Non-Gaussian Cosmological Perturbations from Hybrid Inflation and Preheating

This thesis consists of four research papers and an introduction providing some background. The structure in the universe is generally considered to originate from quantum fluctuations in the very early universe. The standard lore of cosmology states that the primordial perturbations are almost scale-invariant, adiabatic, and Gaussian. A snapshot of the structure from the time when the universe became transparent can be seen in the cosmic microwave background (CMB). For a long time mainly the power spectrum of the CMB temperature fluctuations has been used to obtain observational constraints, especially on deviations from scale-invariance and pure adiabacity. Non-Gaussian perturbations provide a novel and very promising way to test theoretical predictions. They probe beyond the power spectrum, or two point correlator, since non-Gaussianity involves higher order statistics. The thesis concentrates on the non-Gaussian perturbations arising in several situations involving two scalar fields, namely, hybrid inflation and various forms of preheating. First we go through some basic concepts -- such as the cosmological inflation, reheating and preheating, and the role of scalar fields during inflation -- which are necessary for the understanding of the research papers. We also review the standard linear cosmological perturbation theory. The second order perturbation theory formalism for two scalar fields is developed. We explain what is meant by non-Gaussian perturbations, and discuss some difficulties in parametrisation and observation. In particular, we concentrate on the nonlinearity parameter. The prospects of observing non-Gaussianity are briefly discussed. We apply the formalism and calculate the evolution of the second order curvature perturbation during hybrid inflation. We estimate the amount of non-Gaussianity in the model and find that there is a possibility for an observational effect. The non-Gaussianity arising in preheating is also studied. We find that the level produced by the simplest model of instant preheating is insignificant, whereas standard preheating with parametric resonance as well as tachyonic preheating are prone to easily saturate and even exceed the observational limits. We also mention other approaches to the study of primordial non-Gaussianities, which differ from the perturbation theory method chosen in the thesis work.

Geospatial environmental data modelling applications using remote sensing, GIS and spatial statistics

This thesis presents novel modelling applications for environmental geospatial data using remote sensing, GIS and statistical modelling techniques. The studied themes can be classified into four main themes: (i) to develop advanced geospatial databases. Paper (I) demonstrates the creation of a geospatial database for the Glanville fritillary butterfly (Melitaea cinxia) in the Åland Islands, south-western Finland; (ii) to analyse species diversity and distribution using GIS techniques. Paper (II) presents a diversity and geographical distribution analysis for Scopulini moths at a world-wide scale; (iii) to study spatiotemporal forest cover change. Paper (III) presents a study of exotic and indigenous tree cover change detection in Taita Hills Kenya using airborne imagery and GIS analysis techniques; (iv) to explore predictive modelling techniques using geospatial data. In Paper (IV) human population occurrence and abundance in the Taita Hills highlands was predicted using the generalized additive modelling (GAM) technique. Paper (V) presents techniques to enhance fire prediction and burned area estimation at a regional scale in East Caprivi Namibia. Paper (VI) compares eight state-of-the-art predictive modelling methods to improve fire prediction, burned area estimation and fire risk mapping in East Caprivi Namibia. The results in Paper (I) showed that geospatial data can be managed effectively using advanced relational database management systems. Metapopulation data for Melitaea cinxia butterfly was successfully combined with GPS-delimited habitat patch information and climatic data. Using the geospatial database, spatial analyses were successfully conducted at habitat patch level or at more coarse analysis scales. Moreover, this study showed it appears evident that at a large-scale spatially correlated weather conditions are one of the primary causes of spatially correlated changes in Melitaea cinxia population sizes. In Paper (II) spatiotemporal characteristics of Socupulini moths description, diversity and distribution were analysed at a world-wide scale and for the first time GIS techniques were used for Scopulini moth geographical distribution analysis. This study revealed that Scopulini moths have a cosmopolitan distribution. The majority of the species have been described from the low latitudes, sub-Saharan Africa being the hot spot of species diversity. However, the taxonomical effort has been uneven among biogeographical regions. Paper III showed that forest cover change can be analysed in great detail using modern airborne imagery techniques and historical aerial photographs. However, when spatiotemporal forest cover change is studied care has to be taken in co-registration and image interpretation when historical black and white aerial photography is used. In Paper (IV) human population distribution and abundance could be modelled with fairly good results using geospatial predictors and non-Gaussian predictive modelling techniques. Moreover, land cover layer is not necessary needed as a predictor because first and second-order image texture measurements derived from satellite imagery had more power to explain the variation in dwelling unit occurrence and abundance. Paper V showed that generalized linear model (GLM) is a suitable technique for fire occurrence prediction and for burned area estimation. GLM based burned area estimations were found to be more superior than the existing MODIS burned area product (MCD45A1). However, spatial autocorrelation of fires has to be taken into account when using the GLM technique for fire occurrence prediction. Paper VI showed that novel statistical predictive modelling techniques can be used to improve fire prediction, burned area estimation and fire risk mapping at a regional scale. However, some noticeable variation between different predictive modelling techniques for fire occurrence prediction and burned area estimation existed.

