Measurement of the Relativistic Sunyaev-Zeldovich Corrections in RX
  J1347.5-1145
  We present a measurement of the relativistic corrections to the thermal
Sunyaev-Zel'dovich (SZ) effect spectrum, the rSZ effect, toward the massive
galaxy cluster RX J1347.5-1145 by combining sub-mm images from Herschel-SPIRE
with mm-wave Bolocam maps. Our analysis simultaneously models the SZ effect
signal, the population of cosmic infrared background (CIB) galaxies, and
galactic cirrus dust emission in a manner that fully accounts for their spatial
and frequency-dependent correlations. Gravitational lensing of background
galaxies by RX J1347.5-1145 is included in our methodology based on a mass
model derived from HST observations. Utilizing a set of realistic mock
observations, we employ a forward modelling approach that accounts for the
non-Gaussian covariances between observed astrophysical components to determine
the posterior distribution of SZ effect brightness values consistent with the
observed data. We determine a maximum a posteriori (MAP) value of the average
Comptonization parameter of the intra-cluster medium (ICM) within R$_{2500}$ to
be $\langle y \rangle_{2500} = 1.56 \times 10^{-4}$, with corresponding 68~per
cent credible interval $[1.42,1.63] \times 10^{-4}$, and a MAP ICM electron
temperature of $\langle \textrm{T}_{\textrm{sz}} \rangle_{2500} = 22.4$~keV
with 68~per cent credible interval spanning $[10.4,33.0]$~keV. This is in good
agreement with the pressure-weighted temperature obtained from {\it Chandra}
X-ray observations, $\langle \textrm{T}_{\textrm{x,pw}}\rangle_{2500} = 17.4
\pm 2.3$~keV. We aim to apply this methodology to comparable existing data for
a sample of 39 galaxy clusters, with an estimated uncertainty on the ensemble
mean $\langle \textrm{T}_{\textrm{sz}} \rangle_{2500}$ at the $\simeq 1$~keV
level, sufficiently precise to probe ICM physics and to inform X-ray
temperature calibration.
