Second Order Power Spectrum and Nonlinear Evolution at High Redshift

TL;DR

The paper develops a second-order perturbation theory for the cosmological fluid to compute the nonlinear power spectrum, showing that long-wavelength mode coupling largely drives weakly nonlinear evolution and that $P(k, au)=a^2( au)P_{11}(k)+a^4( au)P_2(k)$ with $P_2(k)=P_{22}(k)+2P_{13}(k)$. Applying this to CDM, the authors find suppression of power near the spectrum peak and enhancement at high $k$, implying a transfer of power from large to small scales and a shallower high-$k$ slope, consistent with early-time N-body results but diverging at late times due to previrialization and cutoff effects. The study reveals that nonlinear masses grow more rapidly at high redshift than linear theory would predict, with factors up to $ olinebreak 180$ for $(1+z)=20$ when using a Gaussian window and $oldsymbol{ ho}$-threshold $oldsymbol{ m abla ho/ ho_c=1}$, and shows Press–Schechter underestimates high-mass object abundances at high $z$. These results have significant implications for the formation of the first structures and the interpretation of high-redshift surveys, and they underscore the limits of linear extrapolation and PS predictions in realistic CDM-like spectra.

Abstract

The Eulerian cosmological fluid equations are used to study the nonlinear mode coupling of density fluctuations. We evaluate the second-order power spectrum including all four-point contributions. In the weakly nonlinear regime we find that the dominant nonlinear contribution for realistic cosmological spectra is made by the coupling of long-wave modes and is well estimated by second order perturbation theory. For a linear spectrum like that of the cold dark matter model, second order effects cause a significant enhancement of the high $k$ part of the spectrum and a slight suppression at low $k$ near the peak of the spectrum. Our perturbative results agree well in the quasilinear regime with the nonlinear spectrum from high-resolution N-body simulations. We find that due to the long-wave mode coupling, characteristic nonlinear masses grow less slowly in time (i.e., are larger at higher redshifts) than would be estimated using the linear power spectrum. For the cold dark matter model at $(1+z)=(20,10,5,2)$ the nonlinear mass is about $(180,8,2.5,1.6)$ times (respectively) larger than a linear extrapolation would indicate, if the condition rms $δρ/ρ=1$ is used to define the nonlinear scale. At high redshift the Press-Schechter mass distribution significantly underestimates the abundance of high-mass objects for the cold dark matter model. Although the quantitative results depend on the definition of the nonlinear scale, these basic consequences hold for any initial spectrum whose post-recombination spectral index $n$ decreases sufficiently rapidly with increasing $k$, a feature which arises quite generally during the transition from a radiation- to matter-dominated universe.