Dependence Logic: Investigations into Higher-Order Semantics Defined on Teams

This thesis is a study of a rather new logic called dependence logic and its closure under classical negation, team logic. In this thesis, dependence logic is investigated from several aspects. Some rules are presented for quantifier swapping in dependence logic and team logic. Such rules are among the basic tools one must be familiar with in order to gain the required intuition for using the logic for practical purposes. The thesis compares Ehrenfeucht-Fraïssé (EF) games of first order logic and dependence logic and defines a third EF game that characterises a mixed case where first order formulas are measured in the formula rank of dependence logic. The thesis contains detailed proofs of several translations between dependence logic, team logic, second order logic and its existential fragment. Translations are useful for showing relationships between the expressive powers of logics. Also, by inspecting the form of the translated formulas, one can see how an aspect of one logic can be expressed in the other logic. The thesis makes preliminary investigations into proof theory of dependence logic. Attempts focus on finding a complete proof system for a modest yet nontrivial fragment of dependence logic. A key problem is identified and addressed in adapting a known proof system of classical propositional logic to become a proof system for the fragment, namely that the rule of contraction is needed but is unsound in its unrestricted form. A proof system is suggested for the fragment and its completeness conjectured. Finally, the thesis investigates the very foundation of dependence logic. An alternative semantics called 1-semantics is suggested for the syntax of dependence logic. There are several key differences between 1-semantics and other semantics of dependence logic. 1-semantics is derived from first order semantics by a natural type shift. Therefore 1-semantics reflects an established semantics in a coherent manner. Negation in 1-semantics is a semantic operation and satisfies the law of excluded middle. A translation is provided from unrestricted formulas of existential second order logic into 1-semantics. Also game theoretic semantics are considerd in the light of 1-semantics.

On linear order and computation : The expressiveness of interactive computations on linear orders and computations indexed by ordinals

We solve the Dynamic Ehrenfeucht-Fra\"iss\'e Game on linear orders for both players, yielding a normal form for quantifier-rank equivalence classes of linear orders in first-order logic, infinitary logic, and generalized-infinitary logics with linearly ordered clocks. We show that Scott Sentences can be manipulated quickly, classified into local information, and consistency can be decided effectively in the length of the Scott Sentence. We describe a finite set of linked automata moving continuously on a linear order. Running them on ordinals, we compute the ordinal truth predicate and compute truth in the constructible universe of set-theory. Among the corollaries are a study of semi-models as efficient database of both model-theoretic and formulaic information, and a new proof of the atomicity of the Boolean algebra of sentences consistent with the theory of linear order -- i.e., that the finitely axiomatized theories of linear order are dense.

