Using Lagrangian perturbation theory for precision cosmology

We explore the Lagrangian perturbation theory (LPT) at 1-loop order with Gaussian initial conditions. We present an expansion method to approximately compute the power spectrum in LPT. Our approximate solution has good convergence in the series expansion and enables us to compute the power spectrum in LPT accurately and quickly. Non-linear corrections in the Lagrangian perturbation theory naturally satisfy the law of conservation of mass because the relation between matter density and the displacement vector of dark matter corresponds to the conservation of mass. By matching the 1-loop solution in LPT to the 2-loop solution in standard perturbation theory, we present an approximate solution of the power spectrum which has higher order corrections than the 2-loop order in standard perturbation theory with the conservation of mass satisfied. With this approximation, we can use LPT to compute a non-linear power spectrum without any free parameters, and this solution agrees with numerical simulations at $k=0.2$ $h{\rm Mpc}^{-1}$ and $z=0.35$ to better than 2%.