Exploring the Limits of Spectral Line Stacking in Spectral Line Data and Application Toward the Detection of Bulk $^{13}$C Enrichment of Aromatics in TMC-1
The formation history of polycyclic aromatic hydrocarbons (PAHs) in the interstellar medium remains a topic of active debate, with proposed mechanisms ranging from high-temperature stellar ejecta processes to low-temperature chemistry within molecular clouds. Recently, the identification of small PAHs in the cold, dark cloud TMC-1 has provided some circumstantial evidence of the latter. If these are formed in situ, their isotopic ratios, particularly $^{12}$C/$^{13}$C, should reflect the local, bulk material of the molecular cloud; in contrast, PAHs formed in the circumstellar envelopes of evolved stars may show enhanced $^{13}$C abundances. The expected radio signals of these $^{13}$C substituted species will be faint, and thus we conducted a detailed proof-of-concept analysis examining whether spectral line stacking and matched filtering techniques can robustly retrieve signal from multiple singly substituted $^{13}$C isotopologues in aggregate. We find that while retrieved signal decreases in the limit of spectral line confusion, false-positive detections are exceedingly improbable at the adopted 5$σ$ matched filter response detection threshold. We then demonstrate the technique on actual observational data of isotopologues of HC$_9$N toward TMC-1. Finally, we show using synthetic data that if laboratory rotational spectra of all singly substituted $^{13}$C isotopologues of cyanonaphthalene and cyanopyrene isomers existed, current observations of TMC-1 would be sensitive enough to discriminate between a local, bulk $^{12}$C/$^{13}$C ratio and one with an enhanced $^{13}$C abundance suggestive of inheritance.