Locality and order-invariant logics

Malli on logiikassa käytetty abstraktio monille matemaattisille objekteille. Esimerkiksi verkot, ryhmät ja metriset avaruudet ovat malleja. Äärellisten mallien teoria on logiikan osa-alue, jossa tarkastellaan logiikkojen, formaalien kielten, ilmaisuvoimaa malleissa, joiden alkioiden lukumäärä on äärellinen. Rajoittuminen äärellisiin malleihin mahdollistaa tulosten soveltamisen teoreettisessa tietojenkäsittelytieteessä, jonka näkökulmasta logiikan kaavoja voidaan ajatella ohjelmina ja äärellisiä malleja niiden syötteinä. Lokaalisuus tarkoittaa logiikan kyvyttömyyttä erottaa toisistaan malleja, joiden paikalliset piirteet vastaavat toisiaan. Väitöskirjassa tarkastellaan useita lokaalisuuden muotoja ja niiden säilymistä logiikkoja yhdistellessä. Kehitettyjä työkaluja apuna käyttäen osoitetaan, että Gaifman- ja Hanf-lokaalisuudeksi kutsuttujen varianttien välissä on lokaalisuuskäsitteiden hierarkia, jonka eri tasot voidaan erottaa toisistaan kasvavaa dimensiota olevissa hiloissa. Toisaalta osoitetaan, että lokaalisuuskäsitteet eivät eroa toisistaan, kun rajoitutaan tarkastelemaan äärellisiä puita. Järjestysinvariantit logiikat ovat kieliä, joissa on käytössä sisäänrakennettu järjestysrelaatio, mutta sitä on käytettävä siten, etteivät kaavojen ilmaisemat asiat riipu valitusta järjestyksestä. Määritelmää voi motivoida tietojenkäsittelyn näkökulmasta: vaikka ohjelman syötteen tietojen järjestyksellä ei olisi odotetun tuloksen kannalta merkitystä, on syöte tietokoneen muistissa aina jossakin järjestyksessä, jota ohjelma voi laskennassaan hyödyntää. Väitöskirjassa tutkitaan minkälaisia lokaalisuuden muotoja järjestysinvariantit ensimmäisen kertaluvun predikaattilogiikan laajennukset yksipaikkaisilla kvanttoreilla voivat toteuttaa. Tuloksia sovelletaan tarkastelemalla, milloin sisäänrakennettu järjestys lisää logiikan ilmaisuvoimaa äärellisissä puissa.

Wind drift of snowfall between the radar volume and ground

This study evaluates how the advection of precipitation, or wind drift, between the radar volume and ground affects radar measurements of precipitation. Normally precipitation is assumed to fall vertically to the ground from the contributing volume, and thus the radar measurement represents the geographical location immediately below. In this study radar measurements are corrected using hydrometeor trajectories calculated from measured and forecasted winds, and the effect of trajectory-correction on the radar measurements is evaluated. Wind drift statistics for Finland are compiled using sounding data from two weather stations spanning two years. For each sounding, the hydrometeor phase at ground level is estimated and drift distance calculated using different originating level heights. This way the drift statistics are constructed as a function of range from radar and elevation angle. On average, wind drift of 1 km was exceeded at approximately 60 km distance, while drift of 10 km was exceeded at 100 km distance. Trajectories were calculated using model winds in order to produce a trajectory-corrected ground field from radar PPI images. It was found that at the upwind side from the radar the effective measuring area was reduced as some trajectories exited the radar volume scan. In the downwind side areas near the edge of the radar measuring area experience improved precipitation detection. The effect of trajectory-correction is most prominent in instant measurements and diminishes when accumulating over longer time periods. Furthermore, measurements of intensive and small scale precipitation patterns benefit most from wind drift correction. The contribution of wind drift on the uncertainty of estimated Ze (S) - relationship was studied by simulating the effect of different error sources to the uncertainty in the relationship coefficients a and b. The overall uncertainty was assumed to consist of systematic errors of both the radar and the gauge, as well as errors by turbulence at the gauge orifice and by wind drift of precipitation. The focus of the analysis is error associated with wind drift, which was determined by describing the spatial structure of the reflectivity field using spatial autocovariance (or variogram). This spatial structure was then used with calculated drift distances to estimate the variance in radar measurement produced by precipitation drift, relative to the other error sources. It was found that error by wind drift was of similar magnitude with error by turbulence at gauge orifice at all ranges from radar, with systematic errors of the instruments being a minor issue. The correction method presented in the study could be used in radar nowcasting products to improve the estimation of visibility and local precipitation intensities. The method however only considers pure snow, and for operational purposes some improvements are desirable, such as melting layer detection, VPR correction and taking solid state hydrometeor type into account, which would improve the estimation of vertical velocities of the hydrometeors.

