Distance errors in cosmology
On estimating redshift and luminosity distributions in photometric redshift surveys
Ravi K. Sheth, MNRAS, 378, 709 (2007).
The luminosity functions of galaxies and quasars provide invaluable information about galaxy and quasar formation. Estimating the luminosity function from magnitude-limited samples is relatively straightforward, provided that the distances to the objects in the sample are known accurately; techniques for doing this have been available for about 30 yr. However, distances are usually known accurately for only a small subset of the sample. This is true of the objects in the Sloan Digital Sky Survey, and will be increasingly true of the next generation of deep multicolour photometric surveys. Estimating the luminosity function when distances are only known approximately (e.g. photometric redshifts are available, but spectroscopic redshifts are not) is more difficult. I describe two algorithms which can handle this complication: one is a generalization of the Vmax algorithm, and the other is a maximum likelihood approach. Because these methods account for uncertainties in the distance estimate, they impact a broader range of studies. For example, they are useful for studying the abundances of galaxies which are sufficiently nearby that the contribution of peculiar velocity to the spectroscopic redshift is not negligible, so only a noisy estimate of the true distance is available. In this respect, peculiar velocities and photometric redshift errors have similar effects. The methods developed here are also useful for estimating the stellar luminosity function in samples where accurate parallax distances are not available.
`Eppur Si Muove': On The Motion of the Acoustic Peak in the Correlation
R. E. Smith, R. Scoccimarro & Ravi K. Sheth, Phys. Rev. D,
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 105 [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' shift. Our numerical simulations show evidence of both: in real space and at z=0, we find that for the dark matter the shift is of order a few percent; for haloes the shifts depend on halo mass, with larger shifts being found for the most biased samples, roughly 3-5%. In redshift space these effects are exacerbated, but at higher redshifts are slightly 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.
The scale dependence of halo and galaxy bias: Effects in real space
R. E. Smith, R. Scoccimarro & Ravi K. Sheth, Phys. Rev. D, 75, 063512 (2007).
We examine the scale dependence of dark matter, halo and galaxy clustering on very large scales (0.01
Correlations with environment
Environment and the cosmic evolution of star formation
Ravi K. Sheth, R. Jimenez, B. Panter & A. F. Heavens,
ApJL, 650, L25 (2006).
We present a mark correlation analysis of the galaxies in the Sloan Digital Sky Survey using weights provided by MOPED. The large size of the sample permits statistically significant statements about how galaxies with different metallicities and star formation histories are spatially correlated. Massive objects formed a larger fraction of their stars at higher redshifts and over shorter timescales than did less massive objects (sometimes called down-sizing). We find that those galaxies which dominated the cosmic star formation at z~3 are predominantly in clusters today, whereas galaxies which dominate the star formation at z~0 inhabit substantially lower mass objects in less dense regions today. Hence, our results indicate that star formation and chemical enrichment occured first in the denser regions of the Universe, and moved to less dense regions at later times.
The luminosity weighted or marked correlation function
R. Skibba, Ravi K. Sheth, A. J. Connolly & R. Scranton
MNRAS, 369, 68 (2006).
We present measurements of the redshift-space luminosity-weighted or `marked' correlation function in the Sloan Digital Sky Survey (SDSS). These are compared with a model in which the luminosity function and luminosity dependence of clustering are the same as that observed, and in which the form of the luminosity-weighted correlation function is entirely a consequence of the fact that massive haloes populate dense regions. Our model is in good agreement with the measurements, indicating that the halo mass function in dense regions is top heavy; the correlation between halo mass and large-scale environment is the primary driver for correlations between galaxy properties and environment; and the luminosity of the central galaxy in a halo is different from (in general, brighter than) that of the other objects in the halo. Thus our measurement provides strong evidence for the accuracy of these three standard assumptions of galaxy formation models. These assumptions also form the basis of current halo-model-based interpretations of galaxy clustering.
When the same galaxies are weighted by their u-, g- or r-band luminosities, then the marked correlation function is stronger in the redder bands. When the weight is galaxy colour rather than luminosity, then the data suggest that close pairs of galaxies tend to have redder colours. This wavelength dependence of marked correlations is in qualitative agreement with galaxy formation models, and reflects the fact that the mean luminosity of galaxies in a halo depends more strongly on halo mass in the r-band than in u. The luminosity and colour dependence we find are consistent with models in which the galaxy population in clusters is more massive than the population in the field. If the u-band luminosity is a reliable tracer of star formation, then our results suggest that cluster galaxies have lower star formation rates.
