Comparison of absolute gain photometric calibration between Planck/HFI
  and Herschel/SPIRE at 545 and 857 GHz
  We compare the absolute gain photometric calibration of the Planck/HFI and
Herschel/SPIRE instruments on diffuse emission. The absolute calibration of HFI
and SPIRE each relies on planet flux measurements and comparison with
theoretical far-infrared emission models of planetary atmospheres. We measure
the photometric cross calibration between the instruments at two overlapping
bands, 545 GHz / 500 $\mu$m and 857 GHz / 350 $\mu$m. The SPIRE maps used have
been processed in the Herschel Interactive Processing Environment (Version 12)
and the HFI data are from the 2015 Public Data Release 2. For our study we used
15 large fields observed with SPIRE, which cover a total of about 120 deg^2. We
have selected these fields carefully to provide high signal-to-noise ratio,
avoid residual systematics in the SPIRE maps, and span a wide range of surface
brightness. The HFI maps are bandpass-corrected to match the emission observed
by the SPIRE bandpasses. The SPIRE maps are convolved to match the HFI beam and
put on a common pixel grid. We measure the cross-calibration relative gain
between the instruments using two methods in each field, pixel-to-pixel
correlation and angular power spectrum measurements. The SPIRE / HFI relative
gains are 1.047 ($\pm$ 0.0069) and 1.003 ($\pm$ 0.0080) at 545 and 857 GHz,
respectively, indicating very good agreement between the instruments. These
relative gains deviate from unity by much less than the uncertainty of the
absolute extended emission calibration, which is about 6.4% and 9.5% for HFI
and SPIRE, respectively, but the deviations are comparable to the values 1.4%
and 5.5% for HFI and SPIRE if the uncertainty from models of the common
calibrator can be discounted. Of the 5.5% uncertainty for SPIRE, 4% arises from
the uncertainty of the effective beam solid angle, which impacts the adopted
SPIRE point source to extended source unit conversion factor (Abridged)
