Comparison of ratios in DSFGs for HCN(1–0) to (5–4) observations from the literature (upper). The coloured points indicate high-z measurements; grey ones z = 0 observations. Our HCN(1–0) measurements are outlined in black; the star indicates the limit from the stacked K-band spectrum. Lower: high-z HCN measurements, compared to the predictions of PDR models from Kazandjian et al. (2015). We vary the PDR gas density (n = 10³ − 10⁵ cm⁻³) and the mechanical heating factor (α = 0–10%). The FUV irradiation is fixed to G = 10⁴ G₀. The current high-z HCN detections can not be reproduced by models with n ≤ 10⁴ cm⁻³ without significant mechanical heating. High-z data: HCN(1–0): Carilli et al. (2005), Gao et al. (2007), Oteo et al. (2017); HCN(2–1): G. Jolink (BSc thesis); HCN(3–2): Danielson et al. (2013), Oteo et al. (2017); HCN(4–3) and (5–4): Béthermin et al. (2018), Cañameras et al. (2021). Low-z data: HCN(1–0): Gao & Solomon (2004b), García-Burillo et al. (2012); HCN(2–1): Krips et al. (2008), HCN(3–2): Bussmann et al. (2008), Li et al. (2020), HCN(4–3): Zhang et al. (2014). The squares correspond to the Spilker et al. (2014) stacking analysis.