arXiv subjects

Robert E. Smith

Publications and source records attributed to Robert E. Smith.

54 records · Page 3Linked to original sources

arXiv2011

Nonlinear clustering in models with primordial non-Gaussianity: the halo model approach

We develop the halo model of large-scale structure as an accurate tool for probing primordial non-Gaussianity. In this study we focus on understanding the matter clustering at several redshifts. The primordial non-Gaussianity is modeled as a quadratic correction to the local Gaussian potential, and is characterized by the parameter f_NL. In our formulation of the halo model we pay special attention to the effect of halo exclusion, and show that this can potentially solve the long standing problem of excess power on large scales in this model. The model depends on the mass function, clustering and density profiles of halos. We test these ingredients using a large ensemble of high-resolution Gaussian and non-Gaussian numerical simulations. In particular, we provide a first exploration of how density profiles change in the presence of primordial non-Gaussianities. We find that for f_NL positive/negative high mass halos have an increased/decreased core density, so being more/less concentrated than in the Gaussian case. We also examine the halo bias and show that, if the halo model is correct, then there is a small asymmetry in the scale-dependence of the bias on very large scales, which arises because the Gaussian bias must be renormalized. We show that the matter power spectrum is modified by ~2.5% and ~3.5% on scales k~1.0 h/Mpc at z=0 and z=1, respectively. Our halo model calculation reproduces the absolute amplitude to within 10% and the ratio of non-Gaussian to Gaussian spectra to within 1%. We also measure the matter correlation functions and find similarly good agreement between the model and the data. We anticipate that this modeling will be useful for constraining f_NL from measurements of the shear correlation function in future weak lensing surveys such as Euclid.

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arXiv2011

Testing the Warm Dark Matter paradigm with large-scale structures

We explore the impact of a LWDM cosmological scenario on the clustering properties of large-scale structure in the Universe. We do this by extending the halo model. The new development is that we consider two components to the mass density: one arising from mass in collapsed haloes, and the second from a smooth component of uncollapsed mass. Assuming that the nonlinear clustering of dark matter haloes can be understood, then from conservation arguments one can precisely calculate the clustering properties of the smooth component and its cross-correlation with haloes. We then explore how the three main ingredients of the halo calculations, the mass function, bias and density profiles are affected by WDM. We show that, relative to CDM: the mass function is suppressed by ~50%, for masses ~100 times the free-streaming mass-scale; the bias of low mass haloes can be boosted by up to 20%; core densities of haloes can be suppressed. We also examine the impact of relic thermal velocities on the density profiles, and find that these effects are constrained to scales r<1 kpc/h, and hence of little importance for dark matter tests, owing to uncertainties in the baryonic physics. We use our modified halo model to calculate the non-linear matter power spectrum, and find significant small-scale power in the model. However, relative to the CDM case, the power is suppressed. We then calculate the expected signal and noise that our set of LWDM models would give for a future weak lensing mission. We show that the models should in principle be separable at high significance. Finally, using the Fisher matrix formalism we forecast the limit on the WDM particle mass for a future full-sky weak lensing mission like Euclid or LSST. With Planck priors and using multipoles l<5000, we find that a lower limit of 2.6 keV should be easily achievable.

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arXiv2011

What do cluster counts really tell us about the Universe?

We study the covariance matrix of the cluster mass function in cosmology. We adopt a two-line attack: firstly, we employ the counts-in-cells framework to derive an analytic expression for the covariance of the mass function. Secondly, we use a large ensemble of N-body simulations in the LCDM framework to test this. Our theoretical results show that the covariance can be written as the sum of two terms: a Poisson term, which dominates in the limit of rare clusters; and a sample variance term, which dominates for more abundant clusters. Our expressions are analogous to those of Hu & Kravtsov (2003) for multiple cells and a single mass tracer. Calculating the covariance depends on: the mass function and bias of clusters, and the variance of mass fluctuations within the survey volume. The predictions show that there is a strong bin-to-bin covariance between measurements. In terms of the cross-correlation coefficient, we find r~0.5 for haloes with M<3e14 Msol at z=0. Comparison of these predictions with estimates from simulations shows excellent agreement. We use the Fisher matrix formalism to explore the cosmological information content of the counts. We compare the Poisson likelihood model, with the more realistic likelihood model of Lima & Hu (2004), and all terms entering the Fisher matrices are evaluated using the simulations. We find that the Poisson approximation should only be used for the rarest objects, M>3e14 Msol, otherwise the information content of a survey of size V~13.5 [Gpc/h]^3 would be overestimated, resulting in errors that are ~2 times smaller. As an auxiliary result, we show that the bias of clusters, obtained from the cluster-mass cross-variance, is linear on scales >50 Mpc/h, whereas that obtained from the auto-variance is nonlinear.

