661 research outputs found
The invariant joint distribution of a stationary random field and its derivatives: Euler characteristic and critical point counts in 2 and 3D
The full moments expansion of the joint probability distribution of an
isotropic random field, its gradient and invariants of the Hessian is presented
in 2 and 3D. It allows for explicit expression for the Euler characteristic in
ND and computation of extrema counts as functions of the excursion set
threshold and the spectral parameter, as illustrated on model examples.Comment: 4 pages, 2 figures. Corrected expansion coefficients for orders n>=5.
Relation between Gram-Charlier and Edgeworth expansions is clarified
A local bias approach to the clustering of discrete density peaks
Maxima of the linear density field form a point process that can be used to
understand the spatial distribution of virialized halos that collapsed from
initially overdense regions. However, owing to the peak constraint, clustering
statistics of discrete density peaks are difficult to evaluate. For this
reason, local bias schemes have received considerably more attention in the
literature thus far. In this paper, we show that the 2-point correlation
function of maxima of a homogeneous and isotropic Gaussian random field can be
thought of, up to second order at least, as arising from a local bias expansion
formulated in terms of rotationally invariant variables. This expansion relies
on a unique smoothing scale, which is the Lagrangian radius of dark matter
halos. The great advantage of this local bias approach is that it circumvents
the difficult computation of joint probability distributions. We demonstrate
that the bias factors associated with these rotational invariants can be
computed using a peak-background split argument, in which the background
perturbation shifts the corresponding probability distribution functions.
Consequently, the bias factors are orthogonal polynomials averaged over those
spatial locations that satisfy the peak constraint. In particular, asphericity
in the peak profile contributes to the clustering at quadratic and higher
order, with bias factors given by generalized Laguerre polynomials. We
speculate that our approach remains valid at all orders, and that it can be
extended to describe clustering statistics of any point process of a Gaussian
random field. Our results will be very useful to model the clustering of
discrete tracers with more realistic collapse prescriptions involving the tidal
shear for instance.Comment: 14 pages, 1 figure. (v2): typos fixed + references added. Accepted
for publication in PR
Non-Gaussian Minkowski functionals & extrema counts in redshift space
In the context of upcoming large-scale structure surveys such as Euclid, it
is of prime importance to quantify the effect of peculiar velocities on
geometric probes. Hence the formalism to compute in redshift space the
geometrical and topological one-point statistics of mildly non-Gaussian 2D and
3D cosmic fields is developed. Leveraging the partial isotropy of the target
statistics, the Gram-Charlier expansion of the joint probability distribution
of the field and its derivatives is reformulated in terms of the corresponding
anisotropic variables. In particular, the cosmic non-linear evolution of the
Minkowski functionals, together with the statistics of extrema are investigated
in turn for 3D catalogues and 2D slabs. The amplitude of the non-Gaussian
redshift distortion correction is estimated for these geometric probes. In 3D,
gravitational perturbation theory is implemented in redshift space to predict
the cosmic evolution of all relevant Gram-Charlier coefficients. Applications
to the estimation of the cosmic parameters sigma(z) and beta=f/b1 from upcoming
surveys is discussed. Such statistics are of interest for anisotropic fields
beyond cosmology.Comment: 35 pages, 15 figures, matches version published in MNRAS with a typo
corrected in eq A1
Efficient Local Comparison Of Images Using Krawtchouk Descriptors
It is known that image comparison can prove cumbersome in both computational complexity and runtime, due to factors such as the rotation, scaling, and translation of the object in question. Due to the locality of Krawtchouk polynomials, relatively few descriptors are necessary to describe a given image, and this can be achieved with minimal memory usage. Using this method, not only can images be described efficiently as a whole, but specific regions of images can be described as well without cropping. Due to this property, queries can be found within a single large image, or collection of large images, which serve as a database for search. Krawtchouk descriptors can also describe collections of patches of 3D objects, which is explored in this paper, as well as a theoretical methodology of describing nD hyperobjects. Test results for an implementation of 3D Krawtchouk descriptors in GNU Octave, as well as statistics regarding effectiveness and runtime, are included, and the code used for testing will be published open source in the near future
Non-Gaussian bias: insights from discrete density peaks
Corrections induced by primordial non-Gaussianity to the linear halo bias can
be computed from a peak-background split or the widespread local bias model.
However, numerical simulations clearly support the prediction of the former, in
which the non-Gaussian amplitude is proportional to the linear halo bias. To
understand better the reasons behind the failure of standard Lagrangian local
bias, in which the halo overdensity is a function of the local mass overdensity
only, we explore the effect of a primordial bispectrum on the 2-point
correlation of discrete density peaks. We show that the effective local bias
expansion to peak clustering vastly simplifies the calculation. We generalize
this approach to excursion set peaks and demonstrate that the resulting
non-Gaussian amplitude, which is a weighted sum of quadratic bias factors,
precisely agrees with the peak-background split expectation, which is a
logarithmic derivative of the halo mass function with respect to the
normalisation amplitude. We point out that statistics of thresholded regions
can be computed using the same formalism. Our results suggest that halo
clustering statistics can be modelled consistently (in the sense that the
Gaussian and non-Gaussian bias factors agree with peak-background split
expectations) from a Lagrangian bias relation only if the latter is specified
as a set of constraints imposed on the linear density field. This is clearly
not the case of standard Lagrangian local bias. Therefore, one is led to
consider additional variables beyond the local mass overdensity.Comment: 24 pages. no figure (v2): minor clarification added. submitted to
JCAP (v3): 1 figure added. in Press in JCA
Non-Gaussian statistics of critical sets in 2 and 3D: Peaks, voids, saddles, genus and skeleton
The formalism to compute the geometrical and topological one-point statistics
of mildly non-Gaussian 2D and 3D cosmological fields is developed. Leveraging
the isotropy of the target statistics, the Gram-Charlier expansion is
reformulated with rotation invariant variables. This formulation allows us to
track the geometrical statistics of the cosmic field to all orders. It then
allows us to connect the one point statistics of the critical sets to the
growth factor through perturbation theory, which predicts the redshift
evolution of higher order cumulants. In particular, the cosmic non-linear
evolution of the skeleton's length, together with the statistics of extrema and
Euler characteristic are investigated in turn. In 2D, the corresponding
differential densities are analytic as a function of the excursion set
threshold and the shape parameter. In 3D, the Euler characteristics and the
field isosurface area are also analytic to all orders in the expansion.
Numerical integrations are performed and simple fits are provided whenever
closed form expressions are not available. These statistics are compared to
estimates from N-body simulations and are shown to match well the cosmic
evolution up to root mean square of the density field of ~0.2. In 3D,
gravitational perturbation theory is implemented to predict the cosmic
evolution of all the relevant Gram-Charlier coefficients for universes with
scale invariant matter distribution. The one point statistics of critical sets
could be used to constrain primordial non-Gaussianities and the dark energy
equation of state on upcoming cosmic surveys; this is illustrated on idealized
experiments.Comment: 41 pages, 13 figures, submitted to Phys Rev
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