Strong clustering of underdense regions and the environmental
dependence of clustering from Gaussian initial conditions
U. Abbas & Ravi K. Sheth, MNRAS, 378, 641 (2007).
We discuss two slightly counter-intuitive findings about the environmental dependence of clustering in the Sloan Digital Sky Survey. First, we find that the relation between clustering strength and density is not monotonic: galaxies in the densest regions are more strongly clustered than are galaxies in regions of moderate overdensity; galaxies in moderate overdensities are more strongly clustered than are those in moderate underdensities; but galaxies in moderate underdensities are less clustered than galaxies in the least dense regions. We argue that this is natural if clustering evolved gravitationally from a Gaussian field, since the highest peaks and lowest troughs in Gaussian fields are similarly clustered. The precise non-monotonic dependence of galaxy clustering on density is very well reproduced in a mock catalog which is based on a halo-model decomposition of galaxy clustering.
Second, the distribution of galaxy counts in our sample is rather well described by a Poisson cluster model. We show that, despite their Poisson nature, correlations with environment are expected in such models. More remarkably, the expected trends are very like those in standard models of halo bias, despite the fact that correlations with environment in these models arise purely from the fact that dense regions are dense because they happen to host more massive halos. This is in contrast to the usual analysis which assumes that it is the large scale environment which determines the halo mass function.
- The environmental dependence of galaxy clustering in the Sloan Digital Sky Survey
U. Abbas & Ravi K. Sheth, MNRAS, 372, 1749 (2006).
A generic prediction of hierarchical clustering models is that the mass function of dark haloes in dense regions in the Universe should be top-heavy. We provide a novel test of this prediction using a sample of galaxies drawn from the Sloan Digital Sky Survey. To perform the test, we compare measurements of galaxy clustering in dense and underdense regions. We find that galaxies in dense regions cluster significantly more strongly than those in less dense regions. This is true over the entire 0.1-30 Mpc pair separation range for which we can make accurate measurements. We make similar measurements in realistic mock catalogs in which the only environmental effects are those which arise from the predicted correlation between halo mass and environment. We also provide an analytic halo-model based calculation of the effect. Both the mock catalogs and the analytic calculation provide rather good descriptions of the SDSS measurements. Thus, our results provide strong support for hierarchical models. They suggest that, unless care is taken to study galaxies at fixed mass, correlations between galaxy properties and the surrounding environment are almost entirely due to more fundamental correlations between galaxy properties and host halo mass, and between halo mass and environment.
On the environmental dependence of halo formation
Ravi K. Sheth & Giuseppe Tormen, MNRAS, 350, 1385 (2004).
A generic prediction of hierarchical gravitational clustering models is that the distribution of halo formation times should depend relatively strongly on halo mass, massive haloes forming more recently, and depend only weakly, if at all, on the large-scale environment of the haloes. We present a novel test of this assumption, which uses the statistics of weighted or `marked' correlations, which prove to be particularly well-suited to detecting and quantifying weak correlations with environment. We find that close pairs of haloes form at slightly higher redshifts than more widely separated halo pairs, suggesting that haloes in dense regions form at slightly earlier times than haloes of the same mass in less dense regions. The environmental trends we find are useful for models that relate the properties of galaxies to the formation histories of the haloes that surround them.
The Halo Model
- The halo model description of marked statistics
Ravi K. Sheth, MNRAS, 364, 796 (2005).
Marked statistics allow sensitive tests of how galaxy properties correlate with environment, as well as of how correlations between galaxy properties are affected by environment. A halo-model description of marked correlations is developed, which incorporates the effects which arise from the facts that typical galaxy marks (e.g. luminosity, colour, star formation rate, stellar mass) depend on the mass of the parent halo, and that massive haloes extend to larger radii and populate denser regions. Comparison with measured marked statistics in semi-analytic galaxy formation models shows good agreement on scales smaller than a megaparsec, and excellent agreement on larger scales. On scales smaller than a megaparsec, the halo-model calculation shows that marked statistics allow sensitive tests of whether or not central galaxies in haloes are a special population. A prescription for including more general mark gradients in the halo-model description is also provided. The formalism developed here is particularly well suited to interpretation of marked statistics in astrophysical data sets, because it is phrased in the same language that is currently used to interpret more standard measures of galaxy clustering.