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arXiv2010

Minimizing the stochasticity of halos in large-scale structure surveys

In recent work (Seljak, Hamaus and Desjacques 2009) it was found that weighting central halo galaxies by halo mass can significantly suppress their stochasticity relative to the dark matter, well below the Poisson model expectation. In this paper we extend this study with the goal of finding the optimal mass-dependent halo weighting and use $N$-body simulations to perform a general analysis of halo stochasticity and its dependence on halo mass. We investigate the stochasticity matrix, defined as $C_{ij}\equiv<(δ_i -b_iδ_m)(δ_j-b_jδ_m)>$, where $δ_m$ is the dark matter overdensity in Fourier space, $δ_i$ the halo overdensity of the $i$-th halo mass bin and $b_i$ the halo bias. In contrast to the Poisson model predictions we detect nonvanishing correlations between different mass bins. We also find the diagonal terms to be sub-Poissonian for the highest-mass halos. The diagonalization of this matrix results in one large and one low eigenvalue, with the remaining eigenvalues close to the Poisson prediction $1/\bar{n}$, where $\bar{n}$ is the mean halo number density. The eigenmode with the lowest eigenvalue contains most of the information and the corresponding eigenvector provides an optimal weighting function to minimize the stochasticity between halos and dark matter. We find this optimal weighting function to match linear mass weighting at high masses, while at the low-mass end the weights approach a constant whose value depends on the low-mass cut in the halo mass function. Finally, we employ the halo model to derive the stochasticity matrix and the scale-dependent bias from an analytical perspective. It is remarkably successful in reproducing our numerical results and predicts that the stochasticity between halos and the dark matter can be reduced further when going to halo masses lower than we can resolve in current simulations.

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arXiv2010

An algorithm for the direct reconstruction of the dark matter correlation function from weak lensing and galaxy clustering

The clustering of matter on cosmological scales is an essential probe for studying the physical origin and composition of our Universe. To date, most of the direct studies have focused on shear-shear weak lensing correlations, but it is also possible to extract the dark matter clustering by combining galaxy-clustering and galaxy-galaxy-lensing measurements. In this study we develop a method that can constrain the dark matter correlation function from galaxy clustering and galaxy-galaxy-lensing measurements, by focusing on the correlation coefficient between the galaxy and matter overdensity fields. To generate a mock galaxy catalogue for testing purposes, we use the Halo Occupation Distribution approach applied to a large ensemble of N-body simulations to model pre-existing SDSS Luminous Red Galaxy sample observations. Using this mock catalogue, we show that a direct comparison between the excess surface mass density measured by lensing and its corresponding galaxy clustering quantity is not optimal. We develop a new statistic that suppresses the small-scale contributions to these observations and show that this new statistic leads to a cross-correlation coefficient that is within a few percent of unity down to 5 Mpc/h. Furthermore, the residual incoherence between the galaxy and matter fields can be explained using a theoretical model for scale-dependent bias, giving us a final estimator that is unbiased to within 1%. We also perform a comprehensive study of other physical effects that can affect the analysis, such as redshift space distortions and differences in radial windows between galaxy clustering and weak lensing observations. We apply the method to a range of cosmological models and show the viability of our new statistic to distinguish between cosmological models.

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arXiv2010

Confirmation of general relativity on large scales from weak lensing and galaxy velocities

Although general relativity underlies modern cosmology, its applicability on cosmological length scales has yet to be stringently tested. Such a test has recently been proposed, using a quantity, EG, that combines measures of large-scale gravitational lensing, galaxy clustering and structure growth rate. The combination is insensitive to 'galaxy bias' (the difference between the clustering of visible galaxies and invisible dark matter) and is thus robust to the uncertainty in this parameter. Modified theories of gravity generally predict values of EG different from the general relativistic prediction because, in these theories, the 'gravitational slip' (the difference between the two potentials that describe perturbations in the gravitational metric) is non-zero, which leads to changes in the growth of structure and the strength of the gravitational lensing effect3. Here we report that EG = 0.39 +/- 0.06 on length scales of tens of megaparsecs, in agreement with the general relativistic prediction of EG $\approx$ 0.4. The measured value excludes a model within the tensor-vector-scalar gravity theory, which modifies both Newtonian and Einstein gravity. However, the relatively large uncertainty still permits models within f(R) theory, which is an extension of general relativity. A fivefold decrease in uncertainty is needed to rule out these models.

