We present the analysis of Herschel Spectral and Photometric Imaging Receiver far-infrared (FIR) observations of the z = 2.515 lensed galaxy SMM J163554.2+661225. Combining new 250, 350, and 500 μm observations with existing data, we make an improved fit to the FIR spectral energy distribution of this galaxy. We find a total infrared (IR) luminosity of L(8-1000 μm) = 6.9 ± 0.6 × 10 11 L , a factor of three more precise over previous L IR estimates for this galaxy, and one of the most accurate

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... asurements for any galaxy at these redshifts. This FIR luminosity implies an unlensed star formation rate (SFR) for this galaxy of 119 ± 10 M yr −1 , which is a factor of 1.9 ± 0.35 lower than the SFR derived from the nebular Paα emission line (a 2.5σ discrepancy). Both SFR indicators assume an identical Salpeter initial mass function (IMF) with slope Γ = 2.35 over a mass range of 0.1-100 M ; thus this discrepancy suggests that more ionizing photons may be necessary to account for the higher Paα-derived SFR. We examine a number of scenarios and find that the observations can be explained with a varying star formation history (SFH) due to an increasing SFR, paired with a slight flattening of the IMF. If the SFR is constant in time, then larger changes need to be made to the IMF by either increasing the upper mass cutoff to ∼200 M , or a flattening of the IMF slope to 1.9 ± 0.15, or a combination of the two. These scenarios result in up to double the number of stars with masses above 20 M , which produce the requisite increase in ionizing photons over a Salpeter IMF with a constant SFH.