The impact of halo shapes on the bispectrum in cosmology
R. E. Smith, P. I. R. Watts & Ravi K. Sheth, MNRAS, 365, 214 (2006).
Analytical expressions for the dark matter bispectrum are derived and subsequently evaluated through numerical integration. Two models for the ellipsoidal halo profiles are considered: a toy model designed to isolate the effects of halo shape on the clustering alone; and the more realistic model of Jing & Suto. For equilateral k-space triangles, we show that the predictions of the triaxial model are suppressed, relative to the spherical model, by up to ~7 and ~4 per cent for the two profiles, respectively. When the reduced bispectrum is considered as a function of triangle configuration, it is found to be highly sensitive to halo shapes on small scales. The generic features of our predictions are that, relative to the spherical halo model, the signal is suppressed for k-vector configurations that are close to equilateral triangles and boosted for configurations that are colinear. This appears to be a unique signature of halo triaxiality and potentially provides a means for measuring halo shapes in forthcoming cosmic shear surveys. The galaxy bispectrum is also explored. The functional form of the configuration-dependent bispectrum is, modulo an amplitude shift, not strongly sensitive to the exact form of the HOD, but is mainly determined by the halo shape. However, a combination of measurements made on different scales and for different k-space triangle configurations is sensitive to both halo shape and the HOD.
Substructure in dark matter halos: Towards a model of the abundance
and spatial distribution of subclumps
Ravi K. Sheth, MNRAS, 345, 1200 (2003).
I develop a model for the abundance and spatial distribution of
dark matter subclumps.
The model shows that subclumps of massive parent halos formed at
earlier times than subclumps of the same mass in lower mass parents;
equivalently, halos in dense regions at a given time formed earlier
than halos of the same mass in less dense regions. This may provide
the basis for interpreting recent observations which indicate that
the stellar populations of the most massive elliptical galaxies are
also the oldest.
- Substructure and the halo model of large-scale structure
Ravi K. Sheth & Bhuvnesh Jain, MNRAS, 345, 529 (2003).
We develop the formalism to include substructure in the halo
model of clustering. Real haloes are not likely to be perfectly smooth,
but have substructure which has, to date, been neglected in the halo
model - our formalism allows one to estimate the effects of this
substructure on measures of clustering. We derive expressions for the
two-point correlation function, the power-spectrum, the
cross-correlation between galaxies and mass, as well as higher order
clustering measures. Simple forms of the formulae are obtained for
the limit in which the size of the substructure and the mass fraction
in it is small. Inclusion of substructure allows for a more accurate
analysis of the statistical effects of gravitational lensing. It may
also bring the halo model predictions into better agreement with the
small-scale structure seen in recent high-resolution simulations of
- Halo models of large scale structure (Invited Review)
Asantha Cooray & Ravi K. Sheth, Physics Reports, 372, 1 (2002).
We review the formalism and applications of the halo-based description
of nonlinear gravitational clustering. In this approach, all mass is
associated with virialized dark matter halos; models of the number and
spatial distribution of the halos, and the distribution of dark matter
within each halo, are used to provide estimates of how the statistical
properties of large scale density and velocity fields evolve as a result
of nonlinear gravitational clustering.
We first describe the model, and demonstrate its accuracy
by comparing its predictions with exact results from numerical
simulations of nonlinear gravitational clustering.
We then present several astrophysical applications of the halo model:
these include models of the spatial distribution of galaxies,
the nonlinear velocity, momentum and pressure fields, descriptions
of weak gravitational lensing, and estimates of secondary contributions
to temperature fluctuations in the cosmic microwave background.
PTHalos: A fast method for generating mock galaxy distributions
R. Scoccimarro & Ravi K. Sheth, MNRAS, 329, 629 (2002).
an algorithm which is several orders of magnitude faster than n-body
simulations, but which is, nevertheless, rather accurate. The
algorithm combines perturbation theory with virialized halo models of
the nonlinear density and velocity fields. For two- and three-point
statistics the resulting fields are exact on large scales, and rather
accurate well into the nonlinear regime. We use this algorithm to
generate mock galaxy distributions from halo occupation numbers.