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arXiv2010

Precision cluster mass determination from weak lensing

Weak gravitational lensing has been used extensively in the past decade to constrain the masses of galaxy clusters, and is the most promising observational technique for providing the mass calibration necessary for precision cosmology with clusters. There are several challenges in estimating cluster masses, particularly (a) the sensitivity to astrophysical effects and observational systematics that modify the signal relative to the theoretical expectations, and (b) biases that can arise due to assumptions in the mass estimation method, such as the assumed radial profile of the cluster. All of these challenges are more problematic in the inner regions of the cluster, suggesting that their influence would ideally be suppressed for the purpose of mass estimation. However, at any given radius the differential surface density measured by lensing is sensitive to all mass within that radius, and the corrupted signal from the inner parts is spread out to all scales. We develop a new statistic that is ideal for estimation of cluster masses because it completely eliminates mass contributions below a chosen scale (which we suggest should be about 20 per cent of the virial radius), and thus reduces sensitivity to systematic and astrophysical effects. We use simulated and analytical profiles to quantify systematic biases on the estimated masses for several standard methods of mass estimation, finding that these can lead to significant mass biases that range from ten to over fifty per cent. The mass uncertainties when using our new statistic are reduced by up to a factor of ten relative to the standard methods, while only moderately increasing the statistical errors. This new method of mass estimation will enable a higher level of precision in future science work with weak lensing mass estimates for galaxy clusters.

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arXiv2009

Cosmological perturbation theory for baryons and dark matter I: one-loop corrections in the RPT framework

We generalize the `renormalized' perturbation theory (RPT) formalism of Crocce & Scoccimarro (2006a) to deal with multiple fluids in the Universe and here we present calculations up to the one-loop level. We apply the approach to the non-linear evolution of baryon and cold dark matter (CDM) perturbations, evolving from distinct sets of initial conditions. In current models of structure formation, it is standard practice to treat baryons and CDM as an effective single component fluid. In this approximation, one uses a weighed sum of late-time baryon and CDM transfer functions to set initial conditions. Here, we explore whether this approach can be used for high precision work. We show that, even for a pure linear treatment, there is a large-scale scale-dependent bias between baryons and CDM for WMAP5 cosmology. This bias is >1% until the present day, when it is driven towards unity through gravitational relaxation. Using the RPT formalism we test this approximation in the non-linear regime, and show that the CDM power spectrum in the 2-component fluid differs from that obtained from a 1-component fluid by ~3% on scales of order k~0.05 h/Mpc at z=10, and by ~0.5% at z=0. However, for the case of baryons the situation is worse and we find that the power spectrum is suppressed by ~15% on scales k~0.05 h/Mpc at z=10, and by ~3-5% at z=0, relative to the total matter. Importantly, besides the suppression of the spectrum, baryonic acoustic oscillations (BAO) are amplified for baryon and damped for CDM spectra. Thus, high precision modeling of baryons can not be probed through CDM only simulations; detection significance of BAO will be amplified in probes that study baryonic matter. Total mass can be modeled accurately using a 1-component fluid approach at all times. (Abridged)

