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Register a new userEvolution of skewness and kurtosis of cosmic density fields
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Methods. We perform numerical simulations of the evolution of the cosmic web for the conventional LCDM model. The simulations cover a wide range of box sizes L = 256 - 4000 Mpc/h, mass and force resolutions and epochs from very early moments z = 30 to the present moment z = 0. We calculate density fields with various smoothing lengths to find the dependence of the density field on smoothing scale. We calculate PDF and its moments - variance, skewness and kurtosis. Results. We focus on the third (skewness S) and fourth (kurtosis K) moments of the distribution functions: their dependence on the smoothing scale, the amplitude of fluctuations and the redshift. During the evolution the reduced skewness $S_3= S/sigma$ and reduced kurtosis $S_4=K/sigma^2$ present a complex behaviour: at a fixed redshift curves of $S_3(sigma)$ and $S_4(sigma)$ steeply increase with $sigma$ at $sigmale 1$ and then flatten out and become constant at $sigmage2$. If we fix the smoothing scale $R_t$, then after reaching the maximum at $sigmaapprox 2$, the curves at large $sigma$ start to gradually decline. We provide accurate fits for the evolution of $S_{3,4}(sigma,z)$. Skewness and kurtosis approach at early epochs constant levels, depending on smoothing length: $S_3(sigma) approx 3$ and $S_4(sigma) approx 15$. Conclusions. Most of statistics of dark matter clustering (e.g., halo mass function or concentration-mass relation) are nearly universal: they mostly depend on the $sigma$ with the relatively modest correction to explicit dependence on the redshift. We find just the opposite for skewness and kurtosis: the dependence of moments on evolutionary epoch $z$ and smoothing length $R_t$ is very different, together they determine the evolution of $S_{3,4}(sigma)$ uniquely. The evolution of $S_3$ and $S_4$ cannot be described by current theoretical approximations.
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In a recent paper [textit{M. Cristelli, A. Zaccaria and L. Pietronero, Phys. Rev. E 85, 066108 (2012)}], Cristelli textit{et al.} analysed relation between skewness and kurtosis for complex dynamical systems and identified two power-law regimes of non-Gaussianity, one of which scales with an exponent of 2 and the other is with $4/3$. Finally the authors concluded that the observed relation is a universal fact in complex dynamical systems. Here, we test the proposed universal relation between skewness and kurtosis with large number of synthetic data and show that in fact it is not universal and originates only due to the small number of data points in the data sets considered. The proposed relation is tested using two different non-Gaussian distributions, namely $q$-Gaussian and Levy distributions. We clearly show that this relation disappears for sufficiently large data sets provided that the second moment of the distribution is finite. We find that, contrary to the claims of Cristelli textit{et al.} regarding a power-law scaling regime, kurtosis saturates to a single value, which is of course different from the Gaussian case ($K=3$), as the number of data is increased. On the other hand, if the second moment of the distribution is infinite, then the kurtosis seems to never converge to a single value. The converged kurtosis value for the finite second moment distributions and the number of data points needed to reach this value depend on the deviation of the original distribution from the Gaussian case. We also argue that the use of kurtosis to compare distributions to decide which one deviates from the Gaussian more can lead to incorrect results even for finite second moment distributions for small data sets, whereas it is totally misleading for infinite second moment distributions where the difference depends on $N$ for all finite $N$.
We present results for the ratios of mean ($M_B$), variance ($sigma_B^2$), skewness ($S_B)$ and kurtosis ($kappa_B$) of net baryon-number fluctuations obtained in lattice QCD calculations with a physical light to strange quark mass ratio. Using next-to-leading order Taylor expansions in baryon chemical potential we find that qualitative features of these ratios closely resemble the corresponding experimentally measured cumulants ratios of net proton-number fluctuations for beam energies down to $sqrt{s_{_{NN}}} ge 19.6$ GeV. We show that the difference in cumulant ratios for the mean net baryon-number, $M_B/sigma_B^2=chi_1^B(T,mu_B)/chi_2^B(T,mu_B)$ and the normalized skewness, $S_Bsigma_B=chi_3^B(T,mu_B)/chi_2^B(T,mu_B)$, naturally arises in QCD thermodynamics. Moreover, we establish a close relation between skewness and kurtosis ratios, $S_Bsigma_B^3/M_B=chi_3^B(T,mu_B)/chi_1^B(T,mu_B)$ and $kappa_Bsigma_B^2=chi_4^B(T,mu_B)/chi_2^B(T,mu_B)$, valid at small values of the baryon chemical potential.
We present a Bayesian reconstruction algorithm to generate unbiased samples of the underlying dark matter field from halo catalogues. Our new contribution consists of implementing a non-Poisson likelihood including a deterministic non-linear and scale-dependent bias. In particular we present the Hamiltonian equations of motions for the negative binomial (NB) probability distribution function. This permits us to efficiently sample the posterior distribution function of density fields given a sample of galaxies using the Hamiltonian Monte Carlo technique implemented in the Argo code. We have tested our algorithm with the Bolshoi $N$-body simulation at redshift $z = 0$, inferring the underlying dark matter density field from sub-samples of the halo catalogue with biases smaller and larger than one. Our method shows that we can draw closely unbiased samples (compatible within 1-$sigma$) from the posterior distribution up to scales of about $k$~1 h/Mpc in terms of power-spectra and cell-to-cell correlations. We find that a Poisson likelihood yields reconstructions with power spectra deviating more than 10% at $k$=0.2 h/Mpc. Our reconstruction algorithm is especially suited for emission line galaxy data for which a complex non-linear stochastic biasing treatment beyond Poissonity becomes indispensable.
We present a novel halo painting network that learns to map approximate 3D dark matter fields to realistic halo distributions. This map is provided via a physically motivated network with which we can learn the non-trivial local relation between dark matter density field and halo distributions without relying on a physical model. Unlike other generative or regressive models, a well motivated prior and simple physical principles allow us to train the mapping network quickly and with relatively little data. In learning to paint halo distributions from computationally cheap, analytical and non-linear density fields, we bypass the need for full particle mesh simulations and halo finding algorithms. Furthermore, by design, our halo painting network needs only local patches of dark matter density to predict the halos, and as such, it can predict the 3D halo distribution for any arbitrary simulation box size. Our neural network can be trained using small simulations and used to predict large halo distributions, as long as the resolutions are equivalent. We evaluate our models ability to generate 3D halo count distributions which reproduce, to a high degree, summary statistics such as the power spectrum and bispectrum, of the input or reference realizations.
Applying the Cherny-Shiryaev-Yor invariance principle, we introduce a generalized Jarrow-Rudd (GJR) option pricing model with uncertainty driven by a skew random walk. The GJR pricing tree exhibits skewness and kurtosis in both the natural and risk-neutral world. We construct implied surfaces for the parameters determining the GJR tree. Motivated by Mertons pricing tree incorporating transaction costs, we extend the GJR pricing model to include a hedging cost. We demonstrate ways to fit the GJR pricing model to a market driver that influences the price dynamics of the underlying asset. We supplement our findings with numerical examples.
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