Mapping the spectral index of Cas A: evidence for flattening from radio to infrared

Synchrotron radiation from supernova remnants are caused by electrons accelerated through diffusive shock acceleration (DSA). The standard DSA theory predicts an electron spectral index of $p=2$, corresponding to a radio spectral index of $\alpha=-0.5$. An extension of DSA predicts that the accelerated particles changes the shock structure, resulting in a spectrum that is steeper than $p>2$ ($\alpha<-0.5$) at low energies and flattens with energy. For Cas A, a synchrotron spectral flattening was previously reported for a small part of the remnant in the mid-infrared regime. Here, we present new measurements for spectral flattening using archival radio (4.72~GHz) and mid-infrared (3.6~$\mu$m) data, and produce a complete spectral index map to investigate the spatial variations within the remnant. We compare this to measurements of the radio spectral index from L-(1.285~GHz) and C-(4.64~GHz) band maps. Our result shows overall spectral flattening across the remnant ($\alpha _\mathrm{R-IR} \sim -0.5$ to $-0.7$), to be compared to the radio spectral index of $\alpha_{\rm R}=-0.77$. The flattest values coincide with the locations of most recent particle acceleration. In addition to overall flattening, we detect a relatively steeper region in the southeast of the remnant ($\alpha _\mathrm{R-IR} \sim -0.67$). We explore whether these locally steeper spectra could be the result of synchrotron cooling, which provides constraints on the local magnetic-field strengths/age of the plasma, suggesting $B\lesssim 2$~mG for an age of 100~yr, and even $B\lesssim 1$~mG using the age of Cas A, in agreement with other estimates.