Sheth, Ravi K.
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Publication Large Scale Structure and Galaxies(2009-05-18) Sheth, Ravi K.These notes sketch the motivation for and ingredients of the Halo Model of nonlinear and biased structures in the Universe. A key part of this approach is the relation between halo abundances and their large scale clustering. These come from the excursion set approach, so I have taken the opportunity to collect together all the formulae associated with this approach into one place. These include expressions for: the unconditional mass function, the conditional mass function, the environmental dependence of the mass function, halo bias, merger rates, creation anddestruction rates, the distribution of half-mass assembly times, masses and mass at fixed assembly time. In addition, I discuss how the approach can be used to describe voids, filaments and sheets, as well as the nonlinear counts in cells distribution, and provide analytic formulae for a number of these statistics. Together these formulae show that, in hierarchical models: massive halos assemble their mass later than low mass halos; halos which assemble their mass abnormally late for their mass will tend to have experienced a recent major merger; if one is interested in the mass assembled in pieces which are above some mininum mass, then this happens earlier for the more massive halos; for similar reasons, the mass fraction in pieces which are between a fixed mass range reaches a maximum at higher redshifts for halos which are more massive today. The first trend may explain why the oldest stars tend to sit in massive objects; the second may be why star formation in massive objects ended earlier. This approach also shows that the mass function in dense regions should be ‘top-heavy’, and that more massive halos should be more strongly clustered. If galaxy properties are determined primarily by the mass of their parent halo, then many observed correlations with environment are a simple consequence of these trends. Finally, I summarize the Halo Model of galaxy clustering. I discuss how it describes typedependent clustering, particularly dependence on luminosity and color, and sketch how to use it to build accurate mock catalogs which include information about stellar mass, dust, and star formation history.Publication Modeling Scale-Dependent Bias on the Baryonic Acoustic Scale with the Statistics of Peaks of Gaussian Random Fields(2010-11-23) Desjacques, Vincent; Crocce, Martin; Scoccimarro, Roman; Sheth, Ravi K.Models of galaxy and halo clustering commonly assume that the tracers can be treated as a continuous field locally biased with respect to the underlying mass distribution. In the peak model pioneered by Bardeen et al. [Astrophys. J. 304, 15 (1986)], one considers instead density maxima of the initial, Gaussian mass density field as an approximation to the formation site of virialized objects. In this paper, the peak model is extended in two ways to improve its predictive accuracy. First, we derive the two-point correlation function of initial density peaks up to second order and demonstrate that a peak-background split approach can be applied to obtain the k-independent and k-dependent peak bias factors at all orders. Second, we explore the gravitational evolution of the peak correlation function within the Zel’dovich approximation. We show that the local (Lagrangian) bias approach emerges as a special case of the peak model, in which all bias parameters are scale independent and there is no statistical velocity bias. We apply our formulas to study how the Lagrangian peak biasing, the diffusion due to large scale flows, and the mode coupling due to nonlocal interactions affect the scale dependence of bias from small separations up to the baryon acoustic oscillation (BAO) scale. For 2σ density peaks collapsing at z = 0.3, our model predicts a ~5% residual scale-dependent bias around the acoustic scale that arises mostly from first order Lagrangian peak biasing (as opposed to second order gravity mode coupling). We also search for a scale dependence of bias in the large scale autocorrelation of massive halos extracted from a very large N-body simulation provided by the MICE Collaboration. For halos with mass M ≳ 1014M⊙/h, our measurements demonstrate a scale-dependent bias across the BAO feature which is very well reproduced by a prediction based on the peak model.Publication Scale Dependence of Halo and Galaxy Bias: Effects in Real Space(2007-03-20) Smith, Robert E.; Sheth, Ravi K.; Scoccimarro, RománPublication Spherical Collapse and Cluster Counts in Modified Gravity Models(2009-04-08) Martino, Matthew C.; Sheth, Ravi K.; Stabenau, Hans F.Modifications to the gravitational potential affect the nonlinear gravitational evolution of large scale structures in the Universe. To illustrate some generic features of such changes, we study the evolution of spherically symmetric perturbations when the modification is of Yukawa type; this is nontrivial, because we should not and do not assume that Birkhoff’s theorem applies. We then show how to estimate the abundance of virialized objects in such models. Comparison with numerical simulations shows reasonable agreement: When normalized to have the same fluctuations at early times, weaker large scale gravity produces fewer massive halos. However, the opposite can be true for models that are normalized to have the same linear theory power spectrum today, so the abundance of rich clusters potentially places interesting constraints on such models. Our analysis also indicates that the formation histories and abundances of sufficiently low mass objects are unchanged from standard gravity. This explains why simulations have found that the nonlinear power spectrum at large k is unaffected by such modifications to the gravitational potential. In addition, the most massive objects in models with normalized cosmic microwave background and weaker gravity are expected to be similar to the high-redshift progenitors of the most massive objects in models with stronger gravity. Thus, the difference between the cluster