Stohastička teorija polja u kozmologiji

master's thesis

• Title (english): Stochastic Field Theory in Cosmology
• Author: Eugen Šostik
• Mentor: Zvonimir Vlah (mentor)
• Mentor: Krešimir Kumerički (komentor)
• Committee member: Krešimir Kumerički (član povjerenstva)
• Committee member: Zvonimir Vlah (član povjerenstva)
• Committee member: Ivica Smolić (član povjerenstva)
• Committee member: Sanjin Benić (član povjerenstva)
• Committee member: Maro Cvitan (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Physics); Zagreb
• Defense date and country: 2025-02-05, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Physics

Abstract

Prezentiraju se dvije alternativne matematičke metode za izračunavanje statističkih svojstava raspodjele tvari u svemiru iz evolucijskih jednadžbi za tvar. ”Evolucijskim jednadžbama” se nazivaju prva dva tzv. momenta Vlasovljeve jednadžbe. Kao u već postojećoj metodi, u ovima se primjenjuje matematička teorija stohastičkih (slučajnih) polja, koja spada pod teoriju vjerojatnosti, ali cilj je izbjeći uobičajeni oslonac na postupak rješavanja evolucijskih jednadžbi s pomoću računa smetnje na... More njima. Umjesto toga, ovdje se kao centralni matematički alat koristi integriranje po skupovima funkcija koje bi mogle opisati početne raspodjele tvari - nepoznanice koje predstavljaju značajnu prepreku u teorijskim istraživanjima u suvremenoj kozmologiji i jedan su od razloga za statistički pristup. Pritom se identificira prikladna verzija računa smetnje na spomenutim integralima. ”Lajtmotiv” ovog rada je nastojanje prema postizanju pristojne razine matematičke strogosti u prezentaciji, budući da je u referentnoj literaturi za fiziku primijećena svojevrsna nonšalantnost u tom aspektu koja se manifestira na nimalo trivijalnim matematičkim idejama i objektima ključnim u ovoj temi. U najnižem redu računa smetnje, prezentiranim metodama su generirani rezultati u suglasju s literaturnim, no u višim redovima je uspjeh izostao. Motiv za razvoj alternativnih metoda je primarno ubrzavanje strojnih numeričkih izračuna u kozmologiji, ali smatra se da istražene matematičke ideje mogu biti korisne u drugim teorijama i disciplinama u fizici. Less

Abstract (english)

Two alternative mathematical methods are presented for calculating the statistical properties of the distribution of matter in the Universe from the evolutionary equations for matter. The first two so-called moments of the Vlasov equation are called ”evolutionary equations”. As in the already existing method, these apply the mathematical theory of stochastic (random) fields, which falls under the theory of probability, but the goal is to avoid the usual reliance on the procedure for... More solving evolutionary equations using the calculus of perturbations on them. Instead, the central mathematical tool used here is integration over sets of functions that could describe the initial distributions of matter - unknowns that represent a significant obstacle in theoretical research in contemporary cosmology and so one of the reasons for the statistical approach. In doing so, a suitable version of the perturbation calculus on the aforementioned integrals is identified. The ”leitmotif” of this thesis is the effort to achieve a decent level of mathematical rigor in the presentation, since a certain nonchalance in this aspect has been observed in the reference literature for physics and it manifests on not at all trivial mathematical ideas and objects key to this topic. In the lowest order of perturbation calculus, the presented methods generated results in agreement with those in the literature, but in the higher orders there was no success. The motive for developing alternative methods is primarily to speed up machine numerical calculations in cosmology, but it is believed that the mathematical ideas explored can be useful in other theories and disciplines in physics. Less

• Keywords: kozmologija; ΛCDM; pod-horizontne prostorne skale; kozmičke strukture velikih skala; raspodjela tvari u svemiru; statistička svojstva raspodjele tvari; evolucijske jednadžbe za tvar; račun smetnje; stohastički procesi; stohastička polja; funkcionalni integrali; path integrali; integracija po skupovima funkcija; Wienerova Gausijanska mjera
• Keywords (english): cosmology; ΛCDM; sub-horizon spatial scales; large-scale cosmic structures; distribution of matter in the Universe; statistical properties of matter distribution; evolution equations for matter; perturbation theory; stochastic processes; stochastic fields; functional integrals; path integrals; integration on a set of functions; Wiener Gaussian measure
• Language: croatian
• URN:NBN: urn:nbn:hr:217:296055
• Study programme: Title: The University Integrated Undergraduate and Graduate Programme in Physics; Research Physics; specializations in: Research Physics; Course: Research Physics; Study programme type: university; Study level: integrated undergraduate and graduate; Academic / professional title: sveučilišni magistar fizike (sveučilišni magistar fizike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2025-01-28 13:04:46