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Synchrotron radiation from supernova remnants are caused by electronsaccelerated through diffusive shock acceleration (DSA). The standard DSA theorypredicts an electron spectral index of $p=2$, corresponding to a radio spectralindex of $\alpha=-0.5$. An extension of DSA predicts that the acceleratedparticles changes the shock structure, resulting in a spectrum that is steeperthan $p>2$ ($\alpha<-0.5$) at low energies and flattens with energy. For Cas A,a synchrotron spectral flattening was previously reported for a small part ofthe remnant in the mid-infrared regime. Here, we present new measurements forspectral flattening using archival radio (4.72~GHz) and mid-infrared(3.6~$\mu$m) data, and produce a complete spectral index map to investigate thespatial variations within the remnant. We compare this to measurements of theradio spectral index from L-(1.285~GHz) and C-(4.64~GHz) band maps. Our resultshows 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 mostrecent particle acceleration. In addition to overall flattening, we detect arelatively steeper region in the southeast of the remnant ($\alpha_\mathrm{R-IR} \sim -0.67$). We explore whether these locally steeper spectracould be the result of synchrotron cooling, which provides constraints on thelocal magnetic-field strengths/age of the plasma, suggesting $B\lesssim 2$~mGfor an age of 100~yr, and even $B\lesssim 1$~mG using the age of Cas A, inagreement with other estimates.

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