We calculate the non-linear matter power spectrum using the 3rd-order perturbation theory without ignoring the pressure gradient term. We consider a semi-realistic system consisting of two matter components with and without pressure, and both are expanded into the 3rd order in perturbations in a self-consistent manner, for the first time. While the pressured component may be identified with baryons or neutrinos, in this paper we mainly explore the physics of the non-linear pressure effect using a toy model in which the Jeans length does not depend on time, i.e., the sound speed decreases as 1/a^{1/2}, where a is the scale factor. The linear analysis shows that the power spectrum below the so-called filtering scale is suppressed relative to the power spectrum of the cold dark matter. Our non-linear calculation shows that the actual filtering scale for a given sound speed is smaller than the linear filtering scale by a factor depending on the redshift and the Jeans length. A ~40% change is common, and our results suggest that, when applied to baryons, the temperature of the Inter-galactic Medium inferred from the filtering scale observed in the flux power spectrum of Lyman-alpha forests would be underestimated by a factor of two, if one used the linear filtering scale to interpret the data. The filtering mass, which is proportional to the filtering scale cubed, can also be significantly smaller than the linear theory prediction especially at low redshift, where the actual filtering mass can be smaller than the linear prediction by a factor of three. Finally, when applied to neutrinos, we find that neutrino perturbations deviate significantly from linear perturbations even below the free-streaming scales, and thus neutrinos cannot be treated as linear perturbations.
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