145 research outputs found
The non-Gaussianity of the cosmic shear likelihood - or: How odd is the Chandra Deep Field South?
(abridged) We study the validity of the approximation of a Gaussian cosmic
shear likelihood. We estimate the true likelihood for a fiducial cosmological
model from a large set of ray-tracing simulations and investigate the impact of
non-Gaussianity on cosmological parameter estimation. We investigate how odd
the recently reported very low value of really is as derived from
the \textit{Chandra} Deep Field South (CDFS) using cosmic shear by taking the
non-Gaussianity of the likelihood into account as well as the possibility of
biases coming from the way the CDFS was selected.
We find that the cosmic shear likelihood is significantly non-Gaussian. This
leads to both a shift of the maximum of the posterior distribution and a
significantly smaller credible region compared to the Gaussian case. We
re-analyse the CDFS cosmic shear data using the non-Gaussian likelihood.
Assuming that the CDFS is a random pointing, we find
for fixed . In a
WMAP5-like cosmology, a value equal to or lower than this would be expected in
of the times. Taking biases into account arising from the way the
CDFS was selected, which we model as being dependent on the number of haloes in
the CDFS, we obtain . Combining the CDFS data
with the parameter constraints from WMAP5 yields and for a flat
universe.Comment: 18 pages, 16 figures, accepted for publication in A&A; New Bayesian
treatment of field selection bia
Strong lensing optical depths in a \LambdaCDM universe
We investigate strong gravitational lensing in the concordance CDM
cosmology by carrying out ray-tracing along past light cones through the
Millennium Simulation, the largest simulation of cosmic structure formation
ever carried out. We extend previous ray-tracing methods in order to take full
advantage of the large volume and the excellent spatial and mass resolution of
the simulation. As a function of source redshift we evaluate the probability
that an image will be highly magnified, will be highly elongated or will be one
of a set of multiple images. We show that such strong lensing events can almost
always be traced to a single dominant lensing object and we study the mass and
redshift distribution of these primary lenses. We fit analytic models to the
simulated dark halos in order to study how our optical depth measurements are
affected by the limited resolution of the simulation and of the lensing planes
that we construct from it. We conclude that such effects lead us to
underestimate total strong-lensing cross sections by about 15 percent. This is
smaller than the effects expected from our neglect of the baryonic components
of galaxies. Finally we investigate whether strong lensing is enhanced by
material in front of or behind the primary lens. Although strong lensing
lines-of-sight are indeed biased towards higher than average mean densities,
this additional matter typically contributes only a few percent of the total
surface density.Comment: version accepted for publicatio
Ray-Tracing Simulations of Weak Gravitational Lensing
Weak gravitational lensing, i.e. the distortion of images of distant galaxies due to the deflection of light rays in gravitational fields, is a powerful method to study the matter distribution in the Universe. It can be used to put constraints on cosmological parameters by looking at the statistical properties of the distortion field (cosmic shear), or to study the relation of the galaxy distribution to the underlying distribution of the dark matter (galaxy-galaxy lensing; GGL). We use ray-tracing simulations of light propagation through the large scale structure of the Universe to study several aspects of the statistical analysis of weak lensing data and to test methods to constrain theories of galaxy formation using GGL. In the first part of this thesis, we discuss several issues regarding the estimation of cosmological parameters from weak lensing data. We point out that using a covariance matrix that is estimated from simulations or the data themselves can lead to a severe underestimation of the errors derived for cosmological parameters, and we report on a method to remove this bias. We then use ray-tracing simulations to test the assumption that the likelihood of the shear correlation functions is a multi-variate Gaussian. For that purpose, we develop and test a new method to estimate high-dimensional probability distributions. We find that the likelihood is significantly skewed and steeper than suggested by the Gaussian approximation. We apply this result to investigate possible reasons for the low value of the power spectrum normalization s8 found by Schrabback et al. (2007) in a cosmic shear analysis of the Chandra Deep Field South (CDFS). Both the assumption of a Gaussian likelihood and the way the CDFS was selected are found to bias s8 low, in total by approximately 15-20%. In the second part of this thesis, we use ray-tracing through a very large simulation of structure formation, the Millennium Simulation (MS), to investigate possibilities to constrain galaxy evolution models using galaxy-galaxy lensing. We introduce several improvements of the standard ray-tracing algorithm which allow us to fully exploit the large volume and high resolution of the MS. Galaxies from semi-analytic models of galaxy formation can be included in the ray-tracing, yielding realistic mock galaxy catalogs. Here, we use the model by De Lucia & Blaizot (2007). We compare our simulations to the measurements of the GGL signal in the Sloan Digital Sky Survey (Sheldon et al. 2004) and find an overall good agreement; only for the most luminous blue galaxies, the simulation predicts a signal smaller than observed. A possible reason for this is the insufficient modelling of dust attenuation in the semi-analytic model. We also find tentative evidence for a steepening of the GGL signal due to baryonic physics on small scales. Finally, we explore the possibilities of measuring the linear stochastic bias and correlation parameters (quantifying the relation between galaxies and dark matter) from measurements of weak lensing and angular clustering. We identify several sources of bias, the most important being the limited angular range for which the GGL signal and the angular correlation function can be measured. Furthermore, higher-order effects due to the coupling of lenses at different distances have to be taken into account, for which we provide analytical expressions. These issues depend strongly on scale and the galaxy samples used, and are best modelled using numerical simulations
Constrained correlation functions
We show that correlation functions have to satisfy contraint relations, owing
to the non-negativity of the power spectrum of the underlying random process.