Dark halo abundances, shapes and merger histories
An improved model for the formation times of dark matter halos
C. Giocoli, J. Moreno, Ravi K. Sheth & G. Tormen, MNRAS, 376, 977 (2007).
A dark matter halo is said to have formed when at least half its mass hass been assembled into a single progenitor. With this definition, it is possible to derive a simple but useful analytic estimate of the distribution of halo formation times. The standard estimate of this distribution depends on the shape of the conditional mass function---the distribution of progenitor masses of a halo as a function of time. If the spherical collapse model is used to estimate the progenitor mass function, then the formation times one infers systematically underestimate those seen in numerical simulations of hierarchical gravitational clustering. We provide estimates of halo formation which may be related to an ellipsoidal collapse model. These estimates provide a substantially better description of the simulations. We also provide an alternative derivation of the formation time distribution which is based on the assumption that haloes increase their mass through binary mergers only. Our results are useful for models which relate halo structure to halo formation.
Ellipsoidal collapse and dark halo shapes
G. Rossi, Ravi K. Sheth & G. Tormen, MNRAS, submitted (2006).
The assumption that dark matter halos formed from an ellipsoidal
collapse allows one to model how the abundance of halos depends on
their mass. We show that this model is easily extended to study
halo shapes, thus providing a framework for describing how
halo shapes are correlated with larger-scale structures.
In the model, halos are predicted to be triaxial, with axis ratios
which are related to the shear field of the patch from which they
formed. Although the exact evolution of an ellipsoidal
perturbation must be solved numerically, we provide an accurate
analytic description which aids considerably in understanding
many features of the collapse process.
Comparison of the predicted axis-ratio distributions with those
measured in high resolution simulations show that our model is
accurate for M* halos. However, it predicts that more massive
halos should be more spherical, opposite to the trend seen in
- An excursion set model of hierarchical clustering:
ellipsoidal collapse and the moving barrier
Ravi K. Sheth & G. Tormen, MNRAS, 329, 61 (2002).
The excursion set approach allows one to estimate the abundance and spatial distribution of virialized dark matter haloes efficiently and accurately. The predictions of this approach depend on how the non-linear processes of collapse and virialization are modelled. We present simple analytic approximations that allow us to compare the excursion set predictions associated with spherical and ellipsoidal collapse. In particular, we present formulae for the universal unconditional mass function of bound objects and the conditional mass function which describes the mass function of the progenitors of haloes in a given mass range today. We show that the ellipsoidal collapse based moving barrier model provides a better description of what we measure in the numerical simulations than the spherical collapse based constant barrier model, although the agreement between model and simulations is better at large lookback times. Our results for the conditional mass function can be used to compute accurate approximations to the local-density mass function, which quantifies the tendency for massive haloes to populate denser regions than less massive haloes. This happens because low-density regions can be thought of as being collapsed haloes viewed at large lookback times, whereas high-density regions are collapsed haloes viewed at small lookback times. Although we have applied our analytic formulae only to two simple barrier shapes, we show that they are, in fact, accurate for a wide variety of moving barriers. We suggest how they can be used to study the case in which the initial dark matter distribution is not completely cold.
Ellipsoidal collapse and an improved model for the number and
spatial distribution of dark matter haloes
Ravi K. Sheth, H. J. Mo & G. Tormen, MNRAS, 323, 1 (2001).
The Press-Schechter, excursion set approach allows one to make predictions about the shape and evolution of the mass function of bound objects. The approach combines the assumption that objects collapse spherically with the assumption that the initial density fluctuations were Gaussian and small. The predicted mass function is reasonably accurate, although it has fewer high-mass and more low-mass objects than are seen in simulations of hierarchical clustering. We show that the discrepancy between theory and simulation can be reduced substantially if bound structures are assumed to form from an ellipsoidal, rather than a spherical, collapse. In the original, standard, spherical model, a region collapses if the initial density within it exceeds a threshold value, delta_sc. This value is independent of the initial size of the region, and since the mass of the collapsed object is related to its initial size, this means that delta_sc is independent of final mass. In the ellipsoidal model, the collapse of a region depends on the surrounding shear field, as well as on its initial overdensity. In Gaussian random fields, the distribution of these quantities depends on the size of the region considered. Since the mass of a region is related to its initial size, there is a relation between the density threshold value required for collapse and the mass of the final object. We provide a fitting function to this delta_ec(m) relation which simplifies the inclusion of ellipsoidal dynamics in the excursion set approach. We discuss the relation between the excursion set predictions and the halo distribution in high-resolution N-body simulations, and use our new formulation of the approach to show that our simple parametrization of the ellipsoidal collapse model represents an improvement on the spherical model on an object-by-object basis. Finally, we show that the associated statistical predictions, the mass function and the large-scale halo-to-mass bias relation, are also more accurate than the standard predictions.
