Local Bias Model Research Articles

Toward a robust inference method for the galaxy bispectrum: likelihood function and model selection

The forthcoming generation of galaxy redshift surveys will sample the large-scale structure of the Universe over unprecedented volumes with high-density tracers. This advancement will make robust measurements of three-point clustering statistics possible. In preparation for this improvement, we investigate how several methodological choices can influence inferences based on the bispectrum about galaxy bias and shot noise. We first measure the real-space bispectrum of dark-matter haloes extracted from 298 N-body simulations covering a volume of approximately 1000 Gpc3. We then fit a series of theoretical models based on tree-level perturbation theory to the numerical data. To achieve this, we estimate the covariance matrix of the measurement errors by using 10,000 mock catalogues generated with the PINOCCHIO code. We study how the model constraints are influenced by the binning strategy for the bispectrum configurations and by the form of the likelihood function. We also use Bayesian model-selection techniques to single out the optimal theoretical description of our data. We find that a three-parameter bias model combined with Poissonian shot noise is necessary to model the halo bispectrum up to scales of kmax≲ 0.08 Mpc-1, although fitting formulae that relate the bias parameters can be helpful to reduce the freedom of the model without compromising accuracy. Our data clearly disfavour local Eulerian and local Lagrangian bias models and do not require corrections to Poissonian shot noise. We anticipate that model-selection diagnostics will be particularly useful to extend the analysis to smaller scales as, in this case, the number of model parameters will grow significantly large.

Signatures of cosmic reionization on the 21-cm two- and three-point correlation function I: quadratic bias modelling

The three-point correlation function (3PCF) of the 21cm brightness temperature from the Epoch of Reionization (EoR) probes complementary information to the commonly studied two-point correlation function (2PCF) about the morphology of ionized regions. We investigate the 21cm 2PCF and 3PCF in configuration space using semi-numerical simulations and test whether they can be described by the local quadratic bias model. We find that fits of bias model predictions for the 2PCF and 3PCF deviate from our measurements by $\sim 20\%$ at scales above the typical size of ionized regions ($\simeq 30$ Mpc) and at early times with global neutral fractions of $\langle x_{\rm HI} \rangle \gtrsim 0.7$. At later times and smaller scales these deviations increase strongly, indicating a break down of the bias model. The 2PCF and 3PCF fits of the linear bias parameter agree at the $10\%$ level for different EoR model configurations. This agreement holds, when adding redshift space distortions to the simulations. The relation between spatial fluctuations in the matter density and the 21cm signal, as predicted by the bias model, is consistent with direct measurements of this relation in simulations for large smoothing scales ($\gtrsim 30$ Mpc). From this latter test we conclude that negative amplitudes of the 21cm 3PCF result from negative bias parameters, which describe the anti-correlation between the matter over-densities and the 21cm signal during the EoR. However, a more detailed interpretation of the bias parameters may require a description of non-local contributions to the bias model.

Cylinders out of a top hat: counts-in-cells for projected densities

Large deviation statistics is implemented to predict the statistics of cosmic densities in cylinders applicable to photometric surveys. It yields few percent accurate analytical predictions for the one-point probability distribution function (PDF) of densities in concentric or compensated cylinders; and also captures the density-dependence of their angular clustering (cylinder bias). All predictions are found to be in excellent agreement with the cosmological simulation Horizon Run 4 in the quasi-linear regime where standard perturbation theory normally breaks down. These results are combined with a simple local bias model that relates dark matter and tracer densities in cylinders and validated on simulated halo catalogues. This formalism can be used to probe cosmology with existing and upcoming photometric surveys like DES, Euclid or WFIRST containing billions of galaxies.

Position-dependent correlation function from the SDSS-III Baryon Oscillation Spectroscopic Survey Data Release 10 CMASS sample

