We investigate the effects of an external magnetic helicity production on the evolution of the cosmic axion field. It is shown that a helicity larger than (f ew × 10 −15 G) 2 Mpc, if produced at temperatures above a few GeV, is in contradiction with the existence of the axion, since it would produce too much of an axion relic abundance.PACS numbers: 14.80. Mz, 98.62.En Recently, the topic of cosmic magnetic helicity generation and phenomenology has been widely discussed (see, e.g., [1,2,3,4,5,6,7,8]). An in depth study of this very peculiar quantity could help in understanding the nature of the cosmic magnetic field itself and its generation. Magnetic helicity is defined aswhereis the electromagnetic field. In a flat universe described by a Robertson-Walker metric, ds 2 = dt 2 −R 2 dx 2 , where R(t) is the expansion parameter normalized so that at the present time t 0 , R(t 0 ) = 1, the electric and magnetic fields are defined as E = −R −1Ȧand B = R −2 ∇ × A, where a dot indicates the derivative with respect to the cosmic time t. Hence, the helicity can be written asIt should be noted that in the literature the definition of H B is usually given without the factor R 2 . In any case, the two definitions coincide at the present time. However we note that, since the early universe is a very good conductor, magnetic fields are frozen into the plasma and then B scales in time as B ∝ R −2 [3]. Therefore, magnetic helicity, as defined in Eq. (1), remains constant (after its generation).The magnetic helicity is related to the topological properties of the magnetic field, it is a CP −odd pseudoscalar quantity and, if different from zero, would reveal a macroscopic P and CP violation in the universe. Let us introduce the magnetic energy and magnetic helicity spectra, respectively, where B(k) and A(k) are the magnetic field and the vector potential in Fourier space and k = |k|.In terms of the spectra, the magnetic energy, E B (t) = (1/2V ) V d 3 x B 2 (x, t), and the magnetic helicity are E B = dk E B , and H B = dk H B . From the above definitions it follows that any magnetic field configuration satisfies the inequality. A straightforward integration in k leads to the so-called "realizability" condition, |H B | ≤ H max B = 2R 3 ξ B E B , which fixes the maximal helicity associated with a given magnetic field configuration. Here, ξ B (t) = 2πR dk k −1 E B (k, t)/E B is the magnetic field correlation length. Therefore, bounds on the magnetic field can be directly translated into bounds on magnetic helicity. Before discussing this point more deeply, it is useful to introduce the following parametrization for the helicity produced at the temperature T * : (2) where g * and g * S count the total numbers of effectively massless degrees of freedom referring to the energy and entropy density of the universe, respectively. (From now on, g * and g * S will be considered equal.) The choice r = 1 gives the maximal helicity associated with a magnetic field correlated on the Hubble scale, ξ B = H −1 (H is the Hubble parameter), and ...