Polypolar spherical harmonic decomposition of galaxy correlators in redshift space: Toward testing cosmic rotational symmetry

We present the implications for cosmic inflation of the Planck measurements of the cosmic microwave background (CMB) anisotropies in both temperature and polarization based on the full Planck survey. The Planck full mission temperature data and a first release of polarization data on large angular scales measure the spectral index of curvature perturbations to be \(n_\mathrm{s} = 0.968 \pm 0.006\) and tightly constrain its scale dependence to \(d n_s/d \ln k =-0.003 \pm 0.007\) when combined with the Planck lensing likelihood. When the high-\(\ell\) polarization data is included, the results are consistent and uncertainties are reduced. The upper bound on the tensor-to-scalar ratio is \(r_{0.002} < 0.11\) (95% CL), consistent with the B-mode polarization constraint \(r< 0.12\) (95% CL) obtained from a joint BICEP2/Keck Array and Planck analysis. These results imply that \(V(\phi) \propto \phi^2\) and natural inflation are now disfavoured compared to models predicting a smaller tensor-to-scalar ratio, such as \(R^2\) inflation. Three independent methods reconstructing the primordial power spectrum are investigated. The Planck data are consistent with adiabatic primordial perturbations. We investigate inflationary models producing an anisotropic modulation of the primordial curvature power spectrum as well as generalized models of inflation not governed by a scalar field with a canonical kinetic term. The 2015 results are consistent with the 2013 analysis based on the nominal mission data.

This paper describes the Seventh Data Release of the Sloan Digital Sky Survey (SDSS), marking the completion of the original goals of the SDSS and the end of the phase known as SDSS-II. It includes 11663 deg^2 of imaging data, with most of the roughly 2000 deg^2 increment over the previous data release lying in regions of low Galactic latitude. The catalog contains five-band photometry for 357 million distinct objects. The survey also includes repeat photometry over 250 deg^2 along the Celestial Equator in the Southern Galactic Cap. A coaddition of these data goes roughly two magnitudes fainter than the main survey. The spectroscopy is now complete over a contiguous area of 7500 deg^2 in the Northern Galactic Cap, closing the gap that was present in previous data releases. There are over 1.6 million spectra in total, including 930,000 galaxies, 120,000 quasars, and 460,000 stars. The data release includes improved stellar photometry at low Galactic latitude. The astrometry has all been recalibrated with the second version of the USNO CCD Astrograph Catalog (UCAC-2), reducing the rms statistical errors at the bright end to 45 milli-arcseconds per coordinate. A systematic error in bright galaxy photometr is less severe than previously reported for the majority of galaxies. Finally, we describe a series of improvements to the spectroscopic reductions, including better flat-fielding and improved wavelength calibration at the blue end, better processing of objects with extremely strong narrow emission lines, and an improved determination of stellar metallicities. (Abridged)