and flat $Λ CDM$ cosmologies. Positive $Δ \log Z$ indicates favoring of curved universes. The Occam regularization penalty associated with the additional $Ω_{K}$ parameter is shown in orange, estimated via the difference in KL divergence between the two models.
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Figure 4
One- and two-dimensional marginalized priors and posteriors for the seven cosmological parameters of the $K Λ CDM$ cosmology, plotted with anesthetic [18]. Plots on the diagonal are one-dimensional marginalizations, and off-diagonal plots are contour plots showing marginal isoprobability contours containing 68% and 95% of the marginalized posterior mass. The prior in blue and the Planck posterior in red are included in all plots, below-diagonal elements show lensing posteriors, and above-diagonal elements show posteriors for BAO. The leftmost column and topmost row detail curvature, where the tension between the posteriors can be seen clearly. The remaining parameters show no discernible tensions. Lensing constrains approximately two parameters, a linear combination of $Ω_{c} h^{2}$ and $n_{s}$ , and a linear combination of $\ln (10^{10} A_{s})$ and $τ$ , which is reflected in the Bayesian model dimensionality of ${\tilde{d}}_{lensing} = 1.8 \pm 1$ . BAO has a model dimensionality of ${\tilde{d}}_{BAO} = 2.6 \pm 0.1$ , consistent with its constraints on $\ln (10^{10} A_{s})$ and $Ω_{K}$ and its partial constraint on $Ω_{c} h^{2}$ .
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