Third-order perturbative solutions in Lagrangian perturbation theory with pressure

Lagrangian perturbation theory for cosmological fluid describes structure formation in the quasinonlinear stage well. We present a third-order perturbative equation for Lagrangian perturbation with pressure in both the longitudinal and transverse modes. Then we derive the perturbative solution for simplest case.

Figure 1

Power spectra of density fluctuation computed from Lagrangian first-, second-, and third-order approximations. a) The initial spectrum index is $n=1$. The power spectra are at $a=8000$. The difference between the results by the first-, second-, and third-order Lagrangian approximation appears. b) The power spectra are at $a=3000$ for $n=0$. In this case, the effect of the third-order approximation also appears. c) The power spectra are at $a=1000$ for $n=-1$. The difference from Lagrangian approximation will still be very small just before shell-crossing.