Large scale bias and the peak background split
Ravi K. Sheth, G. Tormen, MNRAS, 308, 119 (1999).
Dark matter haloes are biased tracers of the underlying dark matter distribution. We use a simple model to provide a relation between the abundance of dark matter haloes and their spatial distribution on large scales. Our model shows that knowledge of the unconditional mass function alone is sufficient to provide an accurate estimate of the large scale bias factor. Then we use the mass function measured in numerical simulations of SCDM, OCDM and LCDM to compute this bias. Comparison with these simulations shows that this simple way of estimating the bias relation and its evolution is accurate for less massive haloes as well as massive ones. In particular, we show that haloes which are less/more massive than typical M* haloes at the time they form are more/less strongly clustered than formulae based on the standard Press-Schechter mass function predict.
The forest of merger history trees associated with the formation
of dark matter haloes
Ravi K. Sheth & G. Lemson, MNRAS, 305, 946 (1999).
We describe a simple, efficient algorithm that allows one to construct Monte Carlo realizations of merger histories of dark matter haloes. The algorithm is motivated by the excursion set model for the conditional and unconditional halo mass functions. The forest of trees constructed using this algorithm depends on the underlying power spectrum. For Poisson or white-noise initial power spectra, the forest has exactly the same properties as the ensemble of trees described by Sheth. In this case, many ensemble-averaged higher order statistics of the tree distribution can be computed analytically. For Gaussian initial conditions with more general power spectra, mean properties of our forests closely resemble the mean properties expected from the excursion set approach. For these more general initial conditions, our algorithm shows how to write down simple, analytic approximations to some higher order statistical quantities associated with the forest. These higher order statistics generated using our algorithm, and the associated analytic approximations, are in good agreement with what is measured in numerical simulations of hierarchical gravitational clustering.
The Excursion Set approach
- An excursion set model of the cosmic web
J. Shen, T. Abel, H. J. Mo & Ravi K. Sheth,
ApJ, 645, 783 (2006).
We discuss an analytic approach for modeling structure formation in sheets, filaments, and knots. This is accomplished by combining models of triaxial collapse with the excursion set approach: sheets are defined as objects that have collapsed along only one axis, filaments have collapsed along two axes, and halos are objects in which triaxial collapse is complete. In the simplest version of this approach, which we develop here, large-scale structure shows a clear hierarchy of morphologies: the mass in large-scale sheets is partitioned up among lower mass filaments, which themselves are made up of still lower mass halos. Our approach provides analytic estimates of the mass fraction in sheets, filaments, and halos and its evolution, for any background cosmological model and any initial fluctuation spectrum. In the currently popular LCDM model, our analysis suggests that more than 99 percent of the cosmic mass is in sheets, and 72 percent in filaments, with mass larger than 10^10 Msolar at the present time. For halos, this number is only 46 percent. Our approach also provides analytic estimates of how halo abundances at any given time correlate with the morphology of the surrounding large-scale structure and how halo evolution correlates with the morphology of large-scale structure.
- A hierarchy of voids: Much ado about nothing
Ravi K. Sheth & Rien van de Weygaert, MNRAS, 350, 517 (2004).
We present a model for the distribution of void sizes and its
evolution in the context of hierarchical scenarios of gravitational
structure formation. We find that at any cosmic epoch the voids
have a size distribution which is well-peaked about a characteristic
void size which evolves self-similarly in time. This is in distinct
contrast to the distribution of virialized halo masses which does
not have a small-scale cut-off.
In our model, the fate of voids is ruled by two processes.
The void-in-void process affects those voids which are embedded in
larger underdense regions: the evolution is effectively one in which a
larger void is made up by the mergers of smaller voids, and is
analogous to how massive clusters form from the mergers of less
The void-in-cloud process is unique to voids, and occurs to voids which
happen to be embedded within a larger scale overdensity: these voids
get squeezed out of existence as the overdensity collapses around
them. It is this second process which produces the cut-off at
small scales. We show that the problem can be mapped to a study of
random walks in the presence of two-barriers: one barrier is required
to account for voids-in-voids, and the other for voids-in-clouds.
