Uncorrelated Modes of the Nonlinear Power Spectrum

TL;DR

The paper addresses the challenge of broad, nonlinear covariance in the galaxy power spectrum by introducing a prewhitening transform that makes the noise covariance effectively white. It demonstrates that the prewhitened 4-point and 3-point covariances become nearly diagonal in an eigenbasis, yielding almost uncorrelated nonlinear modes with eigenvalues closely tracking the nonlinear power, and enabling near-minimum-variance estimators and Fisher matrices analogous to the linear Gaussian case. By defining a prewhitened power spectrum $X(r)$ and its estimator, and providing practical recipes (gourmet, fine, fastfood) within the FKP framework, the work offers a scalable path to measuring nonlinear power spectra from galaxy surveys. Although grounded in a hierarchical, constant-amplitude model and neglecting redshift distortions, the approach reveals that nonlinear information can be organized into stable, decorrelated band-powers, with $X(k)$ often resembling the linear spectrum and the methodology adaptable to realistic survey geometries. This has significant implications for robust parameter estimation in large-scale structure analyses using nonlinear scales.

Abstract

Nonlinear evolution causes the galaxy power spectrum to become broadly correlated over different wavenumbers. It is shown that prewhitening the power spectrum - transforming the power spectrum in such a way that the noise covariance becomes proportional to the unit matrix - greatly narrows the covariance of power. The eigenfunctions of the covariance of the prewhitened nonlinear power spectrum provide a set of almost uncorrelated nonlinear modes somewhat analogous to the Fourier modes of the power spectrum itself in the linear, Gaussian regime. These almost uncorrelated modes make it possible to construct a near minimum variance estimator and Fisher matrix of the prewhitened nonlinear power spectrum analogous to the Feldman-Kaiser-Peacock estimator of the linear power spectrum. The paper concludes with summary recipes, in gourmet, fine, and fastfood versions, of how to measure the prewhitened nonlinear power spectrum from a galaxy survey in the FKP approximation. An Appendix presents FFTLog, a code for taking the fast Fourier or Hankel transform of a periodic sequence of logarithmically spaced points, which proves useful in some of the manipulations.

Figures (13)

• Figure 1: Schematic illustration of the 4-point, 3-point, and 2-point contributions to the covariance ${\frak C}_{ijkl}$ of pairs $ij$ with other pairs $kl$. The 3-point and 2-point contributions are shot-noise contributions in which one or both galaxies of the pair $ij$ are the same as one or both of the pair $kl$.
• Figure 2: Correlation coefficient $K(k_\alpha,$$k_\beta)/$$[K(k_\alpha,$$k_\alpha)$$K(k_\beta,$$k_\beta)]^{1/2}$ of the 4-point contribution $K(k_\alpha,k_\beta)$ to the covariance of the power (i.e. the covariance without shot-noise) in the case of a power law power spectrum with correlation function $\xi(r) = (r/5 \, h^{-1} {\rm Mpc})^{- 1.8}$. Each line is the correlation coefficient for a fixed $k_\beta$, and each line peaks at $k_\alpha = k_\beta$, whereat the value is unity. The hierarchical amplitudes are $R_a = - R_b = 4.195$. The resolution is $128$ points per decade, $\Delta\log k = 1/128$.
• Figure 3: Correlation coefficient $M(k_\alpha,k_\beta)/$$[M(k_\alpha,k_\alpha)$$M(k_\beta,$$k_\beta)]^{1/2}$ of the 4-point contribution $M(k_\alpha,k_\beta)$ to the prewhitened covariance of a power law power spectrum with correlation function $\xi(r) = (r/5 \, h^{-1} {\rm Mpc})^{- 1.8}$. Lines are dotted where the correlation coefficient is negative. This is the same as Figure \ref{['M']}, except that the covariance is prewhitened.
• Figure 4: Correlation coefficients (top) $K(k_\alpha,k_\beta)/$$[K(k_\alpha,k_\alpha)$$K(k_\beta,k_\beta)]^{1/2}$ of the covariance, and (bottom) $M(k_\alpha,k_\beta)/$$[M(k_\alpha,k_\alpha)$$M(k_\beta,k_\beta)]^{1/2}$ of the prewhitened covariance, of the power spectrum in four different models of the power spectrum. Each curve is the correlation coefficient at fixed $k_\beta = 1 \, h \, {\rm Mpc}^{-1}$, plotted as a function of $k_\alpha$. The three sets of panels starting from the left are for power law power spectra with correlation functions $\xi(r) = (r/5 \, h^{-1} {\rm Mpc})^{-\gamma}$ with indices $\gamma = 1.1$, $1.8$, and $2.9$, while the rightmost panel is for the $\Lambda$CDM power spectrum of Eisenstein & Hu (1998) with $\Omega_\Lambda = 0.7$, $\Omega_m = 0.3$, $\Omega_b h^2 = 0.02$, and $h = 0.65$, nonlinearly evolved by the procedure of Peacock & Dodds (1996). The two lines on each graph are for 4-point hierarchical amplitudes (solid) $R_b = - R_a$, and (long-dash) $R_b = R_a$. Lines are dotted where the correlation coefficient is negative. The Schwarz inequality, which requires that the correlation coefficient be $\leq 1$, is violated by the hierarchical model with $R_b = R_a$ at values $k \ll k'$ and $k \gg k'$. The resolution is $128$ points per decade, the same as in Figures \ref{['M']} and \ref{['Mw']}.
• Figure 5: Correlation coefficient $L_{\alpha\beta}/$$(L_{\alpha\alpha}$$L_{\beta\beta})^{1/2}$ of the prewhitened 3-point covariance $L_{\alpha\beta}$ in the representation of eigenfunctions $\phi_\alpha$ of the prewhitened 4-point covariance $M_{\alpha\beta}$, for a power law power spectrum with correlation function $\xi(r) = (r/5 \, h^{-1} {\rm Mpc})^{- 1.8}$. Each line is the correlation coefficient at a fixed nominal wavenumber $k_\beta$, plotted against the nominal wavenumber $k_\alpha$, which labels the 4-point eigenfunctions $\phi_\alpha$ ordered by eigenvalue. Each line peaks at $k_\alpha = k_\beta$, whereat the correlation coefficient is unity. The upper panel is for 4-point hierarchical amplitudes $R_b = - R_a$; the lower panel is for $R_b = R_a$. Lines are dotted where the correlation coefficient is negative. The resolution is $\Delta\log k = 1/128$.