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tokens: the particle sampling noise introduced by the LPT. Since most of the particles cluster in high-density regions, voids are impacted by higher uncertainty in the LPT. The field-level approach of the PPT overcomes this problem, showing a lower standard deviation in the void profile. Fig. 9. Comparison of density profiles obtained with PPT and LPT. Upper panel : cluster density profile obtained by evolving the same initial conditions with PPT and LPT. Lower panel : test for a void profile. This demonstrates that PPT and LPT provide equivalent cosmic structures. The shaded regions indicate the uncertainty region of the profiles, corresponding to the standard deviation of 50 realisations. While the PPT and LPT show similar uncertainty regions for the cluster profile, the LPT has a larger standard deviation than the PPT in the void profile. This larger uncertainty is due to the poor sampling of the voids in the LPT model since most of the particles cluster in high-density regions. 6. Inference results In this section, we present the results of applying our algorithm to Ly- α forest data in redshift space. We show that our method infers unbiased density fields and corresponding power-spectra at all scales considered in this work. We also perform a posterior predictive test for quasar spectra, showing that the inferred quantities can explain the data within the noise uncertainty. 6.1. Inferred density fields As discussed above, our method uses a forward modelling approach, fitting a physical dynamical model to Ly- α forest data. This provides the full posterior distribution, from which we draw samples of the initial matter fluctuations and the non-linear spatial matter distribution at z = 2.5. In this section, the dynamical model is the PPT in redshift space, described in Sect. 2.4 . Since the optical depth is conserved under the mapping between physical and redshift space, our inference in redshift space focuses on the optical depth field, while the density field is obtained in real space. Figure 10 shows slices through the true fields, and the ensemble mean and variances of inferred three-dimensional fields, computed from 600 samples. A first visual comparison between ground truth and the inferred ensemble mean final density and optical depth fields shows that the algorithm correctly recovered the large-scale structure from Ly- α forest data. We note that the optical depth field is in redshift space while the final density field is in physical space. The lower right panel of Fig. 10 shows the corresponding standard deviations of the amplitudes, which is estimated from the samples in the Markov chain. The estimated density standard deviation correlates with the inferred density field. The same is true for the optical depth field. This is expected for a non-linear data model, which couples signal and noise. Higher uncertainty regions correspond to over-densities since the absorption saturates at high density, making the signal weaker. Once the light