The non-linear redshift-space power spectrum of galaxies

We study the power spectrum of galaxies in redshift space, with third-order perturbation theory to include corrections that are absent in linear theory. We assume a local bias for the galaxies: i.e., the galaxy density is sampled from some local function of the underlying mass distribution. We find that the effect of the non-linear bias in real space is to introduce two new features: first, there is a contribution to the power which is constant with wavenumber, whose nature we reveal as essentially a shot-noise term. In principle this contribution can mask the primordial power spectrum, and could limit the accuracy with which the latter might be measured on very large scales. Secondly, the effect of second- and third-order bias is to modify the effective bias (defined as the square root of the ratio of galaxy power spectrum to matter power spectrum). The effective bias is almost scale-independent over a wide range of scales. These general conclusions also hold in redshift space. In addition, we have investigated the distortion of the power spectrum by peculiar velocities, which may be used to constrain the density of the Universe. We look at the quadrupole-to-monopole ratio, and find that higher order terms can mimic linear theory bias, but the bias implied is neither the linear bias, nor the effective bias referred to above. We test the theory with biased N -body simulations, and find excellent agreement in both real and redshift space, providing the local biasing is applied on a scale whose fractional rms density fluctuations are < 0.5.     

Effects of destriping errors on estimates of the CMB power spectrum

Destriping methods for constructing maps of the cosmic microwave background (CMB) anisotropies have been investigated extensively in the literature. However, their error properties have been studied in less detail. Here we present an analysis of the effects of destriping errors on CMB power spectrum estimates for Planck -like scanning strategies. Analytic formulae are derived for certain simple scanning geometries that can be rescaled to account for different detector noise. Assuming Planck -like low-frequency noise, the noise power spectrum is accurately white at high multipoles  (ℓ≳ 50)  . Destriping errors, though dominant at lower multipoles, are small in comparison to the cosmic variance. These results show that simple destriping map-making methods should be perfectly adequate for the analysis of Planck data and support the arguments given in an earlier paper in favour of applying a fast hybrid power spectrum estimator to CMB data with realistic '1/ f ' noise.     

Accurate estimators of power spectra in N-body simulations

A method to rapidly estimate the Fourier power spectrum of a point distribution is presented. This method relies on a Taylor expansion of the trigonometric functions. It yields the Fourier modes from a number of fast Fourier transforms (FFTs), which is controlled by the order N of the expansion and by the dimension D of the system. In three dimensions, for the practical value   N = 3  , the number of FFTs required is 20.
We apply the method to the measurement of the power spectrum of a periodic point distribution that is a local Poisson realization of an underlying stationary field. We derive an explicit analytic expression for the spectrum, which allows us to quantify – and correct for – the biases induced by discreteness and by the truncation of the Taylor expansion, and to bound the unknown effects of aliasing of the power spectrum. We show that these aliasing effects decrease rapidly with the order N . For   N = 3  , they are expected to be, respectively, smaller than  ∼10⁻⁴  and 0.02 at half the Nyquist frequency and at the Nyquist frequency of the grid used to perform the FFTs. The only remaining significant source of errors is reduced to the unavoidable cosmic/sample variance due to the finite size of the sample.
The analytical calculations are successfully checked against a cosmological N -body experiment. We also consider the initial conditions of this simulation, which correspond to a perturbed grid. This allows us to test a case where the local Poisson assumption is incorrect. Even in that extreme situation, the third-order Fourier–Taylor estimator behaves well, with aliasing effects restrained to at most the per cent level at half the Nyquist frequency.
We also show how to reach arbitrarily large dynamic range in Fourier space (i.e. high wavenumber), while keeping statistical errors in control, by appropriately 'folding' the particle distribution.     