An excursion set model for the distribution of dark matter and
dark matter haloes
Ravi K. Sheth, MNRAS, 300, 1057 (1998).
A model of the gravitationally evolved dark matter distribution, in the Eulerian space, is developed. It is a simple extension of the excursion set model that is commonly used to estimate the mass function of collapsed dark matter haloes. In addition to describing the evolution of the Eulerian space distribution of the haloes, the model allows one to describe the evolution of the dark matter itself. It can also be used to describe density profiles, on scales larger than the virial radius of these haloes, and to quantify the way in which matter flows in and out of Eulerian cells. When the initial Lagrangian space distribution is white noise Gaussian, the model suggests that the Inverse Gaussian distribution should provide a reasonably good approximation to the evolved Eulerian density field, in agreement with numerical simulations. Application of this model to clustering from more general Gaussian initial conditions is discussed at the end.
Coagulation/fragmentation and branching process models
Models of reversible coagulation and fragmentation
Ravi K. Sheth & J. Moreno, J. Phys. A, submitted (2006).
Merger-fragmentation models in which the fragmentation rate is the
time-reverse process of the coagulation rate are studied.
Explicit expressions for the evolution of the mass spectrum of linear
and branched polymers when coagulation and fragmentation occur by the
creation and deletion of randomly chosen bonds are provided.
At late times, the system evolves to an equilibrium state, provided
that, absent fragmentation, an infinite cluster would not have formed.
The effect of time-reverse fragmentation is to delay the onset of
gelation in systems which, absent fragmentation, would have formed
arbitrarily massive clusters in finite time. However, if the rate of
fragmentation is sufficiently large compared to coagulation, gelation
can be prevented altogether.
When this is the case, the system evolves to an equilibrium state,
the time to, and mass spectrum at, equilibrium being determined by
the ratio of merger and fragmentation rates.
A generalized model of fragmentation as reversible coagulation is
then proposed, and the generalized-linear and multiplicative
coagulation kernels are used as test cases. In these cases, the
effective coagulation rate has the same mass dependence as when
fragmentation was absent, but they are no longer independent of time.
From this we infer that the associated spectrum of masses is identical
to the one when fragmentation is absent, only the rate of evolution
is different. These statements assume monodisperse initial conditions.
We also show the consequences of our generalized model when the system
is initially composed of r-mers only.
Random walks and the additive coagulation equation
Ravi K. Sheth, MNRAS, 295, 869 (1998).
Numerical simulations of hierarchical gravitational clustering in an expanding universe show that, as time evolves, small clusters merge with each other to form larger clusters, whereas fragmentation of clusters is relatively uncommon. Stochastic models of the hierarchical clustering process can provide insight into, as well as useful approximations to, the evolution measured in these simulations. The Poisson random walk excursion model, the Poisson Galton-Watson branching process, and the monodisperse additive coagulation equation are three examples of such stochastic models. When initially identical particles cluster from an initially Poisson spatial distibution, all three approaches give essentially the same description of how clusters grow. This paper shows that clustering from an initially Poisson distribution in which the initial particles do not all have the same mass can be described by simple generalizations of the models above. Such an initial distribution is said to be `compound Poisson'. Therefore, excursions of random walks associated with compound Poisson distributions are studied here. In such an excursion set model, clusters grow in essentially the same way as they do in the polydisperse additive coagulation model. Thus, the interrelations between excursion set, branching process and coagulation models of clustering, associated with the Poisson distribution, also apply to compound Poisson distributions. This means that, within the context of these models, when the initial conditions are compound Poisson then merger and accretion rates, and the entire merger history tree, can all be written analytically, just as for clustering from Poisson initial conditions.
Coagulation and branching process models of gravitational clustering
Ravi K. Sheth & J. Pitman, MNRAS, 289, 66 (1997).