We report on the first measurement of the three-point function with the position-dependent correlation function from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 10 CMASS sample. This new observable measures the correlation between two-point functions of galaxy pairs within different subvolumes, ξ̂(ř,řL), where řL is the location of a subvolume, and the corresponding mean overdensities, δ̄(řL). This correlation, which we call the “integrated three-point function”, iζ(r)≡⟨ξ̂(ř,řL)δ̄(řL)⟩, measures a three-point function of two short- and one long-wavelength modes, and is generated by nonlinear gravitational evolution and possibly also by the physics of inflation. The iζ(r) measured from the BOSS data lies within the scatter of those from the mock galaxy catalogs in redshift space, yielding a ten-percent-level determination of the amplitude of iζ(r). The tree-level perturbation theory in redshift space predicts how this amplitude depends on the linear and quadratic nonlinear galaxy bias parameters (b1 and b2), as well as on the amplitude and linear growth rate of matter fluctuations (σ8 and f). Combining iζ(r) with the constraints on b1σ8 and fσ8 from the global two-point correlation function and that on σ8 from the weak lensing signal of BOSS galaxies, we measure b2=0.41±0.41 (68% C.L.) assuming standard perturbation theory at the tree level and the local bias model.

Understanding higher-order nonlocal halo bias at large scales by combining the power spectrum with the bispectrum

Understanding the relation between underlying matter distribution and biased tracers such as galaxy or dark matter halo is essential to extract cosmological information from ongoing or future galaxy redshift surveys. At sufficiently large scales such as the BAO scale, a standard approach for the bias problem on the basis of the perturbation theory (PT) is to assume the `local bias' model in which the density field of biased tracers is deterministically expanded in terms of matter density field at the same position. The higher-order bias parameters are then determined by combining the power spectrum with higher-order statistics such as the bispectrum. As is pointed out by recent studies, however, nonlinear gravitational evolution naturally induces nonlocal bias terms even if initially starting only with purely local bias. As a matter of fact, previous works showed that the second-order nonlocal bias term, which corresponds to the gravitational tidal field, is important to explain the characteristic scale-dependence of the bispectrum. In this paper we extend the nonlocal bias term up to third order, and investigate whether the PT-based model including nonlocal bias terms can simultaneously explain the power spectrum and the bispectrum of simulated halos in $N$-body simulations. We show that the power spectrum, including density and momentum, and the bispectrum between halo and matter in $N$-body simulations can be simultaneously well explained by the model including up to third-order nonlocal bias terms up to k~0.1h/Mpc. Also, the results seem in a good agreement with theoretical predictions of a simple coevolution picture, although the agreement is not perfect. These demonstration clearly shows a failure of the local bias model even at such large scales, and we conclude that nonlocal bias terms should be consistently included in order to model statistics of halos. [abridged]

Non-Gaussian bias: insights from discrete density peaks

Corrections induced by primordial non-Gaussianity to the linear halo bias can be computed from a peak-background split or the widespread local bias model. However,numerical simulations clearly support the prediction of the former, in which thenon-Gaussian amplitude is proportional to the linear halo bias. To understand better the reasons behind the failure of standard Lagrangian local bias, in whichthe halo overdensity is a function of the local mass overdensity only, we explore the effect of a primordial bispectrum on the 2-point correlation of discrete density peaks. We show that the effective local bias expansion to peak clustering vastly simplifies the calculation. We generalize this approach to excursion set peaks and demonstrate that the resulting non-Gaussian amplitude, which is a weighted sum of quadratic bias factors, precisely agrees with the peak-background split expectation, which is a logarithmic derivative of the halo mass function with respect to the normalisation amplitude. We point out that statistics of thresholded regions can be computed using the same formalism. Our results suggest that halo clustering statistics can be modelled consistently (in the sense that the Gaussian and non-Gaussian bias factors agree with peak-background split expectations) from a Lagrangian bias relation only if the latter is specified as a set of constraints imposed on the linear density field. This is clearly not the case of standard Lagrangian local bias. Therefore, one is led to consider additional variables beyond the local mass overdensity.

Perturbation theory for the non-linear halo power spectrum: the renormalized bias and halo bias

We revisit an analytical model to describe the halo-matter cross-power spectrum and the halo auto-power spectrum in the weakly nonlinear regime, by combining the perturbation theory (PT) for matter clustering, the local bias model, and the halo bias. Nonlinearities in the power spectra arise from the nonlinear clustering of matter as well as the nonlinear relation between the matter and halo density fields. By using the "renormalization" approach, we express the nonlinear power spectra by a sum of the two contributions: the nonlinear matter power spectrum with the effective linear bias parameter, and the higher-order PT spectra having the halo bias parameters as the coefficients. The halo auto-power spectrum includes the residual shot noise contamination that needs to be treated as additional free parameter. The term(s) of the higher-order PT spectra and the residual shot noise cause a scale-dependent bias function relative to the nonlinear matter power spectrum in the weakly nonlinear regime. We show that the model predictions are in good agreement with the spectra measured from a suit of high-resolution $N$-body simulations up to $k\simeq 0.2 h$/Mpc at $z=0.35$, for different halo mass bins.