The Spin-Statistics Relation and Noncommutative Quantum Field Theory

The efforts of combining quantum theory with general relativity have been great and marked by several successes. One field where progress has lately been made is the study of noncommutative quantum field theories that arise as a low energy limit in certain string theories. The idea of noncommutativity comes naturally when combining these two extremes and has profound implications on results widely accepted in traditional, commutative, theories. In this work I review the status of one of the most important connections in physics, the spin-statistics relation. The relation is deeply ingrained in our reality in that it gives us the structure for the periodic table and is of crucial importance for the stability of all matter. The dramatic effects of noncommutativity of space-time coordinates, mainly the loss of Lorentz invariance, call the spin-statistics relation into question. The spin-statistics theorem is first presented in its traditional setting, giving a clarifying proof starting from minimal requirements. Next the notion of noncommutativity is introduced and its implications studied. The discussion is essentially based on twisted Poincaré symmetry, the space-time symmetry of noncommutative quantum field theory. The controversial issue of microcausality in noncommutative quantum field theory is settled by showing for the first time that the light wedge microcausality condition is compatible with the twisted Poincaré symmetry. The spin-statistics relation is considered both from the point of view of braided statistics, and in the traditional Lagrangian formulation of Pauli, with the conclusion that Pauli's age-old theorem stands even this test so dramatic for the whole structure of space-time.

Precipitation-induced runoff and leaching from milled peat mining mires by peat types : a comparative method for estimating the loading of water bodies during peat pruduction