and field galaxy populations is expected to be larger in models with stronger large scale gravity.Publication Nonlocal Lagrangian bias(2013-04-03) Sheth, Ravi K.; Chan, Kwan Chuen; Scoccimarro, RománHalos are biased tracers of the dark matter distribution. It is often assumed that the initial patches from which halos formed are locally biased with respect to the initial fluctuation field, meaning that the halo-patch fluctuation field can be written as a Taylor series in the dark matter density fluctuation field. If quantities other than the local density influence halo formation, then this Lagrangian bias will generically be nonlocal; the Taylor series must be performed with respect to these other variables as well. We illustrate the effect with Monte Carlo simulations of a model in which halo formation depends on the local shear (the quadrupole of perturbation theory) and provide an analytic model that provides a good description of our results. Our model, which extends the excursion set approach to walks in more than one dimension, works both when steps in the walk are uncorrelated, as well as when there are correlations between steps. For walks with correlated steps, our model includes two distinct types of nonlocality: one is due to the fact that the initial density profile around a patch which is destined to form a halo must fall sufficiently steeply around it—this introduces kdependence to even the linear bias factor, but otherwise only affects the monopole of the clustering signal. The other type of nonlocality is due to the surrounding shear field; this affects the quadratic and higher-order bias factors and introduces an angular dependence to the clustering signal. In both cases, our analysis shows that these nonlocal Lagrangian bias terms can be significant, particularly for massive halos; they must be accounted for in, e.g., analyses of higher-order clustering in Lagrangian or Eulerian space. Comparison of our predictions with measurements of the halo bispectrum in simulations is encouraging. Although we illustrate these effects using halos, our analysis and conclusions also apply to the other constituents of the cosmic web—filaments, sheets and voids.Publication Gravity and Large-Scale Nonlocal Bias(2012-04-05) Chan, Kwan C; Scoccimarro, Román; Sheth, Ravi K.For Gaussian primordial fluctuations the relationship between galaxy and matter overdensities, bias, is most often assumed to be local at the time of observation in the large-scale limit. This hypothesis is however unstable under time evolution, we provide proofs under several (increasingly more realistic) sets of assumptions. In the simplest toy model galaxies are created locally and linearly biased at a single formation time, and subsequently move with the dark matter (no velocity bias) conserving their comoving number density (no merging). We show that, after this formation time, the bias becomes unavoidably nonlocal and nonlinear at large scales.We identify the nonlocal gravitationally induced fields in which the galaxy overdensity can be expanded, showing that they can be constructed out of the invariants of the deformation tensor (Galileons), the main signature of which is a quadrupole field in second-order perturbation theory. In addition, we show that this result persists if we include an arbitrary evolution of the comoving number density of tracers.We then include velocity bias, and show that new contributions appear; these are related to the breaking of Galilean invariance of the bias relation, a dipole field being the signature at second order. We test these predictions by studying the dependence of halo overdensities in cells of fixed dark matter density: measurements in simulations show that departures from the mean bias relation are strongly correlated with the nonlocal gravitationally induced fields identified by our formalism, suggesting that the halo distribution at the present time is indeed more closely related to the mass distribution at an earlier rather than present time. However, the nonlocality seen in the simulations is not fully captured by assuming local bias in Lagrangian space. The effects on nonlocal bias seen in the simulations are most important for the most biased halos, as expected from our predictions. Accounting for these effects when modeling galaxy bias is essential for correctly describing the dependence on triangle shape of the galaxy bispectrum, and hence constraining cosmological parameters and primordial non-Gaussianity. We show that using our formalism we remove an important systematic in the determination of bias parameters from the galaxy bispectrum, particularly for luminous galaxies.Publication Motion of the Acoustic Peak in the Correlation Function(2008-02-25) Smith, Robert E.; Sheth, Ravi K.; Scoccimarro, RománThe 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, nonlinear 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 [Gpc3h-3]. A toy-model calculation shows that if the nonlinear corrections simply smooth the acoustic peak, then this gives rise to an ‘‘apparent’’ shifting to smaller scales. However if tilts in the broadband 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 for a LCDM model with σ8 = 0.9. 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.Publication Analytic Model for the Bispectrum of Galaxies in Redshift Space(2008-07-18) Smith, Robert E.; Sheth, Ravi K.; Scoccimarro, RománWe 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 nonlinear redshift-space distortions and also for the nonlinearity 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 measurements of the redshift-space bispectrum of dark matter from an ensemble of numerical simulations. On very large scales, k = 0.05h Mpc-1, we find reasonably good agreement between our halo model, PT and the data, to within the errors. On smaller scales, k = 0.1h Mpc-1, 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 = 0.5–1.0h Mpc-1, 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 ≳ 0.1h Mpc-1. 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, because different galaxy populations are naturally included into our description.