Specifically, for any statistically homogeneous and (for more than one spatial
dimension) isotropic random field with correlation function , we derive
inequalities for the correlation coefficients (for
integer ) of the form , where the lower
and upper bounds on depend on the , with . Explicit expressions
for the bounds are obtained for arbitrary . These constraint equations very
significantly limit the set of possible correlation functions. For one
particular example of a fiducial cosmic shear survey, we show that the Gaussian
likelihood ellipsoid has a significant spill-over into the forbidden region of
correlation functions, rendering the resulting best-fitting model parameters
and their error region questionable, and indicating the need for a better
description of the likelihood function.
We conduct some simple numerical experiments which explicitly demonstrate the
failure of a Gaussian description for the likelihood of . Instead, the
shape of the likelihood function of the correlation coefficients appears to
follow approximately that of the shape of the bounds on the , even if the
Gaussian ellipsoid lies well within the allowed region.
For more than one spatial dimension of the random field, the explicit
expressions of the bounds on the are not optimal. We outline a
geometrical method how tighter bounds may be obtained in principle. We
illustrate this method for a few simple cases; a more general treatment awaits
future work.Comment: 18 pages, 9 figures, submitted to A&
Constrained probability distributions of correlation functions
Context: Two-point correlation functions are used throughout cosmology as a
measure for the statistics of random fields. When used in Bayesian parameter
estimation, their likelihood function is usually replaced by a Gaussian
approximation. However, this has been shown to be insufficient.
Aims: For the case of Gaussian random fields, we search for an exact
probability distribution of correlation functions, which could improve the
accuracy of future data analyses.
Methods: We use a fully analytic approach, first expanding the random field
in its Fourier modes, and then calculating the characteristic function.
Finally, we derive the probability distribution function using integration by
residues. We use a numerical implementation of the full analytic formula to
discuss the behaviour of this function.
Results: We derive the univariate and bivariate probability distribution
function of the correlation functions of a Gaussian random field, and outline
how higher joint distributions could be calculated. We give the results in the
form of mode expansions, but in one special case we also find a closed-form
expression. We calculate the moments of the distribution and, in the univariate
case, we discuss the Edgeworth expansion approximation. We also comment on the
difficulties in a fast and exact numerical implementation of our results, and
on possible future applications.Comment: 13 pages, 5 figures, updated to match version published in A&A
(slightly expanded Sects. 5.3 and 6
Dependence of cosmic shear covariances on cosmology - Impact on parameter estimation
In cosmic shear likelihood analyses the covariance is most commonly assumed
to be constant in parameter space. Therefore, when calculating the covariance
matrix (analytically or from simulations), its underlying cosmology should not
influence the likelihood contours. We examine whether the aforementioned
assumption holds and quantify how strong cosmic shear covariances vary within a
reasonable parameter range. Furthermore, we examine the impact on likelihood
contours when assuming different cosmologies in the covariance. We find that
covariances vary significantly within the considered parameter range
(Omega_m=[0.2;0.4], sigma_8=[0.6;1.0]) and that this has a non-negligible
impact on the size of likelihood contours. This impact increases with
increasing survey size, increasing number density of source galaxies,
decreasing ellipticity noise, and when using non-Gaussian covariances. To
improve on the assumption of a constant covariance we present two methods. The
adaptive covariance is the most accurate method, but it is computationally
expensive. To reduce the computational costs we give a scaling relation for
covariances. As a second method we outline the concept of an iterative
likelihood analysis. Here, we additionally account for non-Gaussianity using a
ray-tracing covariance derived from the Millennium simulation.Comment: 11 pages, 8 figure
A bias in cosmic shear from galaxy selection: results from ray-tracing simulations
We identify and study a previously unknown systematic effect on cosmic shear