Lagrangian bias in the local bias model

It is often assumed that the halo-patch fluctuation field can be written as a Taylor series in the initial Lagrangian dark matter density fluctuation field. We show that if this Lagrangian bias is local, and the initial conditions are Gaussian, then the two-point cross-correlation between halos and mass should be linearly proportional to the mass-mass auto-correlation function. This statement is exact and valid on all scales; there are no higher order contributions, e.g., from terms proportional to products or convolutions of two-point functions, which one might have thought would appear upon truncating the Taylor series of the halo bias function. In addition, the auto-correlation function of locally biased tracers can be written as a Taylor series in the auto-correlation function of the mass; there are no terms involving, e.g., derivatives or convolutions. Moreover, although the leading order coefficient, the linear bias factor of the auto-correlation function is just the square of that for the cross-correlation, it is the same as that obtained from expanding the mean number of halos as a function of the local density only in the large-scale limit. In principle, these relations allow simple tests of whether or not halo bias is indeed local in Lagrangian space. We discuss why things are more complicated inpractice. We also discuss our results in light of recent work on the renormalizability of halo bias, demonstrating that it is better to renormalize than not. We use the Lognormal model to illustrate many of our findings.

Evidence for quadratic tidal tensor bias from the halo bispectrum

The relation between the clustering properties of luminous matter in the form of galaxies and the underlying dark matter distribution is of fundamental importance for the interpretation of ongoing and upcoming galaxy surveys. The so called local bias model, where galaxy density is a function of local matter density, is frequently discussed as a means to infer the matter power spectrum or correlation function from the measured galaxy correlation. However, gravitational evolution generates a term quadratic in the tidal tensor and thus non-local in the density field, even if this term is absent in the initial conditions (Lagrangian space). Because the term is quadratic, it contributes as a loop correction to the power spectrum, so the standard linear bias picture still applies on large scales, however, it contributes at leading order to the bispectrum for which it is significant on all scales. Such a term could also be present in Lagrangian space if halo formation were influenced by the tidal field. We measure the corresponding coupling strengths from the matter-matter-halo bispectrum in numerical simulations and find a non-vanishing coefficient for the tidal tensor term. We find no scale dependence of the bias parameters up to k=0.1 h/Mpc and that the tidal effect is increasing with halo mass. While the Lagrangian bias picture is a better description of our results than the Eulerian bias picture, our results suggest that there might be a tidal tensor bias already in the initial conditions. We also find that the coefficients of the quadratic density term deviate quite strongly from the theoretical predictions based on the spherical collapse model and a universal mass function. Both quadratic density and tidal tensor bias terms must be included in the modeling of galaxy clustering of current and future surveys if one wants to achieve the high precision cosmology promise of these datasets.

Large-scale bias and efficient generation of initial conditions for nonlocal primordial non-Gaussianity

We study the scale dependence of halo bias in generic (nonlocal) primordial non-Gaussian (PNG) initial conditions of the type motivated by inflation, parametrized by an arbitrary quadratic kernel. We first show how to generate nonlocal PNG initial conditions with minimal overhead compared to local PNG models for a general class of primordial bispectra that can be written as linear combinations of separable templates. We run cosmological simulations for the local, and nonlocal equilateral and orthogonal models and present results on the scale dependence of halo bias. We also derive a general formula for the Fourier-space bias using the peak-background split in the context of the excursion-set approach to halos and discuss the difference and similarities with the known corresponding result from local bias models. Our peak-background split bias formula generalizes previous results in the literature to include non-Markovian effects and nonuniversality of the mass function and are in better agreement with measurements in numerical simulations than previous results for a variety of halo masses, redshifts and halo definitions. We also derive for the first time quadratic bias results for arbitrary nonlocal PNG, and show that nonlinear bias loops give small corrections at large scales. The resulting well-behaved perturbation theory paves the way to constrain nonlocal PNG from measurements of the power spectrum and bispectrum in galaxy redshift surveys.