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arXiv2009

The impact of correlated projections on weak lensing cluster counts

Large-scale structure projections are an obstacle in converting the shear signal of clusters detected in weak-lensing maps into virial masses. However, this step is not necessary for constraining cosmology with the shear-peak abundance, if we are able to predict its amplitude. We generate a large ensemble of N-body simulations spanning four cosmological models, with total volume V~1 (Gpc/h)^3 per model. Variations to the matter density parameter and amplitude of fluctuations are considered. We measure the abundance of peaks in the mass density projected in ~100 Mpc/h slabs to determine the impact of structures spatially correlated with the simulation clusters, identified by the 3D friends-of-friends algorithm. The halo model shows that the choice of the smoothing filter for the density field is important in reducing the contribution of correlated projections to individual halo masses. Such contributions are less than 2% in the case of the optimal, compensated filter used throughout this analysis. We measure the change in the mass of peaks when projected in slabs of various thicknesses. Peaks in slabs of 26 Mpc/h and 102 Mpc/h suffer an average mass change of less than 2% compared to their mass in slabs of 51 Mpc/h. We then explore the cosmology dependence of the projected-peak mass function, and find that, for a wide range of slab thicknesses (<500 Mpc/h), it scales with cosmology in exactly the same way as the 3D friends-of-friends mass function and the Sheth-Tormen formula. This extends the earlier result of Marian et al. (2009). Finally, we show that for all cosmological models considered, the low and intermediate mass bins of the peak abundance can be described using a modified Sheth-Tormen functional form to within 10%-20% accuracy.

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arXiv2009

Impact of scale-dependent bias and nonlinear evolution on the ISW

I summarize recent results from Smith, Hernandez-Monteagudo & Seljak (2009), a study of the impact of nonlinear evolution of gravitational potentials in the LCDM model on the Integrated Sachs-Wolfe (ISW) contribution to the cross-power spectrum of the CMB and a set of biased tracers of the mass. We use a large ensemble of N-body simulations to directly follow the potentials and compare the results to analytic perturbation theory (PT) methods. The PT predictions match our results to high precision for k<0.2 Mpc/h. We analyze the CMB-density tracer cross-spectrum using simulations and renormalized bias PT, and find good agreement. The usual assumption is that nonlinear evolution enhances the growth of structure and counteracts the linear ISW on small scales, leading to a change in sign of the CMB-LSS cross-spectrum at small scales. However, PT analysis suggests that this trend reverses at late times when the logarithmic growth rate f=d ln D/d ln a<1/2 or Omega_m (z)<0.3. Numerical results confirm these expectations and we find nonlinear enhancement of the ISW signal on small scales at late-times. On computing the total contribution to the angular spectrum, we find that nonlinearity and scale dependence of the bias are unable to influence the signal-to-noise of the current and future measurements.

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arXiv2009

The cosmology dependence of weak lensing cluster counts

We present the main results of a numerical study of weak lensing cluster counting. We examine the scaling with cosmology of the projected-density-peak mass function. Our main conclusion is that the projected-peak and the three-dimensional mass functions scale with cosmology in an astonishingly close way. This means that, despite being derived from a two-dimensional field, the weak lensing cluster abundance can be used to constrain cosmology in the same way as the three-dimensional mass function probed by other types of surveys.

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arXiv2009

Covariance of cross-correlations: towards efficient measures for large-scale structure

We study the covariance of the cross-power spectrum of different tracers for the large-scale structure. We develop the counts-in-cells framework for the multi-tracer approach, and use this to derive expressions for the full non-Gaussian covariance matrix. We show, that for the usual auto-power statistic, besides the off-diagonal covariance generated through gravitational mode-coupling, the discreteness of the tracers and their associated sampling distribution can generate strong off-diagonal covariance, and that this becomes the dominant source of covariance as k>>k_f=2 pi/L. On comparison with the derived expressions for the cross-power covariance, we show that the off-diagonal terms can be suppressed, if one cross-correlates a high tracer-density sample with a low one. Taking the effective estimator efficiency to be proportional to the signal-to-noise ratio (SN), we show that, to probe clustering as a function of physical properties of the sample, i.e. cluster mass or galaxy luminosity, then the cross-power approach can out perform the auto-power one by factors of a few. We confront the theory with measurements of the mass-mass, halo-mass, and halo-halo power spectra from a large ensemble of N-body simulations. We show that there is a significant SN advantage to be gained from using the cross-power approach when studying the bias of rare haloes. The analysis is repeated in configuration space and again SN improvement is found. We estimate the covariance matrix for these samples, and find strong off-diagonal contributions. The covariance depends on halo mass, with higher mass samples having stronger covariance. In agreement with theory, we show that the covariance is suppressed for the cross-power. This work points the way towards improved estimators for clustering studies.

