In this brief paper we revisit the Fisher information content of cosmological power spectra or two-point functions of Gaussian fields in order to comment on the assumption of Gaussian estimators and ...the use of parameter-dependent covariance matrices for parameter inference in the context of precision cosmology. Even though the assumption of a Gaussian likelihood is motivated by the central limit theorem, we discuss that it leads to Fisher information content that violates the Cramér-Rao bound if used consistently, owing to independent but artificial information from the parameter-dependent covariance matrix. At any fixed multipole, this artificial term is shown to become dominant in the case of a large number of correlated fields. While the distribution of the estimators does indeed tend to a Gaussian with a large number of modes, it is shown, however, that its Fisher information content does not, in the sense that their covariance matrix never carries independent information content, precisely because of the non-Gaussian shape of the distribution. In this light, we discuss the use of parameter-dependent covariance matrices with Gaussian likelihoods for parameter inference from two-point statistics. As a rule of thumb, Gaussian likelihoods should always be used with a covariance matrix fixed in parameter space, since only this guarantees that conservative information content is assigned to the observables, and at the same time, prevents biases appearing.
Recently, several studies proposed non-linear transformations, such as a logarithmic or Gaussianization transformation, as efficient tools to recapture information about the (Gaussian) initial ...conditions. During non-linear evolution, part of the cosmologically relevant information leaks out from the second moment of the distribution. This information is accessible only through complex higher order moments or, in the worst case, becomes inaccessible to the hierarchy. The focus of this work is to investigate these transformations in the framework of Fisher information using cosmological perturbation theory of the matter field with Gaussian initial conditions. We show that at each order in perturbation theory, there is a polynomial of corresponding order exhausting the information on a given parameter. This polynomial can be interpreted as the Taylor expansion of the maximally efficient 'sufficient' observable in the non-linear regime. We determine explicitly this maximally efficient observable for local transformations. Remarkably, this optimal transform is essentially the simple power transform with an exponent related to the slope of the power spectrum; when this is −1, it is indistinguishable from the logarithmic transform. This transform Gaussianizes the distribution, and recovers the linear density contrast. Thus a direct connection is revealed between undoing of the non-linear dynamics and the efficient capture of Fisher information. Our analytical results were compared with measurements from the Millennium Simulation density field. We found that our transforms remain very close to optimal even in the deeply non-linear regime with σ2 ∼ 10.
Beyond the linear regime, the power spectrum and higher order moments of the matter field no longer capture all cosmological information encoded in density fluctuations. While non-linear transforms ...have been proposed to extract this information lost to traditional methods, up to now, the way to generalize these techniques to discrete processes was unclear; ad hoc extensions had some success. We pointed out in Carron and Szapudi's paper that the logarithmic transform approximates extremely well the optimal 'sufficient statistics', observables that extract all information from the (continuous) matter field. Building on these results, we generalize optimal transforms to discrete galaxy fields. We focus our calculations on the Poisson sampling of an underlying lognormal density field. We solve and test the one-point case in detail, and sketch out the sufficient observables for the multipoint case. Moreover, we present an accurate approximation to the sufficient observables in terms of the mean and spectrum of a non-linearly transformed field. We find that the corresponding optimal non-linear transformation is directly related to the maximum a posteriori Bayesian reconstruction of the underlying continuous field with a lognormal prior as put forward in the paper of Kitaura et al.. Thus, simple recipes for realizing the sufficient observables can be built on previously proposed algorithms that have been successfully implemented and tested in simulations.
We discuss an analytical approximation for the matter power spectrum covariance matrix and its inverse on translinear scales, k ∼ 0.1h − 0.8 h Mpc−1 at z = 0. We proceed to give an analytical ...expression for the Fisher information matrix of the non-linear density-field spectrum, and derive implications for its cosmological information content. We find that the spectrum information is characterized by a pair of upper bounds, ‘plateaux’, caused by the trispectrum, and a ‘knee’ in the presence of white noise. The effective number of Fourier modes, normally growing as a power law, is bounded from above by these plateaux, explaining naturally earlier findings from N-body simulations. These plateaux limit best possible measurements of the non-linear power at the per cent level in an h
−3 Gpc3 volume; the extraction of model parameters from the spectrum is limited explicitly by their degeneracy to the non-linear amplitude. The value of the first, supersurvey (SS) plateau depends on the characteristic survey volume and the large-scale power; the second, intra-survey plateau is set by the small-scale power. While both have simple interpretations within the hierarchical Ansatz, the SS plateau can be predicted and generalized to still smaller scales within Takada and Hu's spectrum response formalism. Finally, the noise knee is naturally set by the density of tracers.
Abstract
The cosmological dark matter field is not completely described by its hierarchy of N-point functions, a non-perturbative effect with the consequence that only part of the theory can be ...probed with the hierarchy. We give here an exact characterization of the joint information of the hierarchy within the lognormal field. The lognormal field is the archetypal example of a field where this effect occurs, and, at the same time, one of the few tractable and insightful available models to specify fully the statistical properties of the evolved matter density field beyond the perturbative regime. Non-linear growth in the Universe in that model is set letting the log-density field probability density functional evolve keeping its Gaussian shape, according to the diffusion equation in Euclidean space. We show that the hierarchy probes a different evolution equation, the diffusion equation defined not in Euclidean space but on the compact torus, with uniformity as the long-term solution. The extraction of the hierarchy of correlators can be recast in the form of a non-linear transformation applied to the field, ‘wrapping’, undergoing a sharp transition towards complete disorder in the deeply non-linear regime, where all memory of the initial conditions is lost.
