Efficient estimators of Fourier-space statistics for large number of objects rely on fast Fourier transforms (FFTs), which are affected by aliasing from unresolved small-scale modes due to the finite ...FFT grid. Aliasing takes the form of a sum over images, each of them corresponding to the Fourier content displaced by increasing multiples of the sampling frequency of the grid. These spurious contributions limit the accuracy in the estimation of Fourier-space statistics, and are typically ameliorated by simultaneously increasing grid size and discarding high-frequency modes. This results in inefficient estimates for e.g. the power spectrum when desired systematic biases are well under per cent level. We show that using interlaced grids removes odd images, which include the dominant contribution to aliasing. In addition, we discuss the choice of interpolation kernel used to define density perturbations on the FFT grid and demonstrate that using higher order interpolation kernels than the standard Cloud-In-Cell algorithm results in significant reduction of the remaining images. We show that combining fourth-order interpolation with interlacing gives very accurate Fourier amplitudes and phases of density perturbations. This results in power spectrum and bispectrum estimates that have systematic biases below 0.01 per cent all the way to the Nyquist frequency of the grid, thus maximizing the use of unbiased Fourier coefficients for a given grid size and greatly reducing systematics for applications to large cosmological data sets.
We study the impact of setting initial conditions in numerical simulations using the standard procedure based on the Zel'dovich approximation (ZA). As it is well known from the perturbation theory, ...ZA initial conditions have incorrect second- and higher-order growth and therefore excite long-lived transients in the evolution of the statistical properties of density and velocity fields. We also study the improvement brought by using more accurate initial conditions based on second-order Lagrangian perturbation theory (2LPT). We show that 2LPT initial conditions reduce transients significantly and thus are much more appropriate for numerical simulations devoted to precision cosmology. Using controlled numerical experiments with ZA and 2LPT initial conditions, we show that simulations started at redshift zi= 49 using the ZA underestimate the power spectrum in the non-linear regime by about 2, 4 and 8 per cent at z= 0, 1, and 3, respectively, whereas the mass function of dark matter haloes is underestimated by 5 per cent at m= 1015 M⊙ h−1 (z= 0) and 10 per cent at m= 2 × 1014 M⊙ h−1 (z= 1). The clustering of haloes is also affected to the few per cent level at z= 0. These systematics effects are typically larger than statistical uncertainties in recent mass function and power spectrum fitting formulae extracted from numerical simulations. At large scales, the measured transients in higher-order correlations can be understood from first principle calculations based on perturbation theory.
The standard procedure to generate initial conditions in numerical simulations of structure formations is to use the Zel’dovich approximation (ZA). Although the ZA correctly reproduces the linear ...growing modes of density and velocity perturbations, non‐linear growth is inaccurately represented, particularly for velocity perturbations because of the ZA failure to conserve momentum. This implies that it takes time for the actual dynamics to establish the correct statistical properties of density and velocity fields. We extend the standard formulation of non‐linear perturbation theory (PT) to include transients as non‐linear excitations of decaying modes caused by the initial conditions. These new non‐linear solutions interpolate between the initial conditions and the late‐time solutions given by the exact non‐linear dynamics. To quantify the magnitude of transients, we focus on higher order statistics of the density contrast δ and velocity divergence Θ, characterized by the Sp and Tp parameters. These describe the non‐Gaussianity of the probability distribution through its connected moments 〈δp〉c ≡ Sp〈δ2〉p−1, 〈Θp〉c ≡ Tp 〈Θ2〉p−1. We calculate Sp(a) and Tp(a) to leading order in PT with top‐hat smoothing as a function of the scale factor a. We find that the time‐scale of transients is determined, at a given order p, by the effective spectral index neff. The skewness factor S3 (T3) attains 10 per cent accuracy only after a ≈ 6 (a ≈ 15) for neff ≈ 0, whereas higher (lower) neff demands more (less) expansion away from the initial conditions. These requirements become much more stringent as p increases, always showing slower decay of transients for Tp than Sp. For models with density parameter Ω ≠ 1, the conditions above apply to the linear growth factor; thus an Ω = 0.3 open model requires roughly a factor of 2 larger expansion than a critical density model to reduce transients by the same amount. The predicted transients in Sp are in good agreement with numerical simulations. More accurate initial conditions can be achieved by using second‐order Lagrangian PT (2LPT), which reproduces growing modes up to second order and thus eliminates transients in the skewness parameters. We show that for p > 3 this scheme can reduce the required expansion by more than an order of magnitude compared to the ZA. Setting up 2LPT initial conditions requires only minimal, inexpensive changes to ZA codes. We suggest simple steps for its implementation.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
We present the methodology for a joint cosmological analysis of weak gravitational lensing from the fourth data release of the ESO Kilo-Degree Survey (KiDS-1000) and galaxy clustering from the ...partially overlapping Baryon Oscillation Spectroscopic Survey (BOSS) and the 2-degree Field Lensing Survey (2dFLenS). Cross-correlations between BOSS and 2dFLenS galaxy positions and source galaxy ellipticities have been incorporated into the analysis, necessitating the development of a hybrid model of non-linear scales that blends perturbative and non-perturbative approaches, and an assessment of signal contributions by astrophysical effects. All weak lensing signals were measured consistently via Fourier-space statistics that are insensitive to the survey mask and display low levels of mode mixing. The calibration of photometric redshift distributions and multiplicative gravitational shear bias has been updated, and a more complete tally of residual calibration uncertainties was propagated into the likelihood. A dedicated suite of more than 20 000 mocks was used to assess the performance of covariance models and to quantify the impact of survey geometry and spatial variations of survey depth on signals and their errors. The sampling distributions for the likelihood and the
χ
2
goodness-of-fit statistic have been validated, with proposed changes for calculating the effective number of degrees of freedom. The prior volume was explicitly mapped, and a more conservative, wide top-hat prior on the key structure growth parameter
S
8
=
σ
8
(Ω
m
/0.3)
1/2
was introduced. The prevalent custom of reporting
S
8
weak lensing constraints via point estimates derived from its marginal posterior is highlighted to be easily misinterpreted as yielding systematically low values of
S
8
, and an alternative estimator and associated credible interval are proposed. Known systematic effects pertaining to weak lensing modelling and inference are shown to bias
S
8
by no more than 0.1 standard deviations, with the caveat that no conclusive validation data exist for models of intrinsic galaxy alignments. Compared to the previous KiDS analyses,
S
8
constraints are expected to improve by 20% for weak lensing alone and by 29% for the joint analysis.
