Scalar Field Cosmology -- Geometry of Dynamics

Aleksander Stachowski; Marek Szydlowski; Orest Hrycyna

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Scalar Field Cosmology -- Geometry of Dynamics

Marek Szydlowski, Orest Hrycyna, Aleksander Stachowski

General Relativity and Quantum Cosmology · Submitted 19 Aug 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim This enables us to characterise a structurally stable and therefore a generic class of SFC models.

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What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.17 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualisation of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the Scalar Field Cosmology. We characterise also the class of potentials which reproduces the \Lambda CDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterise a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space.	contribution 0.78	0.93	0.03	asserted 0.97
s1	We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric.	method 0.67	0.93	0.04	asserted 1.00
s2	The metric in the ensemble is used to measure a distance between different models.	method 0.91	0.55	0.03	asserted 1.00
s3	We point out the advantages of visualisation of dynamics in the phase space.	interpretation 0.36	0.49	0.03	asserted 0.99
s4	It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC.	contribution 0.98	0.80	0.06	asserted 0.83
s5	We also study the symmetries of dynamical systems of SFC by searching for their exact solutions.	contribution 0.87	0.93	0.04	asserted 0.96
s6	In this context we stressed the importance of scaling solutions.	interpretation 0.51	0.55	0.05	asserted 0.95
s7	It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points.	result 0.99	0.86	0.16	asserted 1.00
s8	Only critical point itself located on two dimensional stable submanifold can be identified as scaling solution.	result 0.95	0.83	0.11	asserted 1.00
s9	We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the universe.	result 0.93	0.95	0.07	asserted 1.00
s10	A class of potentials forced by scaling (homology) symmetries was given.	result 0.73	0.86	0.07	asserted 1.00
s11	We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the Scalar Field Cosmology.	contribution 0.45	0.67	0.11	asserted 0.97
s12	We characterise also the class of potentials which reproduces the \Lambda CDM model, which is known to be structurally stable.	result 0.51	0.79	0.08	asserted 1.00
s13	We show that the structural stability issue can be effectively used is selection of the scalar field potential function.	interpretation 0.47	0.89	0.22	asserted 0.95
s14	This enables us to characterise a structurally stable and therefore a generic class of SFC models.	interpretation 0.47	0.87	0.34	asserted 0.98
s15	We have found a nonempty and dense subset of structurally stable models.	result 0.99	0.96	0.28	asserted 1.00
s16	We show that these models possess symmetry of homology.	result 0.98	0.92	0.17	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.97 · evidence from proof or derivation 1.00.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper

Original abstract

We study the Scalar Field Cosmology (SFC) using the geometric language of the phase space. We define and study an ensemble of dynamical systems as a Banach space with a Sobolev metric. The metric in the ensemble is used to measure a distance between different models. We point out the advantages of visualisation of dynamics in the phase space. It is investigated the genericity of some class of models in the context of fine tuning of the form of the potential function in the ensemble of SFC. We also study the symmetries of dynamical systems of SFC by searching for their exact solutions. In this context we stressed the importance of scaling solutions. It is demonstrated that scaling solutions in the phase space are represented by unstable separatrices of the saddle points. Only critical point itself located on two dimensional stable submanifold can be identified as scaling solution. We have also found a class of potentials of the scalar fields forced by the symmetry of differential equation describing the evolution of the universe. A class of potentials forced by scaling (homology) symmetries was given. We point out the role of the notion of a structural stability in the context of the problem of indetermination of the potential form of the Scalar Field Cosmology. We characterise also the class of potentials which reproduces the \Lambda CDM model, which is known to be structurally stable. We show that the structural stability issue can be effectively used is selection of the scalar field potential function. This enables us to characterise a structurally stable and therefore a generic class of SFC models. We have found a nonempty and dense subset of structurally stable models. We show that these models possess symmetry of homology.

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References

How the literature uses it

6 arXiv papers cite this in 8 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 6 sentences without a confident label

Cites it as background 2 papers

Such diagrams have found prolific use in the study of the late universe dominated by dark energy [ref] , and have also been used in the study of inflationary scalar fields from early on [ref] , but only occasionally.

Global Portraits of Nonminimal Inflation: Metric and Palatini · Introduction · 0.80
There are many works in the literature applying the methods of dynamical system anaylsis to cosmological models and most of them take the latest observational data into account with different approaches [ref] .

Dynamical System Analysis of Quintessence Models with Exponential Potential -- Revisited · INTRODUCTION · 0.86

What it cites · 14 sentences without a confident label

Uses the method or tool of 1 paper

The phase variables are chosen as energetic variables [ref] .

Scalar field cosmology in the energy phase-space -- unified description of dynamics · Road toward structurally stable cosmological models · 0.88

Cites as background 5 papers

The role of symmetries in the context of the generation of new cosmological solution for the inflation was investigated in [ref] .

Equation of state for Universe from similarity symmetries · Symmetry group of cosmological dynamical systems · 0.88
The role of symmetries in the context of the generation of new cosmological solution for the inflation was investigated in [ref] .

A classification of scalar field potentials with cosmological scaling solutions · Symmetry group of cosmological dynamical systems · 0.88
Scaling solutions are understood in the sense of the same authors [ref] .

Exponential potentials and cosmological scaling solutions · Symmetry group of cosmological dynamical systems · 0.81

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