Observable windows for the QCD axion through $N_{\text{eff}}$

We show that when the QCD axion is directly coupled to quarks with $c_q/f , \partial_\mu a , \bar{q} \gamma^\mu \gamma^5 q$, such as in DFSZ models, the dominant production mechanism in the early universe at temperatures $1 , {\rm GeV}\lesssim T \lesssim 100 ,{\rm GeV}$isobtainedvia$q \bar{q} \leftrightarrow g a$and$q g \leftrightarrow q a$,where$g$ are gluons. Different heavy quarks $q_i$ can produce a thermal axion background that decouples at a temperature $T_i$: (1) top quark at $T_t \lesssim 100 , {\rm GeV}$for$f/c_t \lesssim 3\times 10^8 {\rm GeV}$;(2)bottomquarkat$T_b \lesssim m_b$,for$f/c_b\lesssim 8\times 10^{7} {\rm GeV}$;(3)charmquarkat$T_c \lesssim m_c $for$f/c_c \lesssim 5\times 10^{7} {\rm GeV}$. Each of these cases corresponds to a contribution to the effective number of relativistic degrees of freedom, in the windows given by $0.027 \leq \Delta N_$\text{eff}$\leq 0.031$,$0.037 \leq \Delta N_$\text{eff}$ \leq 0.039$and$0.039 \leq \Delta N_$\text{eff}$$, respectively. These contributions are larger than the one obtained when thermalization happens only above the electroweak phase transition, $\mathrm{\Delta }{N}_{\mathrm{e}\mathrm{f}\mathrm{f}}\lesssim 0.027$, and are within reach of future CMB S4 experiments, thus opening an alternative window to detect the axion and to test the early universe at such temperatures.