Precipitation-induced runoff and leaching from milled peat mining mires by peat types: a comparative method for estimating the loading of water bodies during peat production. This research project in environmental geology has arisen out of an observed need to be able to predict more accurately the loading of watercourses with detrimental organic substances and nutrients from already existing and planned peat production areas, since the authorities capacity for insisting on such predictions covering the whole duration of peat production in connection with evaluations of environmental impact is at present highly limited. National and international decisions regarding monitoring of the condition of watercourses and their improvement and restoration require more sophisticated evaluation methods in order to be able to forecast watercourse loading and its environmental impacts at the stage of land-use planning and preparations for peat production.The present project thus set out from the premise that it would be possible on the basis of existing mire and peat data properties to construct estimates for the typical loading from production mires over the whole duration of their exploitation. Finland has some 10 million hectares of peatland, accounting for almost a third of its total area. Macroclimatic conditions have varied in the course of the Holocene growth and development of this peatland, and with them the habitats of the peat-forming plants. Temperatures and moisture conditions have played a significant role in determining the dominant species of mire plants growing there at any particular time, the resulting mire types and the accumulation and deposition of plant remains to form the peat. The above climatic, environmental and mire development factors, together with ditching, have contributed, and continue to contribute, to the existence of peat horizons that differ in their physical and chemical properties, leading to differences in material transport between peatlands in a natural state and mires that have been ditched or prepared for forestry and peat production. Watercourse loading from the ditching of mires or their use for peat production can have detrimental effects on river and lake environments and their recreational use, especially where oxygen-consuming organic solids and soluble organic substances and nutrients are concerned. It has not previously been possible, however, to estimate in advance the watercourse loading likely to arise from ditching and peat production on the basis of the characteristics of the peat in a mire, although earlier observations have indicated that watercourse loading from peat production can vary greatly and it has been suggested that differences in peat properties may be of significance in this. Sprinkling is used here in combination with simulations of conditions in a milled peat production area to determine the influence of the physical and chemical properties of milled peats in production mires on surface runoff into the drainage ditches and the concentrations of material in the runoff water. Sprinkling and extraction experiments were carried out on 25 samples of milled Carex (C) and Sphagnum (S) peat of humification grades H 2.5 8.5 with moisture content in the range 23.4 89% on commencement of the first sprinkling, which was followed by a second sprinkling 24 hours later. The water retention capacity of the peat was best, and surface runoff lowest, with Sphagnum and Carex peat samples of humification grades H 2.5 6 in the moisture content class 56 75%. On account of the hydrophobicity of dry peat, runoff increased in a fairly regular manner with drying of the sample from 55% to 24 30%. Runoff from the samples with an original moisture content over 55% increased by 63% in the second round of sprinkling relative to the first, as they had practically reached saturation point on the first occasion, while those with an original moisture content below 55% retained their high runoff in the second round, due to continued hydrophobicity. The well-humified samples (H 6.5 8.5) with a moisture content over 80% showed a low water retention capacity and high runoff in both rounds of sprinkling. Loading of the runoff water with suspended solids, total phosphorus and total nitrogen, and also the chemical oxygen demand (CODMn O2), varied greatly in the sprinkling experiment, depending on the peat type and degree of humification, but concentrations of the same substances in the two sprinklings were closely or moderately closely correlated and these correlations were significant. The concentrations of suspended solids in the runoff water observed in the simulations of a peat production area and the direct surface runoff from it into the drainage ditch system in response to rain (sprinkling intensity 1.27 mm/min) varied c. 60-fold between the degrees of humification in the case of the Carex peats and c. 150-fold for the Sphagnum peats, while chemical oxygen demand varied c. 30-fold and c. 50-fold, respectively, total phosphorus c. 60-fold and c. 66-fold, total nitrogen c. 65-fold and c. 195-fold and ammonium nitrogen c. 90-fold and c. 30-fold. The increases in concentrations in the runoff water were very closely correlated with increases in humification of the peat. The correlations of the concentrations measured in extraction experiments (48 h) with peat type and degree of humification corresponded to those observed in the sprinkler experiments. The resulting figures for the surface runoff from a peat production area into the drainage ditches simulated by means of sprinkling and material concentrations in the runoff water were combined with statistics on the mean extent of daily rainfall (0 67 mm) during the frost-free period of the year (May October) over an observation period of 30 years to yield typical annual loading figures (kg/ha) for suspended solids (SS), chemical oxygen demand of organic matter (CODmn O2), total phosphorus (tot. P) and total nitrogen (tot. N) entering the ditches with respect to milled Carex (C) and Sphagnum (S) peats of humification grades H 2.5 8.5. In order to calculate the loading of drainage ditches from a milled peat production mire with the aid of these annual comparative values (in kg/ha), information is required on the properties of the intended production mire and its peat. Once data are available on the area of the mire, its peat depth, peat types and their degrees of humification, dry matter content, calorific value and corresponding energy content, it is possible to produce mutually comparable estimates for individual mires with respect to the annual loading of the drainage ditch system and the surrounding watercourse for the whole service life of the production area, the duration of this service life, determinations of energy content and the amount of loading per unit of energy generated (kg/MWh). In the 8 mires in the Köyhäjoki basin, Central Ostrobothnia, taken as an example, the loading of suspended solids (SS) in the drainage ditch networks calculated on the basis of the typical values obtained here and existing mire and peat data and expressed per unit of energy generated varied between the mires and horizons in the range 0.9 16.5 kg/MWh. One of the aims of this work was to develop means of making better use of existing mire and peat data and the results of corings and other field investigations. In this respect combination of the typical loading values (kg/ha) obtained here for S, SC, CS and C peats and the various degrees of humification (H 2.5 8.5) with the above mire and peat data by means of a computer program for the acquisition and handling of such data would enable all the information currently available and that deposited in the system in the future to be used for defining watercourse loading estimates for mires and comparing them with the corresponding estimates of energy content. The intention behind this work has been to respond to the challenge facing the energy generation industry to find larger peat production areas that exert less loading on the environment and to that facing the environmental authorities to improve the means available for estimating watercourse loading from peat production and its environmental impacts in advance. The results conform well to the initial hypothesis and to the goals laid down for the research and should enable watercourse loading from existing and planned peat production to be evaluated better in the future and the resulting impacts to be taken into account when planning land use and energy generation. The advance loading information available in this way would be of value in the selection of individual peat production areas, the planning of their exploitation, the introduction of water protection measures and the planning of loading inspections, in order to achieve controlled peat production that pays due attention to environmental considerations.

Boundedness and convergence of singular integrals on fractal type sets

The topic of this dissertation lies in the intersection of harmonic analysis and fractal geometry. We particulary consider singular integrals in Euclidean spaces with respect to general measures, and we study how the geometric structure of the measures affects certain analytic properties of the operators. The thesis consists of three research articles and an overview. In the first article we construct singular integral operators on lower dimensional Sierpinski gaskets associated with homogeneous Calderón-Zygmund kernels. While these operators are bounded their principal values fail to exist almost everywhere. Conformal iterated function systems generate a broad range of fractal sets. In the second article we prove that many of these limit sets are porous in a very strong sense, by showing that they contain holes spread in every direction. In the following we connect these results with singular integrals. We exploit the fractal structure of these limit sets, in order to establish that singular integrals associated with very general kernels converge weakly. Boundedness questions consist a central topic of investigation in the theory of singular integrals. In the third article we study singular integrals of different measures. We prove a very general boundedness result in the case where the two underlying measures are separated by a Lipshitz graph. As a consequence we show that a certain weak convergence holds for a large class of singular integrals.