The local bias model in the large-scale halo distribution

We explore the biasing in the clustering statistics of halos as compared to dark matter (DM) in simulations. We look at the second and third order statistics at large scales of the (intermediate) MICEL1536 simulation and also measure directly the local bias relation h = f({\delta}) between DM fluctuations, {\delta}, smoothed over a top-hat radius Rs at a point in the simulation and its corresponding tracer h (i.e. halos) at the same point. This local relation can be Taylor expanded to define a linear (b1) and non-linear (b2) bias parameters. The values of b1 and b2 in the simulation vary with Rs approaching a constant value around Rs > 30 - 60 Mpc/h. We use the local relation to predict the clustering of the tracer in terms of the one of DM. This prediction works very well (about percent level) for the halo 2-point correlation {\xi}(r_12) for r_12 > 15 Mpc/h, but only when we use the biasing values that we found at very large smoothing radius Rs > 30 - 60 Mpc/h. We find no effect from stochastic or next to leading order terms in the f({\delta}) expansion. But we do find some discrepancies in the 3-point function that needs further understanding. We also look at the clustering of the smoothed moments, the variance and skewness which are volume average correlations and therefore include clustering from smaller scales. In this case, we find that both next to leading order and discreetness corrections (to the local model) are needed at the 10 - 20% level. Shot-noise can be corrected with a term {\sigma}e^2/n where {\sigma}e^2 < 1, i.e., always smaller than the Poisson correction. We also compare these results with the peak-background split predictions from the measured halo mass function. We find 5-10% systematic (and similar statistical) errors in the mass estimation when we use the halo model biasing predictions to calibrate the mass.

Testing standard perturbation theory and the Eulerian local biasing scheme against N-body simulations

We test third-order standard perturbation theory (SPT) as an approximation to non-linear cosmological structure formation. A novel approach is used to numerically calculate the three-dimensional dark matter density field using SPT from the initial conditions of two high-resolution cosmological simulations. The calculated density field is compared to the non-linear dark matter field of the simulations both point-by-point and statistically. For smoothing scales above 8 Mpc/h it shows a good agreement up to redshift 0. We present a simple fitting formula to relate the linear and non-linear density contrast that accurately recovers the non-linear time evolution for 0 <= z <= 10 at the per cent level. To address the problem of biasing between the matter field and the haloes identified in the simulation, we employ the Eulerian local bias model (ELB), including non-linear bias up to the third order. The bias parameters are obtained by fitting a scatter plot of halo and matter density (both from the simulation and from SPT). Using these bias parameters, we can reconstruct the halo density field. We find that this reconstruction is not able to capture all the details of the halo distribution. We investigate how well the large scale bias can be described by a constant and if it corresponds to the linear bias parameter b_1 of the local bias model. We also discuss how well the halo-halo power spectrum and the halo-mass cross spectrum from the reconstructed halo density field agree with the corresponding statistics from the simulation. The results show that while SPT is an excellent approximation for the matter field for suitably large smoothing scales even at redshift 0, the ELB model can only account for some of the properties of the halo density field.

Relativistic effects and primordial non-Gaussianity in the galaxy bias

When dealing with observables, one needs to generalize the bias relation between the observed galaxy fluctuation field to the underlying matter distribution in a gauge-invariant way. We provide such relation at second-order in perturbation theory adopting the local Eulerian bias model and starting from the observationally motivated uniform-redshift gauge. Our computation includes the presence of primordial non-Gaussianity. We show that large scale-dependent relativistic effects in the Eulerian bias arise independently from the presence of some primordial non-Gaussianity. Furthermore, the Eulerian bias inherits from the primordial non-Gaussianity not only a scale-dependence, but also a modulation with the angle of observation when sources with different biases are correlated.

Modeling the Galaxy Three‐Point Correlation Function

We present new predictions for the galaxy three-point correlation function (3PCF) using high-resolution dissipationless cosmological simulations of a flat ΛCDM universe which resolve galaxy-size halos and subhalos. We create realistic mock galaxy catalogs by assigning luminosities and colors to dark matter halos and subhalos, and we measure the reduced 3PCF as a function of luminosity and color in both real and redshift space. As galaxy luminosity and color are varied, we find small differences in the amplitude and shape dependence of the reduced 3PCF, at a level qualitatively consistent with recent measurements from the SDSS and 2dFGRS. We confirm that discrepancies between previous 3PCF measurements can be explained in part by differences in binning choices. We explore the degree to which a simple local bias model can fit the simulated 3PCF. The agreement between the model predictions and galaxy 3PCF measurements lends further credence to the straightforward association of galaxies with CDM halos and subhalos.

