tokens: Figure 4. The (0.3–1 versus 1–4 keV) X-ray soft time lag versus frequency (left) and (3–10 keV) energy spectra (right) fitted with the kynxilrev and kynxillver models, respectively, using the same values of the parameters obtained for spin zero (top) and rapidly spinning (bottom) from Table 2 . Both the a = 0 and the a = 0.99 best-fitting probability values are rather low. The poor quality of the model fits is mainly due to the best-fitting residuals that appear at energies below ∼3.3 keV in the LF energy spectra (possibly due to the presence of a blueshifted, S xvi absorption line at these energies, e.g. Jiang et al. 2018 ). Indeed, if we neglect the points below 3.3 keV in the energy spectra the best-fitting improves significantly, with χ 2 = 430.8 and 439.8 for 403 degrees of freedom when a = 0.99 and 0, respectively. We furthermore investigated if we can get a better fit by allowing the inclination to be a free parameter. The best-fitting inclination values are 47.5° ± 1.8 and for the a = 0.99 and 0 cases, respectively. These values are similar to the one we had originally assumed (45°). Since the fit does not improve significantly (Δχ 2 ∼ 2 − 3 for 414 degrees of freedom) and the new model parameter values are consistent (within the errors) with the previous ones, we decided to keep the original best-fitting results. Looking at the residuals of the best-fits to the time lags (left column panels in Fig. 4 ), we can see that the a = 0 model cannot fit the observed time lags as good as the a = 0.99 model. The best-fitting a = 0 model appears to be consistently overestimating the time lags at frequencies higher than ∼3 × 10 −4 Hz. As a result, Δχ 2 = 10 between the best fit of the a = 0.99 and the a = 0 model to the time lags only (Δχ 2 = 18 when we consider the energy spectra as well). It is difficult to quantify the improvement in the quality of the best fit in the high-spin case, one reason being that the number of degrees of freedom is the same in both model fits. One way to do so is to compute the ‘ratio of likelihoods’, which is equal to (Mushotzky 1982 ). In our case, this is identical to the ‘evidence ratio’, which is based on the Akaike (Akaike 1973 ) information criterion (see equation 8 in Emmanoulopoulos et al. 2016 ). This ratio is a measure of the relative likelihood of one versus the other model, and is equal to ∼150 in our case (∼8100 when we consider the energy spectra as well). This result does not mean that we can reject the hypothesis that the spin zero model is correct, but it does mean that the high spin model is much ‘more likely’ (i.e many hundreds to thousand times) to be the ‘real/correct’ model. We therefore conclude that our results strongly support the hypothesis of a rapidly rotating BH in IRAS 13224-3809. The best-fitting iron abundance and disc density values are consistent with the results from Jiang et al. ( 2018 ), who performed a detailed spectral analysis of the 2016 observations using a similar model to
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