Planck 2015 constraints on the non-flat $\Lambda$CDM inflation model
  We study Planck 2015 cosmic microwave background (CMB) anisotropy data using
the energy density inhomogeneity power spectrum generated by quantum
fluctuations during an early epoch of inflation in the non-flat $\Lambda$CDM
model. Unlike earlier analyses of non-flat models, which assumed an
inconsistent power-law power spectrum of energy density inhomogeneities, we
find that the Planck 2015 data alone, and also in conjunction with baryon
acoustic oscillation measurements, are reasonably well fit by a closed
$\Lambda$CDM model in which spatial curvature contributes a few percent of the
current cosmological energy density budget. In this model, the measured Hubble
constant and non-relativistic matter density parameter are in good agreement
with values determined using most other data. Depending on parameter values,
the closed $\Lambda$CDM model has reduced power, relative to the tilted,
spatially-flat $\Lambda$CDM case, and can partially alleviate the low multipole
CMB temperature anisotropy deficit and can help partially reconcile the CMB
anisotropy and weak lensing $\sigma_8$ constraints, at the expense of somewhat
worsening the fit to higher multipole CMB temperature anisotropy data. Our
results are interesting but tentative; a more thorough analysis is needed to
properly gauge their significance.