The three-point function in large-scale structure – I. The weakly non-linear regime in N-body simulations

This paper presents a comparison of the predictions for the two- and three-point correlation functions of density fluctuations, ξ and ζ , in gravitational perturbation theory (PT) against large cold dark matter (CDM) simulations. This comparison is made possible for the first time on large weakly non-linear scales (>10  h ⁻¹ Mpc) thanks to the development of a new algorithm for estimating correlation functions for millions of points in only a few minutes. Previous studies in the literature comparing the PT predictions of the three-point statistics with simulations have focused mostly on Fourier space, angular space or smoothed fields. Results in configuration space, such as those presented here, were limited to small scales where leading-order PT gives a poor approximation. Here we also propose and apply a method for separating the first-order and subsequent contributions to PT by combining different output times from the evolved simulations. We find that in all cases there is a regime where simulations do reproduce the leading-order (tree-level) predictions of PT for the reduced three-point function   Q ₃∼ ζ / ξ ²  . For steeply decreasing correlations (such as the standard CDM model) deviations from the tree-level results are important even at relatively large scales, ≃20 Mpc  h ⁻¹. On larger scales ξ goes to zero and the results are dominated by sampling errors. In more realistic models (such as the ΛCDM cosmology) deviations from the leading-order PT become important at smaller scales   r ≃10 Mpc  h ^-1  , although this depends on the particular three-point configuration. We characterize the range of validity of this agreement and show the behaviour of the next-order (one-loop) corrections.     

Decorrelating the power spectrum of galaxies

We show how to decorrelate the (pre-whitened) power spectrum measured from a galaxy survey into a set of high-resolution uncorrelated band-powers. The treatment includes non-linearity, but not redshift distortions. Amongst the infinitely many possible decorrelation matrices, the square root of the Fisher matrix, or a scaled version thereof, offers a particularly good choice, in the sense that the band-power windows are narrow, approximately symmetric, and well-behaved in the presence of noise. We use this method to compute band-power windows for, and the information content of, the Sloan Digital Sky Survey, the Las Campanas Redshift Survey, and the IRAS 1.2-Jy Survey.     

A fitting formula for the non-linear evolution of the bispectrum

We present a fitting formula for the non-linear evolution of the bispectrum in cold dark matter (CDM) models, obtained from measurements in high-resolution numerical simulations. The formula interpolates between the perturbative and highly non-linear regimes, and generalizes previous results obtained for scale-free initial conditions.     

Parameter information from non-linear cosmological fields

We develop a general formalism for analysing parameter information from non-Gaussian cosmic fields. The method can be adapted to include the non-linear effects in galaxy redshift surveys, weak lensing surveys and cosmic velocity field surveys as part of parameter estimation. It can also be used as a test of non-Gaussianity of the cosmic microwave background. Generalizing maximum-likelihood analysis to second order, we calculate the non-linear Fisher information matrix and likelihood surfaces in parameter space. To this order we find that the information content is always increased by including non-linearity. Our methods are applied to a realistic model of a galaxy redshift survey, including non-linear evolution, galaxy bias, shot-noise and redshift-space distortions to second order. We find that including non-linearities allows all of the degeneracies between parameters to be lifted. Marginalized parameter uncertainties of a few per cent will then be obtainable using forthcoming galaxy redshift surveys.     

Determination of the linear mass power spectrum from the mass function of galaxy clusters

We develop a new method to determine the linear mass power spectrum using the mass function of galaxy clusters. We obtain the rms mass fluctuation  σ( M )  using the expression for the mass function in the Press & Schechter, Sheth, Mo & Tormen and Jenkins et al. formalisms. We apply different techniques to recover the adimensional power spectrum  Δ²( k )  from  σ( M )  namely the   k _eff  approximation, the singular value decomposition and the linear regularization method. The application of these techniques to the τCDM and ΛCDM GIF simulations shows a high efficiency in recovering the theoretical power spectrum over a wide range of scales. We compare our results with those derived from the power spectrum of the spatial distribution of the same sample of clusters in the simulations obtained by application of the classical Feldman, Kaiser & Peacock (FKP) method. We find that the mass function based method presented here can provide a very accurate estimate of the linear power spectrum, particularly for low values of k . This estimate is comparable to, or even better behaved than, the FKP solution.
The principal advantage of our method is that it allows the determination of the linear mass power spectrum using the joint information of objects of a wide range of masses without dealing with specific assumptions on the bias relative to the underlying mass distribution.