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We develop a maximum likelihood based method of reconstructing the band powers of the density and velocity power spectra at each wavenumber bin from the measured clustering features of galaxies in redshift space, including marginalization over uncertainties inherent in the small‐scale, non‐linear redshift distortion, the Fingers‐of‐God (FoG) effect. The reconstruction can be done assuming that the density and velocity power spectra depend on the redshift‐space power spectrum having different angular modulations of μ with μ 2 n   ( n = 0, 1, 2) and that the model FoG effect is given as a multiplicative function in the redshift‐space spectrum.  By using  N ‐body simulations and the halo catalogues, we test our method by comparing the reconstructed power spectra with the spectra directly measured from the simulations. For the spectrum of μ 0  or equivalently the density power spectrum  P  δδ ( k ), our method recovers the amplitudes to an accuracy of a few per cent up to  k ≃ 0.3  h  Mpc −1  for both dark matter and haloes. For the power spectrum of μ 2 , which is equivalent to the density–velocity power spectrum  P  δθ ( k ) in the linear regime, our method can recover, within the statistical errors, the input power spectrum for dark matter up to  k ≃ 0.2  h  Mpc −1  and at both redshifts  z = 0 and 1, if the adequate FoG model being marginalized over is employed. However, for the halo spectrum that is least affected by the FoG effect, the reconstructed spectrum shows greater amplitudes than the spectrum  P  δθ ( k ) inferred from the simulations over a range of wavenumbers 0.05 ≤ k ≤ 0.3  h  Mpc −1 . We argue that the disagreement may be ascribed to a non‐linearity effect that arises from the cross‐bispectra of density and velocity perturbations. Using the perturbation theory and assuming Einstein gravity as in simulations, we derive the non‐linear correction term to the redshift‐space spectrum, and find that the leading‐order correction term is proportional to μ 2  and increases the μ 2 ‐power spectrum amplitudes more significantly at larger  k , at lower redshifts and for more massive haloes. We find that adding the non‐linearity correction term to the simulation  P  δθ ( k ) can fairly well reproduce the reconstructed  P  δθ ( k ) for haloes up to  k ≃ 0.2  h  Mpc −1 .

Likelihood reconstruction method of real‐space density and velocity power spectra from a redshift galaxy survey