Gaussianization: Enhancing the Statistical Power of the Power Spectrum
GGaussianization: Enhancing the StatisticalPower of the Power Spectrum
Mark C. Neyrinck
Abstract
The power spectrum is widely used in astronomy, to analyze temporalor spatial structure. In cosmology, it is used to quantify large-scale structure (LSS)and the cosmic microwave background (CMB). This is because the power spec-trum completely quantifies Gaussian random fields, which the CMB and LSS fieldsseem to be at early epochs. However, at late epochs and small scales, cosmologicaldensity fields become highly non-Gaussian. The power spectrum loses power to de-scribe LSS and CMB fields on small scales, most obviously through high covariancein the power spectrum as a function of scale. Practically, this significantly degradesconstraints that observations can place on cosmological parameters. However, ifa nonlinear transformation that produces a (more) Gaussian 1-point distribution isapplied to a field, the extra covariance in the field’s power spectrum can be dra-matically reduced. In the case of the roughly lognormal low-redshift matter densityfield, a log transform accomplishes this. Applying a log transform to the densityfield before measuring the power spectrum also tightens cosmological parameterconstraints by a factor of several.A Gaussian random field has convenient statistical properties. Its meaningfulinformation is fully quantified by the power spectrum; all connected higher-orderstatistics vanish. Of particular importance for a measurer of (parameters which de-pend on) the power spectrum, all off-diagonal power-spectrum covariance matrixelements vanish for a Gaussian random field.On small scales at late epochs, the cosmological overdensity field δ = ρ (cid:104) ρ (cid:105) − δ → ln ( + δ ) much reduces the power-spectrum covariance [2]. As shown in Fig. 2, it also reduces error bars on inferredparameters, reaching a factor of 5 reduction in the best case of the tilt n s . Mark C. NeyrinckThe Johns Hopkins University, Baltimore, MD 21218 1 a r X i v : . [ a s t r o - ph . C O ] S e p Mark C. Neyrinck k [ h/ Mpc] P ( k ) δ ln(1 + δ ) Fig. 1
Left: δ in a 2- h − Mpc slice of the 500- h − Mpc Millennium simulation (MS), viewedwith an unfortunate linear color scale. Middle: the same slice with a logarithmic color scale. Right:the 2D power spectra P δ and P ln ( + δ ) of δ (black) and ln ( + δ ) (green), in 9 such slices. The wild,coherent fluctuations in P δ illustrate its high (co)variance, absent in P ln ( + δ ) . − − ln σ .
02 0 . − − − k max [Mpc − ] E rr o r b a r ln σ n s P δ . . n s P ln(1 + δ ) . Fig. 2
Fisher-matrix estimates of error-bar (half-)widths and error ellipses for the cosmologicalparameters ln σ and n s , inferred analyzing P δ (black) and P ln ( + δ ) (green). We show how theydepend on the maximum wavenumber k max included in a power-spectrum analysis of a 1-Gpcreal-space matter density field. Diagonal panels show unmarginalized error bars over single pa-rameters. In the lower-left panel, errors in each parameter are marginalized over the other. Theupper-right panel shows how error ellipses contract as k max increases. There is an ellipse shownfor each k max constituting the curves in the other panels. Outside the bold ellipses, analyzing onlylarge scales where k max < . − , P δ and P ln ( + δ ) perform similarly. Inside the bold ellipses,nonlinear scales are included, up to the innermost ellipse that corresponds to k max = . − .Here, P ln ( + δ ) greatly outperforms P δ . See Ref. [3] for more details. References [1] P. Coles, B. Jones, MNRAS , 1 (1991)[2] M.C. Neyrinck, I. Szapudi, A.S. Szalay, ApJL698