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arXiv2009

Impact of Scale Dependent Bias and Nonlinear Structure Growth on the ISW Effect: Angular Power Spectra

We investigate the impact of nonlinear evolution of the gravitational potentials in the LCDM model on the Integrated Sachs-Wolfe (ISW) contribution to the CMB temperature power spectrum, and on the cross-power spectrum of the CMB and a set of biased tracers of the mass. We use an ensemble of N-body simulations to directly follow the potentials and compare results to perturbation theory (PT). The predictions from PT match the results to high precision for k<0.2 h/Mpc. We compute the nonlinear corrections to the angular power spectrum and find them to be <10% of linear theory for l<100. These corrections are swamped by cosmic variance. On scales l>100 the departures are more significant, however the CMB signal is more than a factor 10^3 larger at this scale. Nonlinear ISW effects therefore play no role in shaping the CMB power spectrum for l<1500. We analyze the CMB--density tracer cross-spectrum using simulations and renormalized bias PT, and find good agreement. The usual assumption is that nonlinear evolution enhances the growth of structure and counteracts linear ISW on small scales, leading to a change in sign of the CMB-LSS cross-spectrum at small scales. However, PT analysis suggests that this trend reverses at late times when the logarithmic growth rate f(a)=dlnD/dlna<0.5 or om_m(a)<0.3. Numerical results confirm these expectations and we find no sign change in ISW-LSS cross-power for low redshifts. Corrections due to nonlinearity and scale dependence of the bias are found to be <10% for l<100, therefore below the S/N of the current and future measurements. Finally, we estimate the CMB--halo cross-correlation coefficient and show that it can be made to match that for CMB--dark matter to within 5% for thin redshift shells, mitigating the need to model bias evolution.

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arXiv2008

Statistical properties of the linear tidal shear

Large-scale structures originate from coherent motions induced by inhomogeneities in the primeval gravitational potential. Here, we investigate the two-point statistics of the second derivative of the potential, the tidal shear, under the assumption of Gaussianity. We derive an exact closed form expression for the angular averaged, two-point distribution of the shear components which is valid for an arbitrary Lagrangian separation. This result is used to write down the two-point statistics of the shear eigenvalues in compact form. Next, we examine the large-scale asymptotics of the correlation of the shear eigenvalues, and the alignment of the principal axes. The analytic results are in good agreement with measurements obtained from random realizations of the gravitational potential. Finally, we show that a number of two-point distributions of the shear eigenvalues are well approximated by Gaussian bivariates over a wide range of separation and smoothing scales. We speculate that the Gaussian approximation also holds for multiple point distributions of the shear eigenvalues. It is hoped that these results will be relevant for studies aimed at describing the properties of the (evolved) matter distribution in terms of the statistics of the primordial shear field.

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arXiv2008

An analytic model for the bispectrum of galaxies in redshift space

We develop an analytic theory for the redshift space bispectrum of dark matter, haloes and galaxies. This is done within the context of the halo model of structure formation, as this allows for the self-consistent inclusion of linear and non-linear redshift space distortions and also for the non-linearity of the halo bias. The model is applicable over a wide range of scales: on the largest scales the predictions reduce to those of the standard perturbation theory (PT); on smaller scales they are determined primarily by the nonlinear virial velocities of galaxies within haloes, and this gives rise to the U-shaped anisotropy in the reduced bispectrum -- a finger print of the Finger-Of-God distortions. We then confront the predictions with bispectrum measurements from an ensemble of numerical simulations. On very large scales, k=0.05 h/Mpc, we find reasonably good agreement between our Halo Model, PT and the data, to within the errors. On smaller scales, k=0.1 h/Mpc, the measured bispectra differ from the PT at the level of 10-20%, especially for colinear triangle configurations. The Halo Model predictions improve over PT, but are accurate to no better than 10%. On smaller scales k=0.5-1.0 h/Mpc, our model provides a significant improvement over PT, which breaks down. This implies that studies which use the lowest order PT to extract galaxy bias information are not robust on scales k>0.1 h/Mpc. The analytic and simulation results also indicate that there is no observable scale for which the configuration dependence of the reduced bispectrum is constant--hierarchical models for the higher order correlation functions in redshift space are unlikely to be useful. It is hoped that our model will facilitate extraction of information from large-scale structure surveys of the Universe.