measurements, caused by the selection of galaxies used for shape measurement,
in particular the rejection of close (blended) galaxy pairs. We use ray-tracing
simulations based on the Millennium Simulation and a semi-analytical model of
galaxy formation to create realistic galaxy catalogues. From these, we quantify
the bias in the shear correlation functions by comparing measurements made from
galaxy catalogues with and without removal of close pairs. A likelihood
analysis is used to quantify the resulting shift in estimates of cosmological
parameters. The filtering of objects with close neighbours (a) changes the
redshift distribution of the galaxies used for correlation function
measurements, and (b) correlates the number density of sources in the
background with the density field in the foreground. This leads to a
scale-dependent bias of the correlation function of several percent,
translating into biases of cosmological parameters of similar amplitude. This
makes this new systematic effect potentially harmful for upcoming and planned
cosmic shear surveys. As a remedy, we propose and test a weighting scheme that
can significantly reduce the bias.Comment: 9 pages, 9 figures, version accepted for publication in Astronomy &
Astrophysic
Why your model parameter confidences might be too optimistic -- unbiased estimation of the inverse covariance matrix
AIMS. The maximum-likelihood method is the standard approach to obtain model
fits to observational data and the corresponding confidence regions. We
investigate possible sources of bias in the log-likelihood function and its
subsequent analysis, focusing on estimators of the inverse covariance matrix.
Furthermore, we study under which circumstances the estimated covariance matrix
is invertible. METHODS. We perform Monte-Carlo simulations to investigate the
behaviour of estimators for the inverse covariance matrix, depending on the
number of independent data sets and the number of variables of the data
vectors. RESULTS. We find that the inverse of the maximum-likelihood estimator
of the covariance is biased, the amount of bias depending on the ratio of the
number of bins (data vector variables), P, to the number of data sets, N. This
bias inevitably leads to an -- in extreme cases catastrophic -- underestimation
of the size of confidence regions. We report on a method to remove this bias
for the idealised case of Gaussian noise and statistically independent data
vectors. Moreover, we demonstrate that marginalisation over parameters
introduces a bias into the marginalised log-likelihood function. Measures of
the sizes of confidence regions suffer from the same problem. Furthermore, we
give an analytic proof for the fact that the estimated covariance matrix is
singular if P>N.Comment: 6 pages, 3 figures, A&A, in press, shortened versio
Intrinsic galaxy shapes and alignments II: Modelling the intrinsic alignment contamination of weak lensing surveys
Intrinsic galaxy alignments constitute the major astrophysical systematic of
forthcoming weak gravitational lensing surveys but also yield unique insights
into galaxy formation and evolution. We build analytic models for the
distribution of galaxy shapes based on halo properties extracted from the
Millennium Simulation, differentiating between early- and late-type galaxies as
well as central galaxies and satellites. The resulting ellipticity correlations
are investigated for their physical properties and compared to a suite of
current observations. The best-faring model is then used to predict the
intrinsic alignment contamination of planned weak lensing surveys. We find that
late-type galaxy models generally have weak intrinsic ellipticity correlations,
marginally increasing towards smaller galaxy separation and higher redshift.
The signal for early-type models at fixed halo mass strongly increases by three
orders of magnitude over two decades in galaxy separation, and by one order of
magnitude from z=0 to z=2. The intrinsic alignment strength also depends
strongly on halo mass, but not on galaxy luminosity at fixed mass, or galaxy
number density in the environment. We identify models that are in good
agreement with all observational data, except that all models over-predict
alignments of faint early-type galaxies. The best model yields an intrinsic
alignment contamination of a Euclid-like survey between 0.5-10% at z>0.6 and on
angular scales larger than a few arcminutes. Cutting 20% of red foreground
galaxies using observer-frame colours can suppress this contamination by up to
a factor of two.Comment: 23 pages, 14 figures; minor changes to match version published in
MNRA
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