Locally Biased Galaxy Formation and Large‐Scale Structure

We examine the influence of the morphology-density relation and a wide range of simple models for biased galaxy formation on statistical measures of large-scale structure. We contrast the behavior of local biasing models, in which the efficiency of galaxy formation is determined by the density, geometry, or velocity dispersion of the local mass distribution, with that of nonlocal biasing models, in which galaxy formation is modulated coherently over scales larger than the galaxy correlation length. If morphological segregation of galaxies is governed by a local morphology-density relation, then the correlation function of E/S0 galaxies should be steeper and stronger than that of spiral galaxies on small scales, as observed, while on large scales the E/S0 and spiral galaxies should have correlation functions with the same shape but different amplitudes. Similarly, all of our local bias models produce scale-independent amplification of the correlation function and power spectrum in the linear and mildly nonlinear regimes; only a nonlocal biasing mechanism can alter the shape of the power spectrum on large scales. Moments of the biased galaxy distribution retain the hierarchical pattern of the mass moments, but biasing alters the values and scale dependence of the hierarchical amplitudes S3 and S4. Pair-weighted moments of the galaxy velocity distribution are sensitive to the details of the bias prescription even if galaxies have the same local velocity distribution as the underlying dark matter. The nonlinearity of the relation between galaxy density and mass density depends on the biasing prescription and the smoothing scale, and the scatter in this relation is a useful diagnostic of the physical parameters that determine the bias. While the assumption that galaxy formation is governed by local physics leads to some important simplifications on large scales, even local biasing is a multifaceted phenomenon whose impact cannot be described by a single parameter or function. The sensitivity of galaxy clustering statistics to the details of galaxy biasing is an obstacle to testing cosmological models, but it is an equally significant asset for testing physical theories of galaxy formation against data from redshift and peculiar velocity surveys.

Bias and Hierarchical Clustering

It is now well established that galaxies are biased tracers of the distribution of matter, although it is still not known what form this bias takes. In local bias models, the propensity for a galaxy to form at a point depends only on the overall density of matter at that point. Hierarchical scaling arguments allow one to build a fully specified model of the underlying distribution of matter and to explore the effects of local bias in the regime of strong clustering. Using a generating function method developed by Bernardeau & Schaeffer, we show that hierarchical models lead one directly to the conclusion that a local bias does not alter the shape of the galaxy correlation function relative to the matter correlation function on large scales. This provides an elegant extension of a result first obtained by Coles for Gaussian underlying fields and confirms the conclusions of Scherrer & Weinberg obtained using a different approach. We also argue that particularly dense regions in a hierarchical density field display a form of bias that is different from that obtained by selecting such peaks in Gaussian fields: they are themselves hierarchically distributed with scaling parameters Sp = p(p-2). This kind of bias is also factorizable, thus in principle furnishing a simple test of this class of models.

Stochasticity of Bias and Nonlocality of Galaxy Formation: Linear Scales

If one wants to represent the galaxy number density at some point in terms of only the mass density at the same point, stochasticity appears in such a relation, which is referred to as bias. The stochasticity is there because the galaxy number density is not merely a local function of a mass density field but is also a nonlocal functional. Thus, the phenomenological stochasticity of the bias should be accounted for by nonlocal features of galaxy formation processes. Based on mathematical arguments, we show that there are simple relations between biasing and nonlocality on linear scales of density fluctuations and that the stochasticity in Fourier space does not exist on linear scales under a certain condition, even if galaxy formation itself is a complex nonlinear and nonlocal process. The stochasticity in real space, however, arises from the scale dependence of the bias parameter, b. As examples, we derive the stochastic bias parameters of simple nonlocal models of galaxy formation, i.e., the local Lagrangian bias models, the cooperative model, and the peak model. We show that the stochasticity in real space is also weak, except on scales of nonlocality of galaxy formation. Therefore, we do not have to worry too much about the stochasticity on linear scales, especially in Fourier space, even if we do not know the details of the galaxy formation process.