Combination of two physical phenomena, capillary pressure gradient and wettability gradient, allows a simple two-step fabrication process that yields a reliable hydrophobic self-cleaning condenser ...surface. The surface is fabricated with specific microscopic topography and further treatment with a chemically inert low-surface-energy material. This process does not require growth of nanofeatures (nanotubes) or hydrophilic–hydrophobic patterning of the surface. Trapezoidal geometry of the microfeatures facilitates droplet transfer from the Wenzel to the Cassie state and reduces droplet critical diameter. The geometry of the micropatterns enhances local coalescence and directional movement for droplets with diameter much smaller than the radial length of the micropatterns. The hydrophobic self-cleaning micropatterned condenser surface prevents liquid film formation and promotes continuous dropwise condensation cycle. Upon dropwise condensation, droplets follow a designed wettability gradient created with micropatterns from the most hydrophobic to the least hydrophobic end of the surface. The surface has higher condensation efficiency, due to its directional self-cleaning property, than a plain hydrophobic surface. We explain the self-actuated droplet collection mechanism on the condenser surface and demonstrate experimentally the creation of an effective wettability gradient over a 6 mm radial distance. In spite of its fabrication simplicity, the fabricated surface demonstrates self-cleaning property, enhanced condensation performance, and reliability over time. Our work enables creation of a hydrophobic condenser surface with the directional self-cleaning property that can be used for collection of biological (chemical, environmental) aerosol samples or for condensation enhancement.
The primary science driver for 3D galaxy surveys is their potential to constrain cosmological parameters. Forecasts of these surveys’ effectiveness typically assume Gaussian statistics for the ...underlying matter density, despite the fact that the actual distribution is decidedly non-Gaussian. To quantify the effect of this assumption, we employ an analytic expression for the power spectrum covariance matrix to calculate the Fisher information for Baryon Acoustic Oscillation (BAO)-type model surveys. We find that for typical number densities, at k
max = 0.5h Mpc−1, Gaussian assumptions significantly overestimate the information on all parameters considered, in some cases by up to an order of magnitude. However, after marginalizing over a six-parameter set, the form of the covariance matrix (dictated by N-body simulations) causes the majority of the effect to shift to the ‘amplitude-like’ parameters, leaving the others virtually unaffected. We find that Gaussian assumptions at such wavenumbers can underestimate the dark energy parameter errors by well over 50 per cent, producing dark energy figures of merit almost three times too large. Thus, for 3D galaxy surveys probing the non-linear regime, proper consideration of non-Gaussian effects is essential.
We define and study statistical ensembles of matter density profiles describing spherically symmetric, virialized dark matter haloes of finite extent with a given mass and total gravitational ...potential energy. Our ensembles include spatial degrees of freedom only, a microstate being a spherically symmetric matter density function. We provide an exact solution for the grand canonical partition functional, and show its equivalence to that of the microcanonical ensemble. We obtain analytically the mean profiles that correspond to an overwhelming majority of microstates. All such profiles have an infinitely deep potential well, with the singular isothermal sphere arising in the infinite temperature limit. Systems with virial radius larger than gravitational radius exhibit a localization of a finite fraction of the energy in the very centre. The universal logarithmic inner slope of unity of the Navarro-Frenk-White (NFW) haloes is predicted at any mass and energy if an upper bound is set to the maximal depth of the potential well. In this case, the statistically favoured mean profiles compare well to the NFW profiles. For very massive haloes the agreement becomes exact.
Beyond the linear regime of structure formation, part of cosmological information encoded in galaxy clustering becomes inaccessible to the usual power spectrum. Sufficient statistics, A*, were ...introduced recently to recapture the lost, and ultimately extract all, cosmological information. We present analytical approximations for the A* and traditional power spectra as well as for their covariance matrices in order to calculate analytically their cosmological information content in the context of Fisher information theory. Our approach allows the precise quantitative comparison of the techniques with each other and to the total information in the data, and provides insights into sufficient statistics. In particular, we find that while the A* power spectrum has a similar shape to the usual galaxy power spectrum, its amplitude is strongly modulated by small-scale statistics. This effect is mostly responsible for the ability of the A* power spectrum to recapture the information lost for the usual power spectrum. We use our framework to forecast the best achievable cosmological constraints for projected surveys as a function of their galaxy density, and compare the information content of the two power spectra. We find that sufficient statistics recover significantly more cosmological information, resulting in an approximate factor of ≃ 2 gain for dense projected surveys at low redshift. This increase in the effective volume of projected surveys is consistent with previous numerical calculations.
The aim of this work is to analyze the influence of the guard rings (GRs) in solid state detectors (SSDs). Depending on the applied potential, the collection of charges can be disturbed. A study on ...SSDs with floating and grounded GRs is conducted. The tools needed to do so are presented (GEANT4, Sentaurus, and Garfield++). Experimental measurements have been performed on Micrometer Semiconductor Ltd., SSDs which are composed of multiple guard rings (MGRs). Finally, a first application on the Influence sur les Composants Avancés des Radiations de l'Espace (ICARE_NG2) radiation monitor is proposed with a comparison of the response functions (RFs) according to the potential applied on the GRs. Improved performances are observed when the GRs are grounded. A second application on the ICARE_NG currently embedded on Eutelsat 7C (E7C) is also proposed by using the GRs as active shielding, allowing improvements in the measurement of high-energy particles.
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