Smoluchowski's coagulation equation describes the growth of clustering in a system in which clusters coalesce irreversibly by binary mergers. If the rate with which two objects merge is proportional to the sum of their masses, and the process starts with all clusters having the same mass initially (monodisperse initial conditions), then the growth of clustering in the system can be derived from a Poisson Galton-Watson branching process. The branching process is consistent with, and provides a more detailed description of gravitational clustering from Poisson initial conditions than, the excursion set, Press-Schechter approach. The correspondence between the branching process and additive coagulation schemes is used to efficiently simulate realizations of the merger history tree. These results are easily extended to describe clustering from white-noise Gaussian initial conditions. Extending this approach to describe clustering from Gaussian fields with arbitrary power spectra is more complicated. To illustrate this, Smoluchowski's additive coagulation equation with arbitrary discrete initial conditions is solved. The result is used to argue that if Smoluchowski's equation is to describe the growth of clustering from arbitrary Gaussian fluctuation fields, and if it is to be consistent with the excursion set results, then the merger rule must depend on the power spectrum. However, with the exception of the white-noise case, the excursion set approach is formally inconsistent with the binary merger assumption, although, for all power spectra of current interest, binary mergers remain a reasonable approximation. This means that, except for the white-noise case, an analogue of the Smoluchowski coagulation approach cannot provide a description of clustering that is exactly equivalent to the excursion set description. Nevertheless, since binary mergers remain a good approximation, Smoluchowski's equation, with a merger rate that can be determined from the excursion set approach, so that it is a calculable function of the power spectrum, can provide a useful approximate description of the growth of clustering.
Galton-Watson branching processes and the growth of gravitational clustering
Ravi K. Sheth, MNRAS, 281, 1277 (1996).
The Press-Schechter description of gravitational clustering from an initially Poisson distribution is shown to be equivalent to the well- studied Galton-Watson branching process. This correspondence is used to provide a detailed description of the evolution of hierarchical clustering, including a complete description of the merger history tree. The relation to branching process epidemic models means that the Press-Schechter description can also be understood using the formalism developed in the study of queues. The queueing theory formalism, also, is used to provide a complete description of the merger history of any given Press-Schechter clump. In particular, an analytic expression for the merger history of any given Poisson Press-Schechter clump is obtained. This expression allows one to calculate the partition function of merger history trees. It obeys an interesting scaling relation; the partition function for a given pair of initial and final epochs is the same as that for certain other pairs of initial and final epochs. The distribution function of counts in randomly placed cells, as a function of time, is also obtained using the branching process and queueing theory descriptions. Thus, the Press-Schechter description of the gravitational evolution of clustering from an initially Poisson distribution is now complete. All these interrelations show why the Press-Schechter approach works well in a statistical sense, but cannot provide a detailed description of the dynamics of the clustering articles themselves. One way to extend these results to more general Gaussian initial conditions is discussed.
Some SDSS related publications
The probability distribution of the Ly-alpha transmitted flux from a sample of SDSS quasars
V. Desjacques, A. Nusser & Ravi K. Sheth,
MNRAS, 374, 206 (2007).
We present a measurement of the probability distribution function (PDF) of the transmitted flux in the Lya forest from a sample of 3492 quasars included in the SDSS DR3 data release. Our intention is to investigate the sensitivity of the Lya flux PDF as measured from low resolution and low signal-to-noise data to a number of systematic errors such as uncertainties in the mean flux, continuum and noise estimate. The quasar continuum is described by the superposition of a power law and emission lines. We perform a power law continuum fitting on a spectrum-by-spectrum basis, and obtain an average continuum slope of 0.59 +/- 0.36 in the redshift range 2.5
A feature at z~3.2 in the evolution of the Ly-alpha forest optical depth
M. Bernardi, Ravi K. Sheth, et al., AJ, 125, 32 (2003).
The effective optical depth in the Ly-alpha forest region of 1061 low-resolution QSO spectra drawn from the SDSS database decreases with decreasing redshift over the range 2.5
- The velocity dispersion function of early-type galaxies
Ravi K. Sheth, M. Bernardi, P. L. Schechter, et al.,
ApJ, 594, 225 (2003).
The distribution of early-type galaxy velocity dispersions,
phi(sigma), is measured using a sample drawn from the SDSS database.
Its shape differs significantly from that which one obtains by simply
using the mean correlation between luminosity L and velocity
dispersion sigma to transform the luminosity function into a
velocity function: ignoring the scatter around the mean sigma-L
relation is a bad approximation. An estimate of the contribution from
late-type galaxies is also made, which suggests that phi(sigma) is
dominated by early-type galaxies at velocities larger than 200 km/s.