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Abstract
We compare reduced three-point correlations Q of matter, haloes (as proxies for galaxies) and their cross-correlations, measured in a total simulated volume of ∼100 (h−1 Gpc)3, to ...predictions from leading order perturbation theory on a large range of scales in configuration space. Predictions for haloes are based on the non-local bias model, employing linear (b1) and non-linear (c2, g2) bias parameters, which have been constrained previously from the bispectrum in Fourier space. We also study predictions from two other bias models, one local (g2 = 0) and one in which c2 and g2 are determined by b1 via approximately universal relations. Overall, measurements and predictions agree when Q is derived for triangles with (r1r2r3)1/3 ≳60 h−1 Mpc, where r1 − 3 are the sizes of the triangle legs. Predictions for Qmatter, based on the linear power spectrum, show significant deviations from the measurements at the BAO scale (given our small measurement errors), which strongly decrease when adding a damping term or using the non-linear power spectrum, as expected. Predictions for Qhalo agree best with measurements at large scales when considering non-local contributions. The universal bias model works well for haloes and might therefore be also useful for tightening constraints on b1 from Q in galaxy surveys. Such constraints are independent of the amplitude of matter density fluctuation (σ8) and hence break the degeneracy between b1 and σ8, present in galaxy two-point correlations.
The peak-background split argument is commonly used to relate the abundance of dark matter haloes to their spatial clustering. Testing this argument requires an accurate determination of the halo ...mass function. We present a maximum-likelihood method for fitting parametric functional forms to halo abundances which differs from previous work because it does not require binned counts. Our conclusions do not depend on whether we use our method or more conventional ones. In addition, halo abundances depend on how haloes are defined. Our conclusions do not depend on the choice of link length associated with the friends-of-friends halo finder, nor do they change if we identify haloes using a spherical overdensity algorithm instead. The large-scale halo bias measured from the matter–halo cross spectrum b× and the halo autocorrelation function bξ (on scales k∼ 0.03 h Mpc−1 and r∼ 50 h−1 Mpc) can differ by as much as 5 per cent for haloes that are significantly more massive than the characteristic mass M*. At these large masses, the peak-background split estimate of the linear bias factor b1 is 3–5 per cent smaller than bξ, which is 5 per cent smaller than b×. We discuss the origin of these discrepancies: deterministic non-linear local bias, with parameters determined by the peak-background split argument, is unable to account for the discrepancies we see. A simple linear but non-local bias model, motivated by peaks theory, may also be difficult to reconcile with our measurements. More work on such non-local bias models may be needed to understand the nature of halo bias at this level of precision.
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BFBNIB, FZAB, GIS, IJS, KILJ, NLZOH, NUK, OILJ, SBCE, SBMB, UL, UM, UPUK
We study the behaviour of the three-point correlation function ζ of dark matter and mock galaxies, concentrating on the effects of redshift-space distortions and the determination of galaxy bias ...parameters in current redshift galaxy surveys. On large scales, redshift-space distortions tend to wash out slightly the configuration dependence of the reduced three-point function Q3∼ζ/ξ2. On smaller scales (=10 h−1 Mpc), Q3 develops a characteristic U-shape anisotropy between elongated and open triangles due to the effects of velocity dispersion. We show that this shape is quite universal, very weakly dependent on scale, initial spectral index or cosmological parameters and should be detectable in current galaxy surveys even if affected by shot noise or galaxy bias. We present a detailed method for obtaining constraints on galaxy bias parameters from measurements of Q3 in current galaxy redshift surveys, based on the eigenmode analysis similar to that developed for the bispectrum. We show that our method recovers the bias parameters introduced into mock galaxies by a halo occupation distribution prescription and is also able to handle potential systematics in the case when a smaller number than is ideal for mock catalogues is used to estimate the covariance matrix. We find that current redshift surveys (e.g. SDSS or 2dFGRS) are just about large enough to obtain interesting new constraints on bias.
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.
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