Renormalization methods in KAM theory

It is well known that an integrable (in the sense of Arnold-Jost) Hamiltonian system gives rise to quasi-periodic motion with trajectories running on invariant tori. These tori foliate the whole phase space. If we perturb an integrable system, the Kolmogorow-Arnold-Moser (KAM) theorem states that, provided some non-degeneracy condition and that the perturbation is sufficiently small, most of the invariant tori carrying quasi-periodic motion persist, getting only slightly deformed. The measure of the persisting invariant tori is large together with the inverse of the size of the perturbation. In the first part of the thesis we shall use a Renormalization Group (RG) scheme in order to prove the classical KAM result in the case of a non analytic perturbation (the latter will only be assumed to have continuous derivatives up to a sufficiently large order). We shall proceed by solving a sequence of problems in which theperturbations are analytic approximations of the original one. We will finally show that the approximate solutions will converge to a differentiable solution of our original problem. In the second part we will use an RG scheme using continuous scales, so that instead of solving an iterative equation as in the classical RG KAM, we will end up solving a partial differential equation. This will allow us to reduce the complications of treating a sequence of iterative equations to the use of the Banach fixed point theorem in a suitable Banach space.

Boundedness of weakly singular integral operators on domains

The monograph dissertation deals with kernel integral operators and their mapping properties on Euclidean domains. The associated kernels are weakly singular and examples of such are given by Green functions of certain elliptic partial differential equations. It is well known that mapping properties of the corresponding Green operators can be used to deduce a priori estimates for the solutions of these equations. In the dissertation, natural size- and cancellation conditions are quantified for kernels defined in domains. These kernels induce integral operators which are then composed with any partial differential operator of prescribed order, depending on the size of the kernel. The main object of study in this dissertation being the boundedness properties of such compositions, the main result is the characterization of their Lp-boundedness on suitably regular domains. In case the aforementioned kernels are defined in the whole Euclidean space, their partial derivatives of prescribed order turn out to be so called standard kernels that arise in connection with singular integral operators. The Lp-boundedness of singular integrals is characterized by the T1 theorem, which is originally due to David and Journé and was published in 1984 (Ann. of Math. 120). The main result in the dissertation can be interpreted as a T1 theorem for weakly singular integral operators. The dissertation deals also with special convolution type weakly singular integral operators that are defined on Euclidean spaces.

Luonnollisten lukujen joukon määriteltävyys funktioalgebroissa

Let X be a topological space and K the real algebra of the reals, the complex numbers, the quaternions, or the octonions. The functions form X to K form an algebra T(X,K) with pointwise addition and multiplication. We study first-order definability of the constant function set N' corresponding to the set of the naturals in certain subalgebras of T(X,K). In the vocabulary the symbols Constant, +, *, 0', and 1' are used, where Constant denotes the predicate defining the constants, and 0' and 1' denote the constant functions with values 0 and 1 respectively. The most important result is the following. Let X be a topological space, K the real algebra of the reals, the compelex numbers, the quaternions, or the octonions, and R a subalgebra of the algebra of all functions from X to K containing all constants. Then N' is definable in , if at least one of the following conditions is true. (1) The algebra R is a subalgebra of the algebra of all continuous functions containing a piecewise open mapping from X to K. (2) The space X is sigma-compact, and R is a subalgebra of the algebra of all continuous functions containing a function whose range contains a nonempty open set of K. (3) The algebra K is the set of reals or the complex numbers, and R contains a piecewise open mapping from X to K and does not contain an everywhere unbounded function. (4) The algebra R contains a piecewise open mapping from X to the set of the reals and function whose range contains a nonempty open subset of K. Furthermore R does not contain an everywhere unbounded function.