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arXiv2007

`Eppur Si Muove': On The Motion of the Acoustic Peak in the Correlation Function

The baryonic acoustic signature in the large-scale clustering pattern of galaxies has been detected in the two-point correlation function. Its precise spatial scale has been forwarded as a rigid-rod ruler test for the space-time geometry, and hence as a probe for tracking the evolution of Dark Energy. Percent-level shifts in the measured position can bias such a test and erode its power to constrain cosmology. This paper addresses some of the systematic effects that might induce shifts: namely non-linear corrections from matter evolution, redshift space distortions and biasing. We tackle these questions through analytic methods and through a large battery of numerical simulations, with total volume of the order ~100[Gpc\h]^3. A toy-model calculation shows that if the non-linear corrections simply smooth the acoustic peak, then this gives rise to an `apparent' shifting to smaller scales. However if tilts in the broad band power spectrum are induced then this gives rise to more pernicious `physical' shifts. Our numerical simulations show evidence of both: in real space and at z=0, for the dark matter we find percent level shifts; for haloes the shifts depend on halo mass, with larger shifts being found for the most biased samples, up to 3%. From our analysis we find that physical shifts are greater than ~0.4% at z=0. In redshift space these effects are exacerbated, but at higher redshifts are alleviated. We develop an analytical model to understand this, based on solutions to the pair conservation equation using characteristic curves. When combined with modeling of pairwise velocities the model reproduces the main trends found in the data. The model may also help to unbias the acoustic peak.

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arXiv2007

The Scale Dependence of Halo and Galaxy Bias: Effects in Real Space

We examine the scale dependence of dark matter, halo and galaxy clustering on very large scales (0.01<k[h/Mpc]<0.15), due to non-linear effects from dynamics and halo bias. We pursue a two line offensive: high resolution numerical simulations are used to establish several new results, and an analytic model is developed to understand their origins. Our simulations show: (i) that the z=0 dark matter power spectrum is suppressed relative to linear theory by ~5% on scales (0.05<k[h/Mpc]<0.075); (ii) that, indeed, halo bias is non-linear over the scales we probe and that the scale dependence is a strong function of halo mass. High mass haloes show no suppression of power on scales (k<0.07[h/Mpc]), and only show amplification on smaller scales, whereas low mass haloes show strong, ~5-10%, suppression over the range (0.05 <k[h/Mpc] <0.15). Our results have relevance for studies of the baryon acoustic oscillation features. Non-linear mode-mode coupling: (i) damps these features on progressively larger scales as halo mass increases; (ii) produces small shifts in the positions of the peaks and troughs which depend on halo mass. Our analytic model is described in the language of the `halo-model'. However, for the first time the halo-halo clustering term is propagated into the non-linear regime using `1-loop' perturbation theory and a non-linear halo bias model. We show that, with bias parameters derived from simulations, the model predictions are in agreement with the numerical results. We then use the model to explore the scale dependence of galaxies of different colour and find significant differences between the power spectra of the two populations. Thus understanding the scale dependent bias for a given galaxy sample will be crucial for deriving accurate cosmological constraints. (Abridged)

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arXiv2006

Systematic effects in the sound horizon scale measurements

We investigate three potential sources of bias in distance estimations made assuming that a very simple estimator of the baryon acoustic oscillation (BAO) scale provides a standard ruler. These are the effects of the non-linear evolution of structure, scale-dependent bias and errors in the survey window function estimation. The simple estimator used is the peak of the smoothed correlation function, which provides a variance in the BAO scale that is close to optimal, if appropriate low-pass filtering is applied to the density field. While maximum-likelihood estimators can eliminate biases if the form of the systematic error is fully modeled, we estimate the potential effects of un- or mis-modelled systematic errors. Non-linear structure growth using the Smith et al. (2003) prescription biases the acoustic scale by <0.3% at z>1 under the correlation-function estimator. The biases due to representative but simplistic models of scale-dependent galaxy bias are below 1% at z>1 for bias behaviour in the realms suggested by halo model calculations, which is expected to be below statistical errors for a 1000 sq.degs. spectroscopic survey. The distance bias due to a survey window function errors is given in a simple closed form and it is shown it has to be kept below 2% not to bias acoustic scale more than 1% at z=1, although the actual tolerance can be larger depending upon galaxy bias. These biases are comparable to statistical errors for ambitious surveys if no correction is made for them. We show that RMS photometric zero-point errors (at limiting magnitude 25 mag) below 0.14 mag and 0.01 mag for redshift z=1 (red galaxies) and z=3 (Lyman-break galaxies), respectively, are required in order to keep the